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Instructions to authors Aims and scope Physics Reports keeps the active physicist up-to-date on developments in a wide range of topics by publishing timely reviews which are more extensive than just literature surveys but normally less than a full monograph. Each Report deals with one speciﬁc subject. These reviews are specialist in nature but contain enough introductory material to make the main points intelligible to a non-specialist. The reader will not only be able to distinguish important developments and trends but will also ﬁnd a suﬃcient number of references to the original literature. Submission In principle, papers are written and submitted on the invitation of one of the Editors, although the Editors would be glad to receive suggestions. Proposals for review articles (approximately 500–1000 words) should be sent by the authors to one of the Editors listed below. The Editor will evaluate proposals on the basis of timeliness and relevance and inform the authors as soon as possible. All submitted papers are subject to a refereeing process. Editors J.V. ALLABY (Experimental high-energy physics), EP Division, CERN, CH-1211 Geneva 23, Switzerland. E-mail: [email protected] D.D. AWSCHALOM (Experimental condensed matter physics), Department of Physics, University of California, Santa Barbara, CA 93106, USA. E-mail: [email protected] J.A. BAGGER (High-energy physics), Department of Physics & Astronomy, The Johns Hopkins University, 3400 North Charles Street, Baltimore MD 21218, USA. E-mail: [email protected] C.W.J. BEENAKKER (Mesoscopic physics), Instituut–Lorentz, Universiteit Leiden, P.O. Box 9506, 2300 RA Leiden, The Netherlands. E-mail: [email protected] E. BREZIN (Statistical physics and ﬁeld theory), Laboratoire de Physique The´orique, Ecole Normale Superieure, 24 rue Lhomond, 75231 Paris Cedex, France. E-mail: [email protected] G.E. BROWN (Nuclear physics), Institute for Theoretical Physics, State University of New York at Stony Brook, Stony Brook, NY 11974, USA. E-mail: [email protected] D.K. CAMPBELL (Non-linear dynamics), Dean, College of Engineering, Boston University, 44 Cummington Street, Boston, MA 02215, USA. E-mail: [email protected] G. COMSA (Surfaces and thin ﬁlms), Institut fur . Physikalische und Theoretische Chemie, Universit.at Bonn, Wegelerstrasse 12, D-53115 Bonn, Germany. E-mail: [email protected] J. EICHLER (Atomic and molecular physics), Hahn-Meitner-Institut Berlin, Abteilung Theoretische Physik, Glienicker Strasse 100, 14109 Berlin, Germany. E-mail: [email protected] M.P. KAMIONKOWSKI (Astrophysics), Theoretical Astrophysics 130-33, California Institute of Technology, 1200 East California Blvd., Pasadena, CA 91125, USA. E-mail: [email protected]

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Instructions to authors

M.L. KLEIN (Soft condensed matter physics), Department of Chemistry, University of Pennsylvania, Philadelphia, PA 19104-6323, USA. E-mail: [email protected] A.A. MARADUDIN (Condensed matter physics), Department of Physics and Astronomy, University of California, Irvine, CA 92697-4575, USA. E-mail: [email protected] D.L. MILLS (Condensed matter physics), Department of Physics and Astronomy, University of California, Irvine, CA 92697-4575, USA. E-mail: [email protected] R. PETRONZIO (High-energy physics), Dipartimento di Fisica, Universita" di Roma – Tor Vergata, Via della Ricerca Scientiﬁca, 1, I-00133 Rome, Italy. E-mail: [email protected] S. PEYERIMHOFF (Molecular physics), Institute of Physical and Theoretical Chemistry, Wegelerstrasse 12, D-53115 Bonn, Germany. E-mail: [email protected] I. PROCACCIA (Statistical mechanics), Department of Chemical Physics, Weizmann Institute of Science, Rehovot 76100, Israel. E-mail: [email protected] E. SACKMANN (Biological physics), Physik-Department E22 (Biophysics Lab.), Technische Universit.at Munchen, . D-85747 Garching, Germany. E-mail: [email protected] A. SCHWIMMER (High-energy physics), Department of Physics of Complex Systems, Weizmann Institute of Science, Rehovot 76100, Israel. E-mail: [email protected] R.N. SUDAN (Plasma physics), Laboratory of Plasma Studies, Cornell University, 369 Upson Hall, Ithaca, NY 14853-7501, USA. E-mail: [email protected] W. WEISE (Physics of hadrons and nuclei), Institut fur . Theoretische Physik, Physik Department, Technische Universit.at Munchen, . James Franck Strae, D-85748 Garching, Germany. E-mail: [email protected] Manuscript style guidelines Papers should be written in correct English. Authors with insuﬃcient command of the English language should seek linguistic advice. Manuscripts should be typed on one side of the paper, with double line spacing and a wide margin. The character size should be suﬃciently large that all subscripts and superscripts in mathematical expressions are clearly legible. Please note that manuscripts should be accompanied by separate sheets containing: the title, authors’ names and addresses, abstract, PACS codes and keywords, a table of contents, and a list of ﬁgure captions and tables. – Address: The name, complete postal address, e-mail address, telephone and fax number of the corresponding author should be indicated on the manuscript. – Abstract: A short informative abstract not exceeding approximately 150 words is required. – PACS codes/keywords: Please supply one or more PACS-1999 classiﬁcation codes and up to 4 keywords of your own choice for indexing purposes. PACS is available online from our homepage (http://www.elsevier.com/locate/physrep). References. The list of references may be organized according to the number system or the nameyear (Harvard) system. Number system: [1] M.J. Ablowitz, D.J. Kaup, A.C. Newell and H. Segur, The inverse scattering transform – Fourier analysis for nonlinear problems, Studies in Applied Mathematics 53 (1974) 249–315. [2] M. Abramowitz and I. Stegun, Handbook of Mathematical Functions (Dover, New York, 1965).

Instructions to authors

vii

[3] B. Ziegler, in: New Vistas in Electro-nuclear Physics, eds E.L. Tomusiak, H.S. Kaplan and E.T. Dressler (Plenum, New York, 1986) p. 293. A reference should not contain more than one article. Harvard system:

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Instructions to authors

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Physics Reports 361 (2002) 1–56

The neutrinoless double beta decay from a modern perspective J.D. Vergadosa; b ; ∗ b

a Theoretical Physics Division, University of Ioannina, GR-451 10, Ioannina, Greece Department of Physics, University of Cyprus, P.O. box 20537, CY 1878 Nicosia, Cyprus

Received August 2001; editor : G:E: Brown

Contents 1. Introduction 2. The intermediate Majorana neutrino mechanism 2.1. The Majorana neutrino mass mechanism 2.2. The leptonic left–right interference mechanism ( and terms) 2.3. The majoron emission mechanism 3. Brief description of current experiments 4. The R-parity violating contribution to 0-decay 4.1. The contribution arising from the bilinears in the superpotential 4.2. The contribution arising from the cubic terms in the superpotential 4.3. The case of light intermediate neutrinos 5. The e5ective nucleon current 5.1. Handling the short range nature of the transition operator

3 7 8 9 14 15 19 20 20 26 28 28

5.2. Momentum-dependent corrections to the e5ective nucleon current 6. The exotic double-charge exchange − to e+ conversion in nuclei 6.1. The transition operators at the nuclear level 6.2. Irreducible tensor operators 6.3. The branching ratio for (− ; e+ ) 6.4. Results and discussion for (; e+ ) conversion 6.5. Summary and conclusions 7. Extraction of the lepton violating parameters 7.1. Traditional lepton violating parameters 7.2. R-parity induced lepton violating parameters 8. Conclusions Acknowledgements References

Corresponding author. Theoretical Physics Division, University of Ioannina, GR-451 10, Ioannina, Greece. E-mail address: [email protected] (J.D. Vergados). c 2001 Elsevier Science B.V. All rights reserved. 0370-1573/02/$ - see front matter PII: S 0 3 7 0 - 1 5 7 3 ( 0 1 ) 0 0 0 6 8 - 0

30 33 35 37 39 40 43 44 44 48 49 52 52

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Abstract Neutrinoless double beta decay is a very important process both from the particle and nuclear physics point of view. From the elementary particle point of view it pops up in almost every model, giving rise, among others, to the following mechanisms: (a) The traditional contributions like the light neutrino mass mechanism as well as the jL –jR leptonic interference ( and terms). (b) The exotic R-parity violating supersymmetric (SUSY) contributions. In this scheme, the currents are only left handed and the intermediate particles normally are very heavy. There exists, however, the possibility of light intermediate neutrinos arising from the combination of V–A and P–S currents at the quark level. This leads to the same structure as the above term. Similar considerations apply to its sister lepton and muon number violating muon to positron conversion in the presence of nuclei. Anyway, regardless of the dominant mechanism, the observation of neutrinoless double betas decay, which is the most important of the two from an experimental point of view, will severely constrain the existing models and will signal that the neutrinos are massive Majorana particles. From the nuclear physics point of view it is challenging, because: (1) The nuclei, which can undergo double beta decay, have a complicated nuclear structure. (2) The energetically allowed transitions are suppressed (exhaust a small part of the entire strength). (3) Since in some mechanisms the intermediate particles are very heavy, one must cope with the short distance behavior of the transition operators. Thus novel e5ects, like the double beta decay of pions in Eight between nucleons, have to be considered. In SUSY models this mechanism is more important than the standard two nucleon mechanism. (4) The intermediate momenta involved are quite high (about 100 MeV=c). Thus, one has to take into account possible momentum-dependent terms of the nucleon current, like the modiFcation of the axial current due to PCAC, weak magnetism terms, etc. We Fnd that, for the mass mechanism, such modiFcations of the nucleon current for light neutrinos reduce the nuclear matrix elements by about 25%, almost regardless of the nuclear model. In the case of heavy neutrino the e5ect is much larger and model dependent. Taking the above e5ects into account the needed nuclear matrix elements have become available for all the experimentally interesting nuclei A = 76; 82; 96; 100; 116; 128; 130; 136 and 150. Some of them have been obtained in the large basis shell model but most of them in various versions of QRPA. Then using the best presently available experimental limits on the half-life of the 0-decay, we have extracted new limits on the various lepton violating parameters. In particular we Fnd m ¡ 0:5 eV=c2 and, for reasonable choices of the parameters of SUSY models in the allowed SUSY parameter space, we get a stringent limit on the c 2001 Elsevier Science B.V. All rights reserved. ¡ 0:68 × 10−3 . R-parity violating parameter 111 PACS: 23.40.Hc; 21.60.Jz; 27.50.+e; 27.60.+j

J.D. Vergados / Physics Reports 361 (2002) 1–56

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1. Introduction The nuclear double beta decay can occur whenever the ordinary (single) beta decay is forbidden due to energy conservation or greatly suppressed due to angular momentum mismatch. The exotic neutrinoless double beta decay (0-decay) is the most interesting since it violates lepton number by two units. It is a very old process. It was Frst considered by Furry [1] more than half a century ago as soon as it was realized that the neutrino might be a Majorana particle. It was continued with the work of Primako5 and Rosen [2], especially when it was recognized that kinematically it is favored by 108 compared to its non-exotic sister 2-decay. When the corresponding level of the 1015 yr lifetime was reached and the process was not seen, it was tempting to interpret this as an indication that the neutrino was a Dirac particle. The interest in it was resurrected with the advent of gauge theories which favor Majorana neutrinos, and through the pioneering work of Kotani and his group [3], it was brought again to the attention of the nuclear physics community. Today, 50 years later, 0-decay: (A; Z) → (A; Z + 2) + e− + e−

(0-decay)

(1)

continues to be one of the most interesting processes. If the neutrinos are Majorana particles other related processes in which the charge of the nucleus is deceased by two units may also occur, if they happen to be allowed by energy and angular momentum conservation laws, e.g. (A; Z) → (A; Z − 2) + e+ + e+

(0 positron emission) ;

(2)

(A; Z) + e− → (A; Z − 2) + e+

(0 electron positron conversion) ;

(3)

(A; Z) + e− + e− → (A; Z − 2) + X-rays (0 double electron capture) :

(4)

Double electron capture is always possible, whenever (3) is possible, and proceeds in two steps: In the Frst step the two neutral atoms, (A; Z) and the excited (A; Z-2), get admixed via the lepton number violating interaction [4]. In the second step, the (A; Z-2) atom de-excites emitting two hard X-rays and the nucleus, if it is found in an excited state, de-excites emitting -rays. Decays to excited states, preferably 0+ , are in some cases possible and provide additional experimental information, e.g. -rays following their de-excitation to 2+ states. Another lepton violating process not hindered by energy conservation involves neutrinoless muon capture (A; Z) + − → (A; Z − 2) + e+

(0 muon positron conversion) :

(5)

The above processes are expected to occur whenever one has lepton number violating interactions. Lepton number, being a global quantity, is not sacred, but it is expected to be broken at some level. In short, these processes pop up almost everywhere, in every theory. On the other hand, since if there exist lepton violating interactions, the neutrinos have to be Majorana particles, all the above processes can, in principle, decide whether or not the neutrino is a Majorana particle, i.e., it coincides with its own antiparticle. This is true even if these processes are induced not by intermediate neutrinos but by other mechanisms as we will see below.

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Neutrinoless double beta decay (Eq. (1)) seems to be the most likely to yield this information [5 –10]. For this reason we will focus our discussion on this reaction, but we will pay some attention to muon capture, which recently seems to be of experimental interest. We will only peripherally discuss the other less interesting processes [4]. From a nuclear physics [9 –11] point of view, calculating the relevant nuclear matrix elements, it is indeed a challenge. First almost all nuclei, which can undergo double beta decay, are far from closed shells and some of them are even deformed. One thus faces a formidable task. Second the nuclear matrix elements represent a small fraction of a canonical value, like the matrix element to the energy non-allowed transition double Gamow–Teller resonance or some appropriate sum rule. Thus, e5ects which are normally negligible, become important here. Third in many models the dominant mechanism for 0-decay does not involve intermediate light neutrinos, but very heavy particles. Thus, one must be able to cope with the short distance behavior of the relevant operators and wave functions. From the experimental point of view it is also very challenging to measure perhaps the slowest process accessible to observation. Especially since it is realized that, even if one obtains only lower bounds on the lifetime for this decay, the extracted limits on the theoretical model parameters may be comparable, if not better, and complementary to those extracted from the most ambitious accelerator experiments. The recent superkamiokande results have given the Frst evidence of physics beyond the standard model (SM) and, in particular, they indicate that the neutrinos are massive particles. It is important to proceed further and Fnd out whether the neutrinos are Dirac or Majorana particles. As we have mentioned there might be processes other than the conventional intermediate neutrino mechanism, which may dominate 0-decay. It has, however, been known that whatever the lepton violating process is, which gives rise to this decay, it can be used to generate a Majorana mass for the neutrino [12]. The study of the 0-decay is further stimulated by the development of grand uniFed theories (GUTs) and supersymmetric models (SUSY) representing extensions of the SU (2)L ⊗ U (1) SM. The GUTs and SUSY o5er a variety of mechanisms which allow the 0-decay to occur [13]. The best-known mechanism leading to 0-decay is via the exchange of a Majorana neutrino between the two decaying neutrons [5 –10,14]. Nuclear physics dictates that we study the light and heavy neutrino components separately. In the presence of only left-handed currents and for the light intermediate neutrino components, the obtained amplitude is proportional to a suitable average neutrino mass, which vanishes in the limit in which the neutrinos become Dirac particles. On the other hand, in the case of heavy Majorana neutrino components the amplitude is proportional to the average of the inverse of the neutrino mass, i.e. it is again suppressed. In the presence of right-handed currents one can have a contribution similar to the one above for heavy neutrinos but involving a di5erent (larger) average inverse mass with some additional suppression due to the fact that the right-handed gauge boson, if it exists, is heavier than the usual left-handed one. In the presence of right-handed currents it is also possible to have interference between the leptonic left and right currents, jL –jR interference. In this case the amplitude in momentum space becomes proportional to the four-momentum of the neutrino and, as a result, only the light neutrino components become important. One now has two possibilities. First, the two hadronic currents have a chirality structure of the same kind, i.e. JL –JR . Then one can extract

J.D. Vergados / Physics Reports 361 (2002) 1–56

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from the data a dimensionless parameter , which is proportional to the square of the ratio of the masses of the L and R gauge bosons, = (mL =mR )2 . Second the two hadronic currents are left handed, which can occur via the mixing of the two bosons. The relevant lepton violating parameter is now proportional to this mixing . Both of these parameters, however, also involve the neutrino mixing. They are, in a way, proportional to the mixing between the light and heavy neutrinos. In gauge theories one has, of course, many more possibilities. Exotic intermediate scalars may mediate 0-decay [7]. These are not favored in current gauge theories and are not going to be discussed further. In superstring inspired models one may have singlet fermions in addition to the usual right-handed neutrinos. Not much progress has been made on the phenomenological side of these models and they are not going to be discussed further. In recent years supersymmetric models have been taken seriously and semirealistic calculations are taking place. In standard calculations one invokes universality at the GUT scale, employing a set of Fve independent parameters, and uses the renormalization group equation to obtain all parameters (couplings and particle masses) at low energies. Hence, since such parameters are, in principle, calculable in terms of the Fve input parameters, one can use experimental data to constrain the input parameters. Then one can use the 0-decay experiments to constrain the R-parity violating couplings, which cannot be speciFed by the theory [15 –23]. Recent review articles [9 –11] also give a detailed account of some of the latest developments in this Feld. From the above discussion it is clear that one has to consider the case of heavy intermediate particles. One thus has to tackle problems related to the very short ranged operators in the presence of the nuclear repulsive core. If the interacting nucleons are point-like one gets negligible contributions. We know, however, that the nucleons are not point like, but that they have a structure described by quark bag with a size that can be determined experimentally. It can also be accounted for by a form factor, which can be calculated in the quark model or parameterized in a dipole shape with a parameter determined by the experiment. This approach, Frst considered by Vergados [24], has now been adopted by almost everybody. The resulting e5ective operator has a range somewhat less than the proton mass (see Section 4 below). Another approach in handling this problem consists of considering particles other than the nucleons present in the nuclear soup. For 0+ → 0+ the most important of such particles are the pions. Thus one may consider the double beta decay of pions in Eight between nucleons, like − → + ; e − ; e − ;

n → p; + ; e− ; e− :

(6)

Recognition of such a contribution Frst appeared as a remark by the genius of Pontecorvo [25] in the famous paper in which he suggested that the ratio of the lifetimes of the 128 Te and 130 Te isotopes, which merely di5er by two neutrons, is essentially independent of nuclear physics. He did not perform any estimates of such a contribution. Such estimates and calculations were Frst performed by Vergados [26] in the case of heavy intermediate neutrinos, i.e. vector and axial vector currents. It was found that it yields results of the same order as the nucleon mode with the above recipe for treating the short range behavior. It was revived by the Tuebingen group [20,21] in the context of R-parity violating interactions, i.e. scalar, pseudoscalar and tensor currents arising out of neutralino and gluino exchange, and it was found to dominate.

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J.D. Vergados / Physics Reports 361 (2002) 1–56

In yet another approach one may estimate the presence of six quark clusters in the nucleus. Then, since the change of charge takes place in the same hadron there is no suppression due to the short nature of the operator, even if it is a -function. One only needs a reliable method for estimating the probability of Fnding these clusters in a nucleus [27]. All the above approaches seem reasonable and lead to quite similar results. The matrix elements obtained are not severely suppressed. This gives us a great degree of conFdence that the resulting matrix elements are suQciently reliable, allowing double beta decay to probe very important physics. The other recent development is the better description of nucleon current by including momentum-dependent terms, such as the modiFcation of the axial current due to PCAC and the inclusion of the weak magnetism terms. These contributions have been considered previously [28,14], but only in connection with the extraction of the parameter mentioned above. Indeed, these terms were very important in this case since they compete with the p-wave lepton wave function, which, with the usual currents, provides the lowest non-vanishing contribution. Since in the mass term only s-wave lepton wave functions are relevant such terms have hitherto been neglected. It was recently found [29], however, that for light neutrinos the inclusion of these momentumdependent terms reduces the nuclear matrix element by about 25%, independently of the nuclear model employed. On the other hand for heavy neutrinos, the e5ect can be larger and it depends on the nuclear wave functions. The reason for expecting them to be relevant is that the average momentum q of the exchanged neutrino is expected to be large [30]. In the case of a light intermediate neutrino the mean nucleon–nucleon separation is about 2 fm which implies that the average momentum q is about 100 MeV. In the case of a heavy neutrino exchange the mean inter-nucleon distance is considerably smaller and the average momentum q is supposed to be considerably larger. Since 0–-decay is a two-step process, in principle, one needs to construct and sum over all the intermediate nuclear states, a formidable job indeed in the case of the shell model calculations (SMC). Since, however, the average neutrino momentum is much larger compared to the nuclear excitations, one can invoke closure using some average excitation energy (this does not apply in the case of 2-decays). Thus, one need construct only the initial and Fnal nuclear states. In quasiparticle random phase approximation (QRPA) one must construct the intermediate states anyway. In any case, it was explicitly shown, taking advantage of the momentum space formalism developed by Vergados [31], that this approximation is very good [32,33]. The same conclusion was reached independently by others [34] via a more complicated technique relying on coordinate space. Granted that one takes into account all the above ingredients in order to obtain quantitative answers for the lepton number violating parameters from the results of 0-decay experiments, it is necessary to evaluate the relevant nuclear matrix elements with high reliability. The most extensively used methods are the large basis shell model calculations, SMC (for a recent review see [9]) and QRPA (for a recent review see [10,9]). The SMC is forced to use few single particle orbitals, while this restriction does not apply in the case of QRPA. The latter su5ers, of course, from the approximations inherent in the RPA method. So a direct comparison between them is not possible.

J.D. Vergados / Physics Reports 361 (2002) 1–56

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The SMC has a long history [35 – 41] in double beta decay calculations. In recent years it has led to large matrix calculations in traditional as well as Monte Carlo types of formulations [42– 47]. For a more complete set of references as well as a discussion of the appropriate e5ective interactions see Ref. [9]. There have been a number of QRPA calculations covering almost all nuclear targets [48–59]. These involve a number of collaborations, but the most extensive and complete calculations in one way or another include the Tuebingen group. We also have seen some reFnements of QRPA, like proton neutron pairing and the inclusion of renormalization e5ects due to Pauli principle corrections [60,61]. Other less conventional approaches, like operator expansion techniques have also been employed [62]. The above schemes, in conjunction with the other improvements mentioned above o5er some optimism in our e5orts for obtaining nuclear matrix elements accurate enough to allow us to extract reliable values of the lepton violating parameters from the data. We are going to review this procedure in the case of most of the nuclear targets of experimental interest (76 Ge; 82 Se; 96 Zr ; 100 Mo; 116 Cd ; 128 Te; 130 Te; 136 Xe; 150 Nd ).

2. The intermediate Majorana neutrino mechanism We shall consider the 0-decay process assuming that the e5ective beta decay Hamiltonian acquires the form GF 2

H = √ 2 [(eS L 0eL )JL† + (eS R 0eR )JR† + h:c:] ;

(7)

where eL (eR ) and 0eL (0eR ) are Feld operators representing the left (right)-handed electrons and electron neutrinos in a weak interaction basis, in which the charged leptons are diagonal. We suppose that neutrino mixing does take place and is given as [7] 0eL =

3 k=1

0eR =

3 k=1

Uek(11) kL +

Uek(21) kL +

3 k=1 3 k=1

Uek(12) NkL ;

(8)

Uek(22) NkL ;

(9)

where, k (Nk ) are Felds of light (heavy) Majorana neutrino eigenFelds with masses mk (mk 1 (11) (22) MeV) and Mk (Mk 1 GeV), respectively. The matrices Uek and Uek are approximately (12) (21) unitary, while the matrices Uek and Uek are very small (of order of the up quark divided by the heavy neutrino mass scales), so that the overall matrix is unitary. k ; Nk satisfy the T Majorana condition: k !k = C STk ; Nk #k = C NS k , where C denotes the charge conjugation and !; # are phase factors, which guarantee that the eigenmasses are positive.

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J.D. Vergados / Physics Reports 361 (2002) 1–56

Fig. 1. The Feynmann diagrams at the nucleon level when the leptonic currents are left handed leading to the familiar mass term in the 0-decay.

2.1. The Majorana neutrino mass mechanism This mechanism is the most popular and most commonly discussed in the literature (see Fig. 1). The mass term involving the right-handed bosons, relevant only for the heavy neutrino components, is not shown here but it can readily be deduced from those shown here (see Section 6 for the analogous term in the (− ; e+ ) conversion). + We will consider only 0+ i → 0f transitions. Then both outgoing electrons are in the s1=2 state. Thus for the ground state transition, restricting ourselves to the mass mechanism, we obtain for the 0-decay inverse half-life [5 –10,14] as follows: 0 −1 0 2 [T1=2 ] = G01 |MGT | [|XL |2 + |XR |2 − C˜ 1 XL XR + · · · ] :

(10)

The coeQcient C˜ 1 is negligible, ... indicate other non-traditional modes (SUSY, etc.). The nuclear matrix elements entering the above expression are given in units of MGT and are denoted [6] by ': XL =

m

me

('F − 1) + LN 'H ;

XR = RN 'H with

gV 'F = gA

2

(11) (12)

MF ; MGT

(13)

J.D. Vergados / Physics Reports 361 (2002) 1–56

'H =

gV gA

2

9

MFH − MGTH =MGT ;

(14)

where the subscript H indicates heavy particle (neutrino). The lepton-number non-conserving parameters, i.e. the e5ective neutrino mass m ee , or m for short, and LN ; RN are given as follows [7]: 3 3 mp m = (Uek(11) )2 !k mk ; LN = (Uek(12) )2 #k ; (15) Mk k=1 k=1 3 mp R 2 2 22 2 (Uek ) #k ; (16) N = ( + ) Mk k=1

with mp (me ) being the proton (electron) mass, is the mass squared ratio of WL and WR and their mixing. G01 is the integrated kinematical factor [4,6,14,63]. The nuclear matrix elements associated with the exchange of light and heavy neutrino must be computed in a suitable nuclear model. The ellipses ... mean that Eq. (10) can be generalized to the mass term resulting from any other intermediate fermion. At this point we should stress that the main suppression in the case of light neutrinos comes from the smallness of neutrino masses. In the case of heavy neutrino not only from the large values of neutrino masses but the small couplings, U (12) for the left-handed neutrinos or and for the right-handed ones. 2.2. The leptonic left–right interference mechanism ( and terms) As we have already mentioned in the presence of right-handed currents one can have interference between the leptonic currents of opposite chirality (see Fig. 2). The elementary amplitude is now proportional to the four-momentum transfer. We thus have a space component and a time component in the relevant amplitude. This leads to di5erent kinematical functions and yields two new lepton violating parameters [7] and deFned by 3 = RL ; = RL ; RL = (Uek(21) Uek(11) )!k : (17) k=1

The parameters and are small not only due to the smallness of the parameters and but, in addition, due to the smallness of U (21) . As we have already mentioned the can also have a di5erent origin (see Section 4.3). All the above contributions, even though the relevant amplitudes are not explicitly dependent on the neutrino mass, vanish in the limit in which the neutrino is a Dirac particle. The above expression, Eq. (10), for the lifetime is now modiFed to yield 0 −1 0 0 2 [T1=2 ] = G01 |MGT | |XL |2 + |XR |2 − C˜ 1 XL XR + · · · + C˜ 2 ||XL cos

+ C˜ 4 ||2 + C˜ 5 ||2 + C˜ 6 ||||cos( 1 − 2 ) + Re(C˜ 2 XR + C˜ 3 XR ) ; 1

+ C˜ 3 ||XL cos

2

(18)

10

J.D. Vergados / Physics Reports 361 (2002) 1–56

Fig. 2. The Feynmann diagrams at the nucleon level when the leptonic currents are of opposite chirality leading to the dimensionless lepton violating parameters (part (a) of the Fgure) and (part (b) of the Fgure) of 0-decay. Note that in part (a) the process proceeds via the right-handed vector boson, while in part (b) through the mixing of the left- and right-handed bosons.

where XL and XR are deFned in Eqs. (11), and (12). 1 and 2 are the relative phases between XL and and XL and , respectively. The ellipses {: : :} indicate contributions arising from other particles, e.g., intermediate SUSY particles or unusual particles which are predicted by superstring models or exotic Higgs scalars, etc. (see below Section 4). Many nuclear matrix elements appear in this case, but they are fairly well known and they are not going to be reviewed here in detail (see e.g. [5 –10]). For the reader’s convenience we are only going to brieEy discuss in our notation [14] the additional nuclear matrix elements, not ; ' , ' where encountered in the mass mechanism. These are: 'F! ; 'GT! , 'R ; '1± ; '2± 'F ; 'GT T P 2 gV MF! 'F! = ; (19) gA MGT 'GT! = 'R =

MGT! ; MGT

MR MGT

(20) (21)

J.D. Vergados / Physics Reports 361 (2002) 1–56

11

Table 1 The integrated kinematical factors G0k for 0+ → 0+ transition of ()0 -decaya ()0 − decay : 0+ → 0+ transition 48

(Ei − Ef ) (MeV) G01 G02 G03 G04 G05 G06 G07 G08 G09

(10−14 1= yr ) (10−13 1= yr ) (10−14 1= yr ) (10−14 1= yr ) (10−12 1= yr ) (10−11 1= yr ) (10−10 1= yr ) (10−11 1= yr ) (10−9 1= yr )

G01; '0 (10−14 1= yr ) a

Ca

76

Ge

82

Se

96

Zr

100

Mo

116

Cd

5.294

3.067

4.027

4.372

4.055

3.830

8.031 5.235 6.037 1.705 1.265 1.398 11.46 5.247 6.262

0.7928 0.1296 0.4376 0.1538 0.253 0.196 1.248 0.793 0.491

3.524 1.221 2.413 0.724 0.931 0.665 5.523 3.852 1.980

7.362 3.173 5.380 1.530 2.009 1.226 12.07 9.886 3.686

5.731 2.056 4.036 1.178 1.718 1.009 9.563 8.109 2.819

6.233 1.957 4.305 1.269 2.118 1.103 10.69 10.20 2.800

2.425

0.0763

0.6202

1.5315

1.0230

0.9879

128

Te

130

Te

136

Xe

1.891

3.555

3.503

2:207 × 10−1 6:309 × 10−3 6:177 × 10−2 3:368 × 10−2 1:390 × 10−1 6:969 × 10−2 4:363 × 10−1 4:227 × 10−1 1:125 × 10−1

5.543 1.441 3.669 1.113 2.083 1.011 9.544 9.749 2.335

5.914 1.483 3.890 1.183 2.298 1.077 10.25 10.84 2.424

5:206 × 10−3

0.7487

0.7734

The deFnition of G0k is given in the literature [6,14] and in the text.

and '1± = ± 3'F + 'GT − 6'T ;

(22)

'2± = ± 'F! + 'GT! − 19 '1±

(23)

('F = MF =MGT ,

etc., for the space part) and ('F! = MF! =MGT , etc., for in an obvious notation the time component). In the limit in which the average energy denominator can be neglected [14], we obtain 'F = 'F = 'F! ;

(24)

'GT = 'GT = 'GT! = 1 :

(25)

0 have been tabulated [4,6,14,63], see also [9] for a recent review. For The quantities G01 the readers convenience the most important ones are presented in Table 1. The coeQcients C˜ 1 ; C˜ i ; i = 2; : : : ; 6 are combinations of kinematical functions and the nuclear matrix elements discussed in the previous section. They are deFned as follows: C˜ 2 = − (1 − 'F )('2− G˜ 03 − '1+ G˜ 04 ) ;

C˜ 3 = − (1 − 'F )('2+ G˜ 03 − '1− G˜ 04 − 'P G˜ 05 + 'R G˜ 06 ) ; C˜ 4 = '22− G˜ 02 + 19 '12+ G˜ 04 − 29 '1+ '2− G˜ 03 ; C˜ 5 = '22+ G˜ 02 + 19 '12− G˜ 04 − 29 '1− '2+ G˜ 03 + ('P )2 G˜ 08 − 'P 'R G˜ 07 + 'R2 G˜ 09 ) ; C˜ 6 = − 2['2− '2+ G˜ 02 − 19 ('1+ '2+ + '2− '1− )G˜ 03 + 19 '1+ '1− G˜ 04 ] :

(26)

12

J.D. Vergados / Physics Reports 361 (2002) 1–56

Table 2 The kinematical functions G˜ 0i , i = 2–9a Nuclear transition

G˜ 02

G˜ 03

G˜ 04

G˜ 05

G˜ 06

G˜ 07

G˜ 08

G˜ 09

48

6.518 1.635 3.465 4.310 3.588 3.140 0.286 2.599 2.507

0.752 0.552 0.685 0.731 0.704 0.691 0.280 0.662 0.658

0.212 0.194 0.205 0.208 0.206 0.204 0.153 0.201 0.200

31.50 63.93 52.9 54.58 59.97 67.94 126.0 75.15 77.72

0.450 0.745 0.584 0.543 0.582 0.614 1.133 0.658 0.667

73.87 95.01 96.99 107.0 110.3 119.1 141.9 124.2 127.0

2613.0 4001.0 4372.0 5371.0 5660.0 6547.0 7662.0 7035.0 7331.0

0.522 0.563 0.538 0.5324 0.5375 0.5419 0.6565 0.5483 0.5497

Ca →48Ti Ge →76Se 82 Se →82Kr 96 Zr →96Mo 100 Mo →100Ru 116 Cd →116Sn 128 Te →128Xe 130 Te →130Xe 136 Xe →136Ba 76

a

The kinematical functions are given in the notation of Pantis et al. [4].

∼ 10(02 + 60 + 6)=(04 + 1003 + 1002 + 600 + 30); 0 is the available energy in electron Here C˜ 1 = mass units. C1 is ¡ 10% and it can be safely neglected. The quantities G˜ 0i are deFned as follows: G˜ 0i = G0i =G01 (i = 2; 3; 4) ;

G˜ 05 = 2G05 =G01 ; G˜ 06 = 14 me R0 G06 =G01 ; G˜ 07 = 2( 14 me R0 )G07 =G01 ; G˜ 08 = 4G08 =G01 ; G˜ 09 = ( 14 me R0 )2 G09 =G01 :

(27)

The values of the parameters G˜ 0i ; i = 2; : : : ; 6 are presented in Table 2. The coeQcients C˜ i ; i = 2; : : : ; 6 with and without p–n pairing can be found in the literature [14]. For a more conventional formulation, restricted, however, in the light neutrino sector, see Suhonen and Civitarese [9]. Some nuclear matrix elements obtained previously [14] are shown in Tables 3 and 4. It is worth mentioning that in the case of the , in addition to the usual Fermi Gamow–Teller and tensor terms, we have additional contributions coming from the nucleon recoil term ('R ) and the kinematically favored spin antisymmetric term ('P ). Due to these two e5ects the limit extracted for is much smaller than that for [14] (see [3,4]). E5ective operators of a similar structure also appear in the context of R-parity violating interactions when a neutrino appears in the intermediate states (see below). There seem to be signiFcant changes in the nuclear matrix elements, when the p–n pairing is incorporated (see Table 4). This point needs special care and further exploration is necessary. It has only been examined in some exactly soluble models, e.g. SO(8), or better approximation

J.D. Vergados / Physics Reports 361 (2002) 1–56 Table 3 The matrix elements of 0-decay for 48 Ca; the framework of QRPA without p–n pairing Nucleus

48

Ca

76

Ge

82

76

Ge;

96

Se

82

Se;

96

Zr ,

100

Zr

100

Mo;

116

116

Mo

Cd ;

128

13

128

Cd

130

Te;

Te and

136

130

Te

Xe calculated in

Te

136

Xe

QRPA without p–n pairing 0 MGT −0:785 −0:468 'F 0 |MGT (1 − 'F )| 1.152 −134:9 'H

2.929

−2:212 −0:008

−0:038

3.040

−68:37

2.097

0.615

−0:149

2.230

2.409

−44:27

−47:24

1.086

−0:504

−0:035

−0:004

−0:168

−0:817

'R

172.1

193.0

124.2

113.8

105.1

1.077 0.244 −0:038 0.916 −1:147

1.050 0.079 −0:013 0.960 −0:049

Table 4 The matrix elements of 0-decay for 48 Ca; the framework of QRPA with p–n pairing Nucleus

48

Ca

76

Ge

QRPA with p–n pairing 0 MGT −0:405 1.846 0.158 0.274 'F 0 |MGT (1 − 'F )| 0.341 1.340 6.075 −32:75 'H

82

1.143 0.121 −0:130 0.845 −0:836

76

Ge;

96

Se −1:153 −0:416

1.633

−57:20

82

Se;

Zr

0.280 2.282 0.358 −41:64

'VF 'VGT 'VT 'F! 'GT! 'VP

0.184 1.226 0.130 0.131 0.775 −0:009

0.322 1.124 0.214 0.235 0.876 −0:479

−0:467

1.082 0.179 −0:379 0.927 −1:621

2.601 1.587 0.209 2.069 0.335 −4:802

'R

57.32

129.3

131.1

157.3

−1:103

0.944

−124:8

'VF 'VGT 'VT 'F! 'GT! 'VP

0.975 −0:212 −0:437 1.057 0.168

0.449

−0:766

−47:06

−1:173

1.174 −0:477 −0:709 0.683 −3:843

96

Zr ,

100

100

1.074 −0:812 −1:032 0.859 −3:891 −151:5

Mo;

Mo

116

116

Cd ;

128

Cd

−0:584

0.119 0.939 −6:784 0.036 0.926 −14:22 −453:8 1.067 0.934 0.853 0.812 1.142 2.519 162.2

−7:400

0.927 −3:991 −6:170 0.938 −7:592 −333:6

2.437

2.327

−0:0179

−0:004

2.480

2.335

−41:54

−39:82

1.598 0.028 1.553 −21:92

−0:022

−0:007

1.097 0.282 0.001 0.895 −1:451

0.022 1.123 0.349 0.036 0.875 −1:627

157.1

149.0

124.8

1.097 0.307 −0:012 0.894 −1:400

130

Te; 128

Te and

Te

136

130

Xe calculated in

Te

136

Xe

1.270 0.308 0.879 −34:02

1.833 0.184 1.495 −55:72

1.346 0.066 1.257 −35:37

0.370 1.159 0.343 0.260 0.831 −2:907

0.218 1.115 0.411 0.159 0.879 −0:993

0.082 1.167 0.332 0.052 0.832 −2:441

158.6

192.6

138.4

schemes [64], but only in connection with the 2-decay, or shell model calculations but for systems, which do not double beta decay [65]. Returning back to the question of the availability of nuclear matrix elements relevant for neutrinoless double beta decay, we refer once again to two excellent recent reviews [9,10]. These reviews also provide a more detailed description of the nuclear models employed.

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J.D. Vergados / Physics Reports 361 (2002) 1–56

Fig. 3. The Feynmann diagrams at quark level leading to majoron emission in the 0-decay instead of the more well-known mass term. Here '0 stands for the majoron, not to be confused with the neutralino, which we will encounter later in connection with supersymmetry. Fig. 4. The same process as in Fig. 3, but written at the nucleon level in the case of the isosinglet majoron, which couples to the right-handed neutrinos.

2.3. The majoron emission mechanism It is well known that in some theories lepton number is associated with a global, not a local, symmetry. When such theories are broken spontaneously, one encounters physical Nambu– Goldstone bosons, called majorons. These bosons only couple to the neutrinos. So in any model which gives rise to mass term for the light neutrino (mass insertion in the neutrino propagator), one may naturally have a competing majoron–neutrino–antineutrino coupling. Such a mechanism is shown at the quark level in Fig. 3. The majoron, which couples to the left-handed neutrinos, comes from the neutral member of the isotriplet. Such a multiplet, however, cannot easily be accommodated theoretically. So this type of Majoron is not present in the usual models. On the other hand, there is a Majoron '0 , which couples to the right-handed neutrino, the imaginary part of an isosinglet scalar. This gives rise to the mechanism shown in Fig. 4 at the nucleon level. The right-handed neutrino, however, has a small component of light neutrinos (see Eq. (9)): L'0 = gij [SiL 5 jL ]'0 (28) i¡j

with gij =

i¡j

(21) 0 U.i(21) Uj g. !i ;

(29)

J.D. Vergados / Physics Reports 361 (2002) 1–56

15

0 the coupling of the Majoron to the corresponding neutrino Eavors. The expression for with g. the half-life takes the form light ' 0 −1 ] = G01 |gMm |2 (30) [T1=2 with g = i¡j Uei(11) Uej(11) gij . Notice that, even if gij0 takes natural values, the coupling gij is very small due to the smallness of the mixing matrix U (21) . Thus the e5ective couplingg is very small. So, even though we do not su5er in this case from the suppression due to the smallness of the mass of the neutrino, the majoron emission mechanism is perhaps unobservable. There exist, however, exotic models, which, in principle, may allow majoron emission, like the bulk majoron [66] and others [67,68], which we are not going to theoretically pursue any further. It is, however, straightforward to extract the limits on the e5ective coupling g, since the nuclear matrix elements are the same as in the light neutrino mass mechanism and only the kinematical function is di5erent. Before proceeding further with our theoretical analysis we will brieEy discuss the currently planned experiments, since most experimenters are motivated by the extraction of the above lepton violating parameters.

3. Brief description of current experiments In this section we will brieEy present the essential ingredients of current experimental activities, which have culminated after a long history of heroic experimental e5orts, which began in 1948. Before we proceed further with the discussion of the experiments, we should mention that one would like to explore all possible experimental signatures accompanying the simultaneous and from the same space point emission of two electrons (or positrons depending on the target). These include the energy distributions f(1 ) and f(2 ) of each electron as well as the energy distribution f(1 + 2 ) of the two electrons in coincidence as well as the vertex angle 0 between them. The main motivation of 0 double beta decay experiments is to extract the (average) neutrino mass m , even though, as we have already mentioned, neutrinoless double beta decay, if observed, will shed light on many lepton violating mechanisms allowed by current theoretical particle models. In such a scenario Eq. (10) becomes 0 −1 [T1=2 ] = G01 |

m

me

light 2 Mm | :

(31)

From this equation we see that the best choice of a target is dictated by as large as possible kinematical function G01 (see Table 1) and as large as possible nuclear matrix elements. The latter are, of course, a matter of detailed nuclear calculations. The kinematical function, however, depends only slightly on gross properties of the nucleus like A and Z, and it is mainly an increasing function of the available energy. The same is true for the mass-independent amplitude. So a nuclear target should have as large as possible available energy. An additional and, perhaps, much more important reason for selecting a large available energy is to minimize the background problems, since natural radioactivity peaks at low

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J.D. Vergados / Physics Reports 361 (2002) 1–56

energies. The crucial value is 2:614 MeV, which is the energy of the most dangerous ’s originating from the 208 Tl decay. Among the 35 target candidates only six pass this test. These are: 48 Ca (Q = 4:272 MeV, natural abundance = 0:187%); 82 Se (Q = 3:034 MeV, natural abundance = 8:73%); 96 Zr (Q = 3:350 MeV, natural abundance = 2:80%), 100 Mo (Q = 3:054 MeV, natural abundance = 9:63%); 116 Cd (Q = 2:805 MeV, natural abundance = 7:49%) and 150 Nd (Q = 3:367 MeV, natural abundance = 5:6%). The kinematical functions relative to that of the popular Ge, which holds the world record of the longest lifetime limit, i.e. G01 (A; Z)=G01 (Ge), are, respectively, 10:7; 4:7; 9:5; 7:7; 8:2 and 33:5 for A = 48; 82; 96; 100; 116 and 150, respectively. The last value deviates from the simple rule and is somewhat uncertain. Since one would like to have as many double beta decaying nuclei as possible in a source of a given size, one would like to enrich it in the desirable isotope. So the parameter is also very important. One would like to have as high as possible energy resolution, to distinguish 0-decay in the tail of the much faster 2-decay. These considerations are summarized by the following formula:

T1=2 ∼

mt : R Bgd

(32)

In the above expression t is the measuring time, is the eQciency, R is the resolution, m is the source mass and Bgd is the background. The most important quantities are those, which are outside the square root ( and ). We thus need a high purity detector of perhaps hundreds of kg mass with good energy resolution operating at the maximum detection eQciency. The latter can be maximized (100%) only for an active source technique. Clearly, one cannot go very far by only increasing the size of the target, due to the square root. In addition, very large mass can only be achieved at non-prohibitive costs only for few systems, namely 76 Ge;82 Se;116 Cd ;130 Te and 136 Xe. Obviously, one should try to reduce the background to its ultimate limit, i.e. the reaction induced 2 decay, which, however, is in principle calculable and can be removed. In any case, to accomplish this goal one must achieve very high resolution, e.g. R = 4% at 3 MeV for 100 Mo (see [72]). Since the measuring times are of order of ∼ 10 yr , one would like the equipment to have the simplest possible design. The experiment can consist of either an “active” source, i.e. the source serves also as the detector, or a “passive” source, i.e. an experiment in which the source is introduced into the detector. Since the best limits have up to now been obtained by an active source, it is natural to expect that such experiments will continue to lead the Feld. With the above criteria in mind, di5erent experimental groups intend to utilize di5erent targets, see, e.g., the recent reviews by Morales [73] and Tretyak and Zdesenko [74]. As a result, the obtained lifetime limits on one of the most popular targets 76 Ge have improved from 1:2 × 1021 [75] to 1:9 × 1025 [76]. This improvement of 4 orders of magnitude in a period of about 30 years is characteristic of experiments on almost all targets. The main activities are: (1) The GENIUS and MAJORANA projects: The GENIUS experiment [76] intends to construct a large source (about 10 t) of enriched Ge, which also serves as the detector. If and when the 10 t target–detector is implemented, a limit on the neutrino mass m , in the range 10−2 − 10−3 eV will be reached. This ambitious goal, which will shed light on a number

J.D. Vergados / Physics Reports 361 (2002) 1–56

17

of important issues in physics, has a good chance of being achieved, given suQcient funding, since this group already has the world record in the longest lifetime extracted from the 0 = 5:7 × 1025 yr , from which these authors extract the limit on the neutrino mass of data, T1=2 m 6 0:1 eV [76]. This is not universally accepted, however, not only because somewhat favorable nuclear matrix elements were employed, but also because it was based on a statistical technique not previously employed in this type of experiment. As a result the extracted lifetime is viewed as quite optimistic [77]. In any case these authors have subsequently made their 0 = 1:9 × 1025 yr (T 0 = 3:17 × 1025 yr ) at the 90%(68%) C.L. results more conservative [78], T1=2 1=2 Another project is the MAJORANA collaboration [79], which will utilize 0:5 t of HP 76 Ge detectors in conventional Cu cryostats, achieving very low background. It will be conducted by a strong collaboration with long experience in handling and processing 76 Ge crystals, which will include the Duke University, the University of North Carolina, North Carolina State University, ITEP (Moscow), INR (Dubna), PaciFc North West University, Argonne National Laboratory, the University of S. Carolina and New Mexico State University. It is ideally located and it can be realistically expected to reach a lifetime T1=2 ¿ 1027 yr leading to neutrino masses m 6 0:05 eV. (2) The CAMEO project [80]: The BOREXINO Counting Test Facility (CTF), CAMLAND, SNO, etc., characterized by unique features (super-low background and large sensitivity volume), are used in the CAMEO project in Gran Sasso underground laboratory to detect 0-decay. Members of this group, in particular the Kiev group headed by Zdesenko, a pioneer in the Feld, have a long history in the pursuit of 0 and 2 double beta decay [81] (see these recent references for citations to their earlier work). Pilot measurements with 116 Cd and Monte Carlo simulations indicate that the sensitivity of the CAMEO experiment is in the range (3–5) × 1024 yr utilizing only 1 kg of “passive” source. Limits have also been put on the half-lives to excited states. This program will further evolve to CAMEO II [83], utilizing 100 kg of enriched 116 CdWO4 crystals placed in the CTF, and is expected to reach T1=2 ¿ 1026 yr leading to a mass limit of m 6 0:05 eV. With 1 t of material this limit can become m 6 0:02. In fact, recently obtained results with four enriched scintillators with a total mass of 0:339 kg yield a half-life ¿ 0:7(2:5) × 1023 yr at 90%(68%) C.L. Another target, 160 Gd , with a small Q-value of 1:73 MeV, occurring with relative abundance of 11:9% in natural gadolinium, can also be used in the double beta decay search [82], employing GSO crystals [84]. The obtained limits at this stage, e.g. the best limit 1021 yr obtained by Danevich et al. [82], are not as stringent as those obtained by experiments on other targets. In the context of the CAMEO program, two future projects [72] involving the 76 Ge detector are being developed. The Frst GEM-I will involve 1 t of natural HP 76 Ge detectors aiming at 1026 yr leading to m 6 0:05 eV while the second GEM-II with 1 t of enriched HP 76 Ge semiconductor detectors aim at 1028 yr or m 6 0:015 eV. (3) Thermal detectors-thermal bolometers (Milano group) [85,86]: In the continuous struggle against background and towards improving the energy resolution the work of the Milano group, headed by Fiorini, one of the pioneers in the Feld of double beta decay, stands out as the most promising for the future. The idea is to use various types of low temperature devices. Thus using the bolometric technique with 20 natural tellurite crystals, i.e. a total cryogenic mass of about 6:8 kg, at a temperature of 10 mK in the Gran Sasso Underground Laboratory, they

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J.D. Vergados / Physics Reports 361 (2002) 1–56

have obtained the limits 8:6 × 1022 and 1:44 × 1023 yr for the two isotopes 128 Te and 130 Te, respectively, at the 90% conFdence level. It is remarkable that even at this stage of development they have obtained the most restrictive lepton violating parameters ever obtained in a direct experiment, with the exception of the above-mentioned limit from the Ge Heidelberg–Moscow and IGEX experiments. These limits depend, of course, on the nuclear matrix elements employed. A somewhat larger system (56 detectors of 0:76 kg each (CUORICINO)) is an encouraging step towards the ultimate big and complex detector CUORE [85], which is one of the promising future projects. (4) Tracking detectors
(33)

The half-life limits associated with the parameters and have been obtained from decays to 2+ , which cannot be reached with leptonic currents of the same chirality. Strictly speaking, however, even the mass term can contribute, if the nucleon recoil terms are included in the nucleon current [95]. They also have obtained limits for other lepton violating processes. There have been attempts to relate the -decay to other processes and to use the data from the latter to make predictions about the nuclear matrix elements entering the former [96]. The MOON project [94] aims at simultaneously carrying out the study of 0 of 100 Mo and the real-time measurement of low-energy solar neutrinos by inverse -decay. The sensitivity of this experiment, which will utilize 3:3 t of the enriched isotope, is such that it will reach m 6 0:05 eV. (7) Laser tagging techniques [97,98]: A novel approach towards detecting neutrinoless double beta decay with sensitivity down to neutrino masses of 10−3 eV has recently been proposed.

J.D. Vergados / Physics Reports 361 (2002) 1–56

19

It utilizes TPC counters with the possibility of detecting (direct tagging) Ba+ ions by partially neutralizing the Ba2+ in the Fnal state of the 136 Xe decay. These ions are detected through their laser induced Euorescence [99]. So for the future we have two projects: The EXO project [97] utilizes a 40 m3 TPC operated at 10 atm of enriched 136 Xe (1–2 t of natural Xe). The 136 Xe − CTF (2000) involves the dissolution of ∼ 80 kg (∼ 1:5 t ) of enriched (natural) Xe in the liquid scintillator of the CTF to reach limits of 1025 yr . Unfortunately, the 2-decay cannot be discriminated against in this way. The proposers, however, do not consider this a serious background at the sought sensitivities and anyway, if necessary, it can be eliminated by kinematical reconstruction in a Xenon TPC. (8) The COBRA Project: Another interesting proposal for double beta decay experiments, called COBRA [100], consists of utilizing CdTe or CdZnTe semiconductor detectors. The source and detector material is the same, the expected energy resolution is high and the experiment can be done at room temperature or slightly below. It can also achieve the simultaneous detection of a number of negaton and positon emitters. In its Frst stage 10 kg of CdTe or CdZnTe detectors will be employed leading to an order of magnitude improvement in the present lifetime limit or m 6 0:1 eV. This, of course, can be further improved in the future.

4. The R-parity violating contribution to 0-decay In SUSY theories R-parity is deFned as R = (−1)3B+L+2s ;

(34)

with B being the baryon, L the lepton numbers and s the spin. It is +1 for ordinary particles and −1 for their superpartners. R-parity violation has recently been seriously considered in SUSY models. It allows additional terms in the superpotential given by W = i Lai H2a + ijk Lai Lbj Ekc ab + ijk Lai Ujb Dkc ab + ijk Uic Ujc Dkc ;

(35)

where a summation over the Eavor indices i; j; k and the isospin indices a; b is understood (ijk is antisymmetric in the indices i and j). The last term has no bearing in our discussion, but we will assume that it vanishes due to some discrete symmetry to avoid very fast proton decay. The Frst term is a lepton violating bilinear and, since it cannot be rotated away, it can lead to neutrinoless double beta decay. The ’s are dimensionless couplings not predicted by the theory. The couplings are assumed to be given in the basis in which the charged fermions are diagonal. In the above notation L; Q are isodoublet and E c ; Dc isosinglet chiral superFelds, i.e. they represent both the fermion and the scalar components. The above R-parity violating superpotential can lead to Majorana neutrino masses without the need of introducing the right-handed neutrino and invoking the see-saw mechanism [101,23]. One can then have contributions to neutrinoless double beta decay in the usual way via intermediate massive neutrinos as discussed above.

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J.D. Vergados / Physics Reports 361 (2002) 1–56

Fig. 5. The R-parity violating contribution to 0-decay mediated by neutralinos arising from the bilinear terms in the superpotential. For comparison we give the neutrino mediated process of Fig. 1 expressed at the quark level.

4.1. The contribution arising from the bilinears in the superpotential The Frst term in the superpotential can directly lead to neutrinoless double beta decay [23,102,22] via the W -charged lepton–neutralino interaction: g 0 L = − √ n W− eS L 'nL ; (36) 2 where n is a dimensionless quantity, associated with each of the four neutralinos, which arises due to neutrino–neutralino mixing. This term gives rise to a diagram analogous to that of Fig. 1 with the intermediate particle now being the neutralino, which is heavy and leads to a short ranged operator. For the reader’s convenience this is shown in Fig. 5 at the quark level. One thus obtains an analogous lepton violating parameter: mp LN → L'0 = |n |2 : (37) M'n0 n 4.2. The contribution arising from the cubic terms in the superpotential It has also been recognized quite sometime ago that the second and third terms (cubic terms) in the superpotential could lead to neutrinoless double beta decay [15,16]. This has been re-examined quite recently [20]. Typical diagrams at the quark level are shown in Fig. 6. Note that as intermediate states, in addition to the s-leptons and s-quarks, one must consider the neutralinos, four states which are linear combinations of the gauginos and higgsinos, and the colored gluinos (supersymmetric partners of the gluons). Whenever the process is mediated by gluons a Fierz transformation is needed to lead to a colorless combination. The same thing is necessary whenever the fermion line connects a quark to a lepton. As a result one gets at the quark level not only scalar (S) and pseudoscalar (P) couplings, but tensor (T) couplings

J.D. Vergados / Physics Reports 361 (2002) 1–56

21

Fig. 6. The R-parity violating contribution to 0-decay mediated by s-fermions and neutralinos (gluinos) arising from the cubic terms in the superpotential.

as well. This must be contrasted to the V and A structure of the traditional mechanisms. One, therefore, must face the problem of how to transform these operators from the quark to the nucleon level. 4.2.1. The lepton number violating parameters with R-parity non-conservation As we have mentioned the e5ective lepton number violating parameter arising from the bilinear terms in the superpotential is analogous to that arising from the heavy intermediate neutrinos and, thus, it will not be discussed further. The e5ective lepton violating parameter, arising from the cubic terms in the superpotential, assuming that pion exchange mode dominates, as the authors of Refs. [20,10] claim, can be written as SUSY = (111 )2 38 ('PS PS + T )

(38)

with PS (T ) associated with the scalar and pseudoscalar (tensor) quark couplings given by PS = ';˜ e˜ + ';˜ q˜ + ';˜ f˜ + ˜g˜ + 7g˜ ;

(39)

T = ';˜ q˜ − ';˜ f˜ + ˜g˜ − g˜ :

(40)

These authors Fnd 'PS = (5=3), but, as we shall see below, it depends, in general, on ratios of nuclear matrix elements. For the diagram of Fig. 4a one Fnds

2. 2 mp ';˜ e˜ = ( ) : (41) e˜ m'˜ e˜e˜ (GF m2W )2

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J.D. Vergados / Physics Reports 361 (2002) 1–56

For the diagram of Fig. 4b one Fnds

. 2 mp 2 mp ( ) + ( ) ; ˜';˜ q˜ = u˜ d˜ m'˜ d˜d˜ m'˜ u˜u˜ 2(GF m2W )2 mp 1 [( ˜ )2 + (u˜)2 ] : ˜g˜ = .s 6 (GF m2W )2 d mg˜ For the diagram of Fig. 4c one Fnds

mp mp mp . e˜d˜ + e˜u˜ + d˜ u˜ ; ˜';˜ f˜ = m'˜ e˜d˜ m'˜ e˜u˜ m'˜ d˜u˜ 2(GF m2W )2 ˜g˜ = where

mp 1 ; .s ˜ u ˜ d 2 12 (GF mW )2 mg˜

2 mW 2 mW X = ; X = e˜ L ; u˜ L ; d˜ = : mX md˜R

4 mp mp = '˜ ;f˜ '˜ ;f˜ ; i i m'˜ f˜ f˜ m'˜i

(42) (43)

(44) (45)

(46) (47)

i=1

where '˜ ;f˜ and '˜ ;f˜ are the couplings of the ith neutralino to the relevant fermion–sfermion. i i These are calculable (see e.g. Ref. [7]). Thus, ignoring the small Yukawa couplings coming via the Higgsinos and taking into account only the gauge couplings, we Fnd '˜i ;e˜ =

Z2i + tan 0W Z1i ; sin 0W

(48)

'˜i ;u˜ =

Z2i + (tan 0W =3)Z1i ; sin 0W

(49)

'˜ ;d˜ = − i

Z1i ; 3 cos 0W

(50)

˜ W˜ 3 in terms of the neutralino mass where Z1i ; Z2i are the coeQcients in the expansion of the B; eigenstates. Note that in this convention some of the masses m'˜i may be negative. 4.2.2. The pion mode in R-parity induced 0-decay Even though the pion model, (6), may be important in other cases when the intermediate particles are heavy, giving rise to short range operators, in this section we will elaborate a bit further on its application in the extraction of the R-parity violating parameters associated with the processes discussed in the previous subsection. We will Frst evaluate the relevant amplitude, using harmonic oscillator wave functions after adjusting the parameters to Ft related experiments, and then compare this amplitude to that obtained by other techniques.

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23

Fig. 7. The pion mediated 0-decay as a contact interaction (a). It arises as a 1, (b) (an analogous -vertex in the other nucleon is understood), and 2 exchange contributions (c). The lepton violation occurs either in one nucleon (b) or in the pions (c).

Fig. 8. The double beta decay of a neutron into a proton with simultaneous emission of a + . f stands for a neutral fermion, but this process is important when it is very massive, such as heavy neutrino, neutralino or gluino.

(a) The 1 mode: Let us begin with the second process of Eq. (6) (see diagrams (a) and (b) of Fig. 7). This process is further analyzed in Fig. 8 and it involves a direct term and an exchange term. In this case it is clear that the amplitude must be of the PS type only. The tensor contribution cannot lead to a pseudoscalar coupling at the nucleon level. Such a coupling is needed to be combined with the usual pion nucleon coupling in the other vertex to get the relevant operator for a 0+ → 0+ decay. The direct term is nothing but a decay of the pion into two leptons with a simultaneous change of a neutron to a proton by the relevant nucleon current decay. By working out the

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algebra associated with Fig. 8 directly one Fnds the amplitude A1 (direct) = 4.˜1

<1 · ˜q 2 m : 2mN

(51)

The exchange contribution, in which the produced up quark of the meson is not produced from the “vacuum” but comes from the initial nucleon, is a bit more complicated. The harmonic oscillator quark model, however can be used to get its relative magnitude (including the sign) with respect to the direct term. In this way we Fnd A1 (exchange) = − 3.˜1

<1 · ˜q 2 m 2mN

(52)

with .˜1 = (2)3

mN '(0) : 3mq

(53)

Thus the e5ective two-body transition operator in momentum space at the nucleon level becomes <1 · ˜q<2 · ˜q (qb)2 m2 PS exp − (54) ?1 = c1 (2mN )2 6 q2 + m2 with c1 = gr .˜1 i.e. mN c1 = (2)3 gr '(0) ; 3mq

(55)

where gr = 13:5 is the pion nucleon coupling and mq is the constituent quark mass. We see that, in going from the quark to the nucleon level, the factor of three coming from the mass gain is lost due to the momentum being reduced by a factor of three. The quantity '(0) is essentially the meson wave function at the origin given by √ 6 '(0) = 1=4 m−3=2 (0) : (56) 2 The quantity '(0) can be obtained from the → ; decay via the expression 2 2 m 1 1 GF 2 √ m m2 1 − 2 '2 (0) : = A m 2

(57)

From the measured lifetime A = 2:6 × 10−8 we obtain '(0) = 0:46. (b) The 2 mode: The Frst process of Eq. (6) is described by diagram (c) of Fig. 7 and is further illustrated in Fig. 9. Here, we show only the case of intermediate heavy Majorana neutrinos. In the R-parity violating mechanisms the situation is similar. Instead of heavy neutrinos one encounters neutralinos and gluinos. Also instead of the gauge bosons one has scalars (s-leptons or s-quarks). In this case, however, both the PS and T terms contribute. We thus get A2 (T) = 38 .˜2 m4 ; A2 (PS) = 18 .˜2 m4 ;

(58)

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Fig. 9. The 0-decay of pions in Eight illustrated at the quark level in the case of intermediate heavy neutrinos. A similar diagram appears in R-parity violating SUSY models with intermediate s-quarks, s-leptons as well as neutralinos and gluinos.

.˜2 = 4(2)3 '2 (0) ; T ?2 = c2

(59)

<1 · ˜q<2 · ˜q m4 ; (2mN )2 (q2 + m2 )2

(60)

PS T = 23 ?2 ?2

(61)

with c2 = gr2 .˜2 , i.e. c2 = 4(2)3 gr2 '2 (0) :

(62)

Using the above value of '(0) and gr = 13:5 we get c1 = 109 and c2 = 198 which are in good agreement with the values 132.4 and 170.3, respectively, obtained by Faessler et al. [21]. It is now customary, even though it can be avoided [31], to go to coordinate space and express the nuclear matrix elements in the same scale with the standard matrix elements involving only nucleons. Thus, we get mp k mA 2 MEk = .k [MGT + MTk ] ; (63) mp me where the two above matrix elements are the usual GT and T matrix elements with the additional radial dependence given by 1 FGT = e−x ;

FT1 = (3 + 3x + x2 )

e−x

x

;

(64)

26

J.D. Vergados / Physics Reports 361 (2002) 1–56 2 FGT = (x − 2)e−x ;

FT2 = (1 + x)e−x ;

.1 = − c1 C; .2 = c2 C with x = x = m r and 1 m 4 mp 2 : C= mA 48fA2 mp

(65) (66)

(67)

In the above formulas we have tried to stick to the deFnition of SUSY given above (see [38]), but since the tensor (at the quark level) does not contribute to the 1 diagram the “e5ective” nuclear matrix element is not the sum of the two matrix elements of Eq. (63). It can, however, be expressed only in terms of ME2 provided, of course, that we allow 'PS to depend on the nuclear matrix elements, i.e. 2 ME1 MEe5 = ME2 ; 'PS = 4 +1 : (68) 3 ME2 'PS is, of course, independent of the nuclear structure, if ME2 is dominant, as is actually the case. Indeed, for the experimentally derived harmonic oscillator parameter for the meson, b = 1:8 fm, .2 is dominant and 'PS approaches the value of 2=3. In fact, we Fnd .1 = − 1:2 × 10−2 and .2 = 0:15 which are in good agreement with the values −4:4 × 10−2 and 0.20, respectively, obtained by Faessler et al. [21]. Furthermore from the nuclear matrix elements of Ref. [21], one can see that the M 2 is favored, since, among other things, its tensor and Gamow–Teller components are of the same magnitude and sign (in the 1 mode they are opposite). Thus nuclear physics also favors the 2 mode. 4.3. The case of light intermediate neutrinos It is also possible to have light neutrino mediated 0-decay originating from R-parity violating interactions. In this case one has the usual -decay coupling of the V–A type in one vertex and the s-fermion mediated coupling, of the S–P type, in the other end [102,103,23] (see Fig. 10). The lepton violation is achieved via the mixing of isodoublet and isosinglet

Fig. 10. The light neutrino mediated 0-decay in R-parity violating SUSY models. In addition to the usual gauge vertex one has a scalar vertex mediated by s-leptons or down-type s-quarks. The lepton violation proceeds via the mixing between the isodoublet and isosinglet s-fermions. We do not indicate it by a × on the scalar line, as it is customary, since we do not apply perturbation theory, but use the s-fermion mass eigenstates.

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27

s-fermions. The simplest diagram, which involves intermediate s-leptons, can arise from the following interactions: ∗ LLQDc → 111 Vek∗L u L dR ‘˜k (69) k

for the d-quark vertex and LLLEc → 111 VekR Uej SjL eRc ‘˜k

(70)

j; k

for the neutrino vertex. In the above expression V L (V R ) are the mixing matrices which express the doublet (singlet) s-electrons in terms of the mass eigenstates. U is the usual neutrino mixing matrix. The e5ective s-lepton propagator is 2 VLVR L R Mk ek ek P= = V V : (71) ek ek Me4 Mk2 k

k

The last equation follows, since the splitting Mk2 is small compared to the average s-lepton mass Me2 . If, further, the mixing between generations can be ignored we get sin 20e˜ YMe2 4 P= (72) Me˜ : 2 : Combining the above results with the usual V–A coupling one gets GF 2 k. . M= √ Ge˜u L dR eS L 2 eRc u L dL (73) k 2 with √ 2 sin 20e˜ YMe2 4 Ge˜ = 111 111 Uej2 (74) Me˜ : GF 2 : j In the case of s-quark exchange in Fig. 10 the above expressions become ∗ ∗L LLQDc → 111 Vdk u L eRc d˜k

(75)

k

for the u-quark vertex and R LLQDc → 111 Vdk Uej SjL dR d˜ k

(76)

k

for the neutrino vertex. By combining them, we get GF 2 k. . M= √ Gd˜ u L eRc eS L 2 dR u L dL k 2 with √ YMd2˜ 2 2 2 sin 20d˜ Gd˜ = ( ) (Uej ) 2 GF 111 Md4˜ j in a rather obvious notation.

(77)

(78)

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J.D. Vergados / Physics Reports 361 (2002) 1–56

In the case of e˜ and d˜ contributions, in the context of perturbation theory, one can simplify the above expressions by explicitly using the coupling between the isosinglet and the isodoublet s-fermions of the lower charge. In this case sin 20x˜ (79) YMx2˜ = ( + A tan )mx ; x = e; d : 2 Before proceeding farther we have to perform a Fierz transformation: u L eRc eS L k. . dR = − 12 [u L dR eS L eRc k + u L dR eS L i< k eRc + u L i< k dR eS L eRc ] − 18 u L i<. dR eS L i<. eRc k :

(80)

We must now go to the nucleon level and perform a Fourier transform to the coordinate space. For 0+ → 0+ transitions the space component yields GF 2 M=− √ (fA )2 [MT + MGT + rF MF ]e S 0 q · S(1 + 5 )ec (81) 2 with 3 Ge˜ rF = 2 −2 +1 : (82) = Gd˜ =96; Gd˜ 4fA The parameter as well as the quantities M and M˜ have the same meaning as in the mass-independent contribution in the conventional approach (see Section 2.2). Note, however, that in the present mechanism there is no term analogous to the of Section 2.2. We should stress that this novel mechanism can lead to transitions J + , J = 0. So, contrary to conventional wisdom, from the observation of such transitions one cannot deFnitely infer the existence of right-handed currents. 5. The e&ective nucleon current As we have already mentioned, the relevant transition operators are deFned at the quark level, but all computations are carried out at the nucleon level. One thus must have a rigorous and well-deFned procedure on how to go from the quark to the nucleon level. Such a procedure is well understood for the isovector V–A currents entering the one step weak interaction processes, whereby the momenta involved are relatively low. The same cannot be said about neutrinoless double beta decay, which is a two-step process with intermediate momenta not necessarily small. So one must also consider the possible momentum-dependent terms in the hadronic current. A second problem has to do with the short range nature of the operators, if the intermediate particles are all very massive. In this case, the momentum dependence of the nucleon form factor cannot be ignored. The handling of such issues is the subject of this section. 5.1. Handling the short range nature of the transition operator The hadronic current at the quark level takes the form JL† = qA S + [1 − 5 ]q :

(83)

J.D. Vergados / Physics Reports 361 (2002) 1–56

29

Going to the nucleon level is by now a straightforward procedure. In the context of V–A theory, i.e. ignoring the induced pseudoscalar and weak magnetism terms, to be discussed below in Section 5.2, one writes S + [gV (q2 ) − gA (q2 )5 ]H ; JL† = HA

(84)

where q = (p − p ) is the momentum transferred from hadrons to leptons (p and p are the four momenta of neutron and proton, respectively) and gV (q2 ), gA (q2 ) are real functions of a Lorenz scalar q2 , known as the vector and axial vector form factors. The values gV and gA of these form factors in the zero-momentum transfer limit are known as the vector and axial coupling constants, respectively, and take the values gV = 1; gA = 1:254. The needed form factors can be calculated in a given quark model. The non-relativistic harmonic oscillator quark model has certain advantages. One, e.g., can separate out the center of mass motions. Setting gV (q2 ) = gV (q2 ) and gA (q2 ) = gA (q2 ) one Fnds

gA (q2 ) gV (q2 ) (|q|b)2 = = exp − ; (85) gA gV 6 where b is the size of the nucleon. For momentum transfers which are not very large we can approximate these form factors with a dipole shape gV (q2 ) = gV =(1 + q2 =G2V )2 ;

gA (q2 ) = gA =(1 + q2 =G2A )2 :

In previous calculations only one general cut-o5 GV = GA = mA ≈ 0:85 GeV was used (see the next subsection for a more up-to-date choice of the parameters). The above currents lead to the following two-body transition operator: ?a = A+ (i)A+ (j)!a

R0 Fa (rij ); rij

a = V; GT ;

(86)

with !V = 1;

!GT = (i) · (j) :

(87)

If the mass of the exchanged particle between the two nucleons is much heavier than mA one Fnds FV (rij ) = FGT (rij ) = F(r), where F(r) =

m2A FN (x); 48mp me

FN (x) = x[x2 + 3x + 3] exp(−x) ;

(88)

where x = mA rij . Note that in the limit mA → ∞, i.e. if the form factor is neglected, the radial functions become a function and the contribution of the heavy particles becomes zero in the presence of a hard core repulsion. In the presence of the form factor, however, the e5ective operator is characterized by a size if 1=mA and gives a non-zero contribution even in the presence of a repulsive hard core. The form factor, however, causes some damping of the matrix elements compared to those without the form factor and without a hard core. The range of the e5ective operator is still large, compared to those resulting from pion exchange.

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J.D. Vergados / Physics Reports 361 (2002) 1–56

As a result the obtained matrix elements may be a bit more uncertain than the typical matrix elements encountered in the usual nuclear physics. In any case one should calculate the resulting matrix elements as accurately as possible, even though the above uncertainties are typically much smaller than those of the underlying particle model, which gives rise to such mechanisms. The problems associated with the behavior of the transition operator at short distances can also be avoided, if the 0-decay is induced by pions in Eight between the two nucleons (see previous Section 4:3:2). This is comparable to the contribution of the two nucleon mode in the context of V–A theory [27] with mA = 0:85 GeV. Before concluding this subsection we should mention that such short range problems can be avoided, if one utilizes the fact that there is a probability that two nucleons inside the nucleus may behave as six quark clusters. In other words, the two nucleon wave function at short distances, rij ¡ a0 , is replaced by a colorless wave function of six quarks [27]. Then the transition matrix element between the two relative nucleon states (n; l) and (n l ) is proportional to Pnl (a0 )Pn l (a0 ), where Pnl is the probability of Fnding the six quark clusters in the internal region of the two relative nucleon wave functions with quantum numbers (n; l). The results, depend of course on the matching parameter a0 . By employing reasonable approximations one Fnds [27] that this contribution is also comparable to the two nucleon contribution with mA = 0:85 discussed above. In conclusion since in the above approaches underlying physics is di5erent, the obtained agreement allows one to have conFdence in the method employed. 5.2. Momentum-dependent corrections to the eBective nucleon current As we have mentioned the e5ective nucleon current in addition to the usual V and A terms (P,S,T in SUSY contributions) contains momentum-dependent terms [29]. Within the impulse approximation the hadronic current JL in Eq. (7) expressed with nucleon Felds H takes the form

† + 2 2 < 2 2 S JL = HA gV (q ) − igM (q ) q − gA (q ) 5 + gP (q )q 5 H ; (89) 2mp where mp is the nucleon mass and < = (i=2)[ ; ]. gV (q2 ), gM (q2 ), gA (q2 ) and gP (q2 ) are the four nucleon form factors. The axial and vector form factors were discussed above. The values gM (q2 ) and gP (q2 ) in the zero-momentum transfer limit are known as weak-magnetism and induced pseudoscalar coupling constants, respectively. For nuclear structure calculations it is necessary to reduce the nucleon current to its nonrelativistic form. We shall neglect small energy transfers between nucleons in the non-relativistic expansion. Then the form of the nucleon current coincides with those in the Breit frame and we arrive at [69],

J (x) =

A n=1

0 0 2 k k 2 A+ n [g J (q ) + g Jn (q )](x − rn );

k = 1; 2; 3

(90)

J.D. Vergados / Physics Reports 361 (2002) 1–56

with 0

2

2

2

J (q ) = gV (q );

2

Jn (q ) = gM (q )i

n × q

2M

q n · q ; + gA (q ) − 2 2 2

q + m

31

(91)

rn is the coordinate of the nth nucleon. For the two weak magnetism terms we shall use the following parameterization [29]: gM (q2 ) = (p − n )gV (q2 ) ; where (p − n ) = 3:70; GV = MA = 0:85 GeV [70] and GA = 1:09 GeV [71]. In the previous calculations only one general cut-o5 GV = GA ≈ 0:85 GeV was used. In our discussion of the e5ects of the modiFed nucleon current we take the empirical value of GA deduced from the antineutrino quasielastic reaction S p → + n. A larger value of the cut-o5 GA is expected to slightly increase the values of corresponding nuclear matrix elements. It is worth noting that with these modiFcations of the nucleon current one gets a new contribution in the neutrino mass mechanism, namely the tensor contribution. The two-body e5ective transition operator takes in momentum space the form ? = A+ A+ (−hF + hGT <12 − hT S12 ) ;

(92)

where the three terms correspond to Fermi (F), Gamow–Teller (GT) and Tensor (T) with S12 = 3(1 · qˆ2 · q) ˆ − <12 ;

<12 = 1 · 2 :

(93)

One Fnds that 2 hF = gV (q2 ) ;

2 2 2 2 (q2 )q2 1 2 2 q q g 2 M hGT (q2 ) = gA (q2 ) 1 − + + ; 3 q2 + m2 3 q2 + m2 3 4m2p 2 2 (q2 )q2 2 q2 1 q2 1 gM 2 2 2 − + : hT (q ) = gA (q ) 3 q2 + m2 3 q2 + m2 3 4m2p

(94)

The exact results will depend on the details of the nuclear model, since the new operators have di5erent momentum (radial) dependence than the traditional ones and the tensor component is entirely new. We can get a crude idea of what is happening by taking the above average momentum q = 100 MeV=c. Then we Fnd that the GT ME is reduced by 22%. Then assuming that, after setting hGT = hT = 1 the tensor matrix element is about half the Gamow–Teller one, we Fnd that the total reduction is 28%. This is in perfect agreement with the exact results for the A = 76 system, 29%, but a bit smaller than the 38% obtained for the A = 130 system. We will now summarize the results obtained with the above modiFcations of the nucleon current and refer the reader to the published work [29] for more details. The Fermi, Gamow–Teller and Tensor light contributions to the full nuclear matrix element Mm in Eq. (10) for the two representative 76 130 Te are presented in Table 5. One notices signiFcant additional 0-decay nuclei Ge and contributions to GT (AP and PP) and tensor (AA and PP) nuclear matrix elements coming

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Table 5 The Fermi, Gamow–Teller and Tensor nuclear matrix elements for the light Majorana neutrino exchange of the 0-decay of 76 Ge and 130 Te with (rows 2 and 4) and without (rows 1 and 3) short range correlations. The subscripts VV, AA, AP and PP refer to the terms arising from the current combinations vector–vector, axial–axial, axial–pseudoscalar and pseudoscalar–pseudoscalar. Notice that the axial and pseudoscalar currents can interfere Transition 76

Gamow–Teller

Ge

130

Te

Tensor

MFlight

light MGT

MTlight

AA

AP

PP

AP

PP

5.132 2.797

−1:392 −0:790

0.302 0.176

−0:243 −0:246

0.054 0.055

−2:059 −1:261

4.042 2.183

−0:188 −0:190

4.158 1.841

−1:173 −0:578

0.258 0.134

−0:329 −0:333

0.074 0.075

−1:837 −1:033

3.243 1.397

−0:255 −0:258

Table 6 Nuclear matrix elements for the light and heavy Majorana neutrino exchange modes of the 0-decay for the nuclei studied in this work calculated within the renormalized pn-QRPA ()0 -decay: 0+ → 0+ transition ME

76

Ge

82

Se

96

Zr

Light Majorana neutrino (I = light) I MVV 0.80 0.74 0.45 I MAA 2.80 2.66 1.54 I MPP 0.23 0.22 0.15 I MAP −1:04 −0:98 −0:65 MIm

2.80

2.64

1.49

Heavy Majorana neutrino (I = heavy) I MVV 23.9 22.0 16.1 I MMM −55:4 −51:6 −38:1 I MAA 106.0 98.3 68.4 I MPP 13.0 12.0 9.3 I MAP −55:1 −50:7 −41:1 MIN

32.6

30.0

14.7

100

Mo

116

Cd

128

Te

130

Te

136

Xe

150

Nd

0.82 3.30 0.26 −1:17

0.50 2.08 0.15 −0:69

0.75 2.21 0.24 −1:04

0.66 1.84 0.21 −0:91

0.32 0.70 0.11 −0:48

1.14 3.37 0.35 −1:53

3.21

2.05

2.17

1.80

0.66

3.33

28.3

17.2

25.8

23.4

13.9

39.4

−67:3

−39:8

−60:4

−54:5

−31:3

58.3 7.9 −34:8

167.0 23.0 −101:0

29.7

21.5

26.6

23.1

14.1

35.6

123.0 16.1 −70:1

74.0 9.1 −39:0

111.0 14.9 −64:9

100.0 13.6 −59:4

−92:0

from higher order nucleon current terms. AP and PP originate from the second (Frst) and third (second) terms in hGT (hT ) of Eq. (94). By glancing at Table 5 we also see that, with proper treatment of short range two-nucleon correlations (see e.g. [7]), all matrix elements are strongly suppressed. The e5ect is even stronger in the case of heavy intermediate particles. Detailed results [29] for various nuclei are presented in Table 6.

J.D. Vergados / Physics Reports 361 (2002) 1–56

33

Fig. 11. The Feynmann diagrams at the nucleon level for (− ; e+ ) conversion when the leptonic currents are both left handed for light neutrino or both right handed for heavy neutrino. The average neutrino mass extracted is di5erent from the muon number conserving 0-decay.

6. The exotic double-charge exchange − to e+ conversion in nuclei As we have mentioned in the introduction the (− ; e+ ) conversion occurs in all models which permit 0-decay to occur. It is, in fact, the Eavor violating analog of e− ; e+ conversion. The traditional neutrino mediated process is shown in Fig. 11. It is enhanced kinematically compared to the e− ; e+ conversion, but, unfortunately, while the ordinary atom is otherwise absolutely stable, the muonic atom does not enjoy the same advantage, since the muon is captured by the nucleus via the ordinary weak interaction emitting a neutrino. Thus, we can only talk about a branching ratio Re+ = M(− → e+ )=M(− → ) ; which is expected to be extremely small. The best limit found is for the and PSI [104 –107], i.e. Re+ 6 4:6 × 10−12

(95) 48 Ti

nucleus at TRIUMF (96)

(see [107]) and Re+ 6 4:4 × 10−12

(97)

(see [108]), respectively. This limit is expected to be further improved by future experiments, at PSI (SINDRUM II experiment), which aims at pushing the sensitivity of the branching ratio Re+ to 10−14 , and at Brookhaven (MECO experiment) with expected sensitivity about three to four orders of magnitude below the existing experimental limits [109,110].

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J.D. Vergados / Physics Reports 361 (2002) 1–56

Traditionally, − → e± exotic processes were searched by employing medium heavy (like 48 Ti and 63 Cu) [106,108] or very heavy (like 208 Pb and 197 Au) [106,111,107] targets. For technical reasons the MECO target has been chosen to be the light nucleus 27 Al. The MECO experiment, which is planned to start soon at the alternating gradient synchrotron (AGS), is going to use a new very intense − beam and a new detector [109]. The basic feature of this experiment is the use of a pulsed − beam to signiFcantly reduce the prompt background from − and e− contaminations. The best upper limit for the − → e− conversion branching ratio Re− set up to the present has been extracted at PSI (SINDRUM II experiment) as Re− 6 6:1 × 10−13 For the

208 Pb

for

48

Ti target [106] :

(98)

target the determined best limit is

Re− 6 4:6 × 10−11

for

208

Pb target [108] :

(99)

In spite of their many similarities, however, the two lepton number violating processes also have some signiFcant di5erences, which can be brieEy summarized as follows: (i) As we have already mentioned, due to the nuclear masses involved, neutrinoless double beta decay occurs only in speciFc nuclear systems. These systems, with the possible exception of 48 Ca, have a complicated nuclear structure. Furthermore neutrinoless double beta decay can lead mainly to the ground state and, only in exceptional cases, to few low-lying excited states of the Fnal nucleus. Such constraints are not imposed on muon positron conversion, due to the rest energy of the disappearing muon. (ii) From − ; e+ experiments, in conjunction with appropriate nuclear matrix elements as input, one may extract lepton violating parameters, which depend on Eavor. These are related to neutrinoless double beta decay. One merely takes the complex conjugates of neutrino mixing matrices and replaces e → in the initial vertex. Thus Eqs. (8) and (9) become m

e+

=

(LN )e+ =

3 k 3 k

(RN )e+

2

(11) ∗ (Uek(11) Uk ) !k mk ;

(100)

mp ; Mk

(101)

12 ∗ (Uek(12) Uk ) #k

2

= ( + )

3 k

22 ∗ (Uek(22) Uk ) #k

mp Mk

(102)

with mp (me ) being the proton (electron) mass, is the mass squared ratio of WL and WR and their mixing. In the case of SUSY one suitably modiFes one of the indices of the couplings in the cubic terms. One, however, does not know a priori which Eavor combination is favored, see, e.g., a recent review by Kosmas et al. [112]. (iii) The long-wavelength approximation does not hold in the case of (− ; e+ ) conversion, since the momentum of the outgoing e+ is high. Thus, the e5ective two-body operator responsible for the (− ; e+ ) conversion is strongly energy dependent and more complicated than the corresponding one for the 0-decay. This inconvenience may be circumvented by choosing

J.D. Vergados / Physics Reports 361 (2002) 1–56

35

in this case a target with the simplest possible nuclear structure, provided, of course, that it meets some standard experimental requirements. (iv) Neutrinoless double beta decay has an experimental advantage in that there exists no other competing channel for the decay of the initial nucleus. Thus, we view the two processes as providing useful complementary information and both, if possible, should be pursued. Strictly speaking (− ; e+ ) and neutrinoless double beta decay should be treated as two-step processes by explicitly constructing the states of the intermediate (A; Z ± 1) systems. It has been found [32,33], however, that, for neutrinoless double beta decay, since the energy denominators are dominated by the momentum of the virtual neutrino, closure approximation with some average energy denominator works very well. We expect this approximation to also hold in the case of (− ; e+ ) to suQcient accuracy. We will, therefore, obtain the rate to a given Fnal state by replacing the intermediate nuclear energies by some suitable average. Then by summing over the partial rates of all allowed Fnal states of the nucleus (A; Z − 2) we obtain the total rate. The results obtained in this way can be compared to that obtained by invoking closure [113] with some appropriate mean energy Ef of the Fnal states. So far, theoretically the (− ; e+ ) process has been investigated [113,114] on the exclusive reactions 40

Ca + − → e+ + 40Ar (gs) ;

(103)

58

Ni + − → e+ +58 Fe(gs) :

(104)

In these studies the partial g:s: → g:s: transition rate was calculated by performing microscopic calculations of these nuclear matrix elements. On the other hand, the total transition strength to all Fnal states (inclusive process) was estimated along the lines of closure approximation and ignoring four-body terms [113]. It has recently become possible [115] to develop o more complete formalism and perform shell model calculations to investigate the (− ; e+ ) conversion on the reaction 27

Al + − → e+ +27 Na :

(105)

To make the calculation tractable the Fnal states have been limited to those up to ≈ 25 MeV in excitation energy of the Fnal nucleus 27 Na. 6.1. The transition operators at the nuclear level Since the muon–positron transition operators are less well known compared to those of 0-decay we are going to brieEy discuss them here. The isospin and spin structure of the relevant transition operators is the same as those entering neutrinoless double beta decay. Their radial part is, of course, quite di5erent. It can be expressed in terms of the following quantities: (a) The momentum (pe ) of the emitted positron obtained from the kinematics of reaction (5). One Fnds that pe ≡ |pe | = m − b + Q − Ex ;

(106)

36

J.D. Vergados / Physics Reports 361 (2002) 1–56

where Q = M (Z) − M (Z − 2) is the atomic mass di5erence between the initial (A; Z) and Fnal, (A; Z − 2) nucleus, b is the binding energy of the muon at the muonic atom (b ≈ 0:5 MeV for our light target), Ex is the excitation energy (Ex = Ef − Egs ) of the Fnal nucleus and m is the muon mass (m = 105:6 MeV). (b) The relative (rij ) and center of mass (Rij ) coordinates of the two nucleon system rij ; rij = |rij | (107) rij = ri − rj ; rˆ = |rij | and Rij 1 Rij = (ri + rj ); Rˆ = ; Rij = |Rij | : (108) 2 |Rij | The momentum dependence comes through the spherical Bessel functions jl (pe rij =2) and jL (pe Rij ) resulting from the decomposition of the plane wave of the outgoing positron. The additional radial function f(r) of the relative coordinate given by R0 f(r) = F(r)Hcor (r) ; (109) r where the constant R0 represents the nuclear radius. The function Hcor (r) is some reasonable two-nucleon correlation function (7) of the type 2

Hcor (r) = 1 − e−ar (1 − br 2 ) 2

(110) 2

with a = 1:1 1= fm and b = 0:68 1= fm . This correlation function is routinely employed in neutrinoless double beta decay as well. Notice that in the above radial function the muon wave function is not included. This is, because the muon is in 1s state and it varies very slowly inside the nucleus. Thus it can be replaced by its average value. Anyway this average value drops out of the branching ratio. As we have already indicated the radial function F(r) depends on the speciFc mechanism assumed for the − → e+ conversion process to occur. The following cases are of interest: (i) Light Majorana neutrinos, with left-handed leptonic currents. Then F(r) takes the form [115] 2 ∞ sin x 2 ∞ sin x F(r) = dx + dx : (111) 0 x − . + i 0 x + e The quantities e and ., which are functions of r, are given in terms of the nuclear masses and the average excitation energy of the intermediate states as e = [Exn + M (Z − 1) − M (Z) + pe ]r ; . = [m + M (Z) − M (Z − 1) − Exn ]r : The above expression must be contrasted with the expressions entering neutrinoless double beta decay in which e is zero and . is negative, i.e.

M (Z − 2) − M (Z) − M (Z − 1) − Exn r : .= 2 In other words, the Frst integral is regular and in the case of 0-decay, since . ∼ 0, F(r) ∼ 1. This signiFcantly simpliFes the calculations in the case of 0-decay.

J.D. Vergados / Physics Reports 361 (2002) 1–56

37

Note that e depends on the positron momentum. Since it is positive the corresponding integral is Fnite. On the other hand, the Frst term of F(r) in Eq. (111) can be written as ∞ 2 2 ∞ sin x sin x dx = P d x − i2 sin . : (112) 0 x − . + i 0 x−. Note that the amplitude has an imaginary part [115,116], a fact that was missed in the earlier calculations. As we shall see below, however, the imaginary part is dominant. The principal value integral can be written in an equivalent form as follows: ∞ ∞ 2 sin x sin x 2 d x = 2 cos . − 1 + . dx : (113) P 0 x−. 0 x(x + .) The latter expression is more convenient for numerical integration techniques. (ii) Light Majorana neutrinos, with leptonic currents of opposite chirality or R-parity violating supersymmetric interactions (mediated by light Majorana neutrinos in addition to other SUSY ˜ particles). Then one Fnds F(r) → F (r) = r(d = d r)F(r) or F(r) → F(r). (iii) Heavy intermediate particles, e.g. heavy Majorana neutrinos, gauginos neutralinos, etc. Now the energy denominators are determined by the masses of these particles. In this case the operator becomes extremely short ranged. Then, as we have already mentioned, either one assumes that the nucleon has an internal structure or considers the 1 and 2 modes discussed above. In the case of the pionic mode only the radial function associated with the 2 mode is modiFed due to the fact that the momentum carried away by the positron is not in this i ; i = GT; T (see case negligible. In other words, the one pion mode F(r) is replaced by F1 Eq. (64)). In the case of the 2 mode the relevant radial function cannot be obtained analytically but via the substitution: 1 F(r)jl (xe =2) → jl ((! − 12 )xe )Fi2 ([!(1 − !)xe2 + x2 ]1=2 ) d ! ; (114) 0

where xe = pe r; x = m r and Fi2 ; i = GT; T are given by Eq. (65). 6.2. Irreducible tensor operators In this section we are going to exhibit the structure of the various irreducible tensor operators relevant to our calculation characterized by the set of quantum numbers l; ; L; ; L; S; J; G. In the above notation l characterizes the multipole of the plane wave in the relative coordinates, L is the corresponding multipole in the CM system, is the total orbital tensor rank in the relative system, is the total orbital rank in the CM system, L is the total orbital rank obtained by combining the relative and CM ranks and S; J are the spin and the spherical tensor ranks of the operator, respectively. G is the rank of the spherical harmonic characterizing the momentum function of the outgoing positron. J and G combine to give a total rank k. The latter combines with the leptonic currents to give an overall scalar. Clearly, some of these labels may be redundant in most cases. In writing down the operators it is understood, of course, that the usual selection rules apply, i.e. |L − S | 6 J 6 J + S. The quantum number G will not be exhibited.

38

J.D. Vergados / Physics Reports 361 (2002) 1–56

We will begin with operators appearing when the chiralities of the two leptonic currents involved are the same. One encounters Fermi-type operators of the form ?F =

A− (i)A− (j)f(rij )jl

p r e ij

i¡j

2

√ √ jL (pe Rij )[ 4Y (rˆij ) ⊗ 4Y (Rˆ ij )]J

(115)

( = l; = L; S = 0). The Gamow–Teller operators, ?GT , are similarly written as ?GT =

A− (i)A− (j)f(rij )jl

i¡j

p r e ij

2

jL (pe Rij )

√ √ √ [[ 4Y (rˆij ) ⊗ 4Y (Rˆ ij )]L ⊗ (− 3)[
(116)

√ ( = l; = L; S = 0). Note that
?A1 = √

A− (i)A− (j)f(rij )jl

i¡j

[[ 4Y (rˆij ) ⊗

√

p r e ij

2

jL (pe Rij )

√ 4Y (Rˆ ij )]L ⊗ (− 2)[
(117)

( = l; = L; S = 1). Note that √ [
(118)

The second spin antisymmetric operator is ?A2 =

i¡j

A− (i)A− (j)f(rij )jl

p r e ij

2

jL (pe Rij )

√ √ [[ 4Y (rˆij ) ⊗ 4Y (Rˆ ij )]L ⊗ (
(119)

( = l; = L; S = 1). Note that each operator must be overall symmetric with respect to interchange of the particle indices. So in the special case of 0+ → 0+ neutrinoless double beta decay the lowest non-vanishing multipole in the case of the spin antisymmetric operators is l = 1. Thus one gets no contribution in the long-wavelength approximation. In other words in the long-wavelength approximation, as it is well known, only the Fermi and Gamow–Teller operators contribute. We are now going to consider the case in which the theory contains both R (right) and L (left) currents and in particular the L–R interference in the leptonic sector. This may be important only in the case of light neutrinos. As we have already mentioned, similar operators are obtained in the context of R-parity violating supersymmetric interactions, which involve intermediate light Majorana neutrinos. The amplitude is now proportional to the four-momentum of the intermediate neutrino. The time component has a structure similar to the above, but it

J.D. Vergados / Physics Reports 361 (2002) 1–56

39

will not be further discussed, since it is suppressed. Its space component, after the Fourier transform, gives an amplitude proportional to the gradient of the Fourier transform of the previous , and ? case. We thus get the above operators, to be denoted by ?F ?GT A1 (associated with the term linear in the spin), with f(r) replaced by f (r) (note the absence of the quadratic spin antisymmetric operator). In this case, in addition to operators of the above form, we encounter an operator of spin rank two (tensor operator), which is of the form p r e ij ?T = A− (i)A− (j)f (rij )jl jL (pe Rij ) 2 i¡j √ √ (120) [[ 4Y (rˆij ) ⊗ 4Y L (Rˆ ij )]L ⊗ [
?GT =

A− (i)A− (j)f(rij )
(Gamow–Teller ) ;

(123)

i =j

?A2 =

ˆ ; A− (i)A− (j)f (rij ) (
(124)

i =j

?T =

A− (i)A− (j)f (rij )[3(
(Tensor ) :

(125)

i =j

6.3. The branching ratio for (− ; e+ ) The branching ratio Re+ of the (− ; e+ ) reaction deFned in Eq. (95) contains the lepton and Eavor violating (LFV) parameters associated with the speciFc gauge model assumed. These parameters, are entered in Re+ via a single lepton-violating parameter ne5 . Under some reasonable assumptions these parameters can be separated from the nuclear physics aspects of the problem. As has been pointed out [114], the branching ratio Re+ takes the form 1 pe (gs) 2 pe (f) 2 2 Re+ = C|e5 | 2=3 |Mi→f |2 : (126) m pe (gs) A ZfPR (A; Z) f

40

J.D. Vergados / Physics Reports 361 (2002) 1–56

The parameter C is tiny (C = 1:5 × 10−21 ) due to the fact that − → e+ conversion is a second order weak process. In this deFnition, the total muon capture rate has been written in terms of the well-known Primako5 function fPR (A; Z) [123], which is approximately given by fPR = 1:62(Z=A) − 0:62 and takes into account the e5ect of the nucleon-nucleon correlations on the total muon capture rate. |Mi→f |2 denotes the square of the partial transition nuclear matrix element between an initial |Ji and a Fnal |Jf state. This can be written as 1 |Mi→f |2 = |J M |?|Ji Mi |2 : (127) 2Ji + 1 M M f f f

i

In our case |Ji = |g:s:, i.e. the ground state of the initial nucleus. The summation in Eq. (126) runs over all states of the Fnal nucleus lying up to ≈ 25 MeV. For the Fermi and Gamow–Teller contribution, the square of the matrix element |Mi→f |2 takes the form 2 1 fV 2 2 |Mi→f | = Jf ||?F ||Ji (gs) − Jf ||?GT ||Ji (gs) ; (128) fA 2Ji + 1 L

where fV and fA are the usual vector and axial vector coupling constants (fA =fV = 1:25). By combining Eqs. (128) and (126) we see that, for the evaluation of the branching ratio Re+ , we have to calculate the reduced matrix elements Jf ||?F ||Ji and Jf ||?GT ||Ji for |Ji = |g:s: and |Jf any accessible state of the Fnal nucleus. In the present work these states have been constructed in the framework of the shell model as is described in the original publication [115]. The reduced matrix elements Jf ||?F ||Ji and Jf ||?GT ||Ji , which sensitively depend on the nuclear wave functions, were obtained in the 1s–0d shell model basis. As an e5ective interaction the universal s–d shell interaction of Wildenthal [124] which has been tested over many years, has been employed. This interaction is known to accurately reproduce many nuclear observables for s–d shell nuclei. The Wildenthal two-body matrix elements as well as the single particle energies are determined by least square Fts to experimental data [125] in the region of the periodic table with A = 17–39. The needed eigenstates were obtained in the isospin basis [126]. The initial state of 27 Al is (5=2)+ with T = 1=2. The states of the daughter nucleus 27 Na are characterized by Tf = 5=2 and Jf = 1=2–13=2. The Frst 250 states for each spin Jf with T = 5=2 were calculated reaching up to Ex = 25 MeV, in excitation energy. The low lying spectrum of 27 Al was well reproduced in this calculation. On the other hand, in the case of the unstable 27 Na isotope a comparison between theory and experiment cannot be accomplished due to lack of experimental data. For the special case of reaction (105) studied in the present work, since M (Al) − M (Na) = − 10:6 MeV, the momentum transfer at which our matrix elements must be computed, is given by pe = 94:5 − Ex

(MeV) :

(129)

6.4. Results and discussion for (; e+ ) conversion As we have already mentioned, the primary purpose of the calculation described in the previous section was to obtain the transition rates for − → e+ to all individual Fnal states as

J.D. Vergados / Physics Reports 361 (2002) 1–56

41

Fig. 12. The matrix elements |MF |2 and |MGT |2 as a function of the excitation associated with the dominant multipole L = 0 (for the deFnitions see text). Only the real part is included in the plot. The relative strengths are not modiFed if the imaginary part of the amplitude is included.

well as the total rate by summing up all partial transition strengths. In practice, only positive parity states up to 25 MeV in excitation energy were included. The calculation was restricted only to the case of light Majorana neutrinos. Surprisingly the imaginary part of the Frst term of Eq. (111), given by Eq. (112), missed in the earlier calculations, is dominant (115), (116). In Fig. 12, the distribution of the square of the reduced matrix elements, i.e. the strengths |MF |2 and |MGT |2 , for the multipolarity L = 0. As it can be seen, the Gamow–Teller contribution (solid-line) is more pronounced than the Fermi one (dot-line). The total Gamow–Teller contribution, represented by the area included between the energy axis and the histogram of Fig. 12, is almost 2.5 times greater than that of the Fermi type. SpeciFcally for the ground state transition, the calculated reduced matrix elements for Fermi and Gamow–Teller components are −0:12 and 1:09, respectively. As can be seen from Fig. 12 the main part of the Gamow–Teller contribution comes from the g:s: → g:s: transition. On the contrary for the Fermi component the main contribution comes from the Frst excited (5=2)+ state which appears at Ex = 2:67 MeV. Furthermore, 51% of the total Gamow–Teller strength is distributed among all the excited states up to 25 MeV while the other 49% goes to the ground state. On the other hand, 46% of the total Fermi strength is distributed to the Frst excited (5=2)+ and only 3.7% of the total strength to the ground state. The contribution of the remaining multipolarities L = 2 and 4 is, in general, quite small compared to that of L = 0. This becomes obvious by glancing at Table 7 where the total Fermi and Gamow–Teller strengths with respect to multipolarities L are listed.

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J.D. Vergados / Physics Reports 361 (2002) 1–56

Table 7 The individual Fermi and Gamow–Teller transition strengths in terms of multipolarities contributing to the total rate of the process, 27 Al(− ; e+ )27 Na, i.e. the − → e+ conversion in 27 Ala Multipole

Fermi contr.

Gamow–Teller contr.

L=0 L=2 L=4

8.291 0.076 1.553 ×10−4

77.272 0.426 5.515 ×10−4

a

The sum of the real and imaginary parts is shown.

Table 8 The total strength and the fraction of it for − → e+ conversion in of the Fnal systema Jf

Real part

gs → gs 1=2 3=2 5=2 7=2 9=2 11=2 13=2 Total Total

1.38 3:63 × 10−3 6:09 × 10−2 2:24 3:39 × 10−2 3:37 × 10−2 — — 3.75 closure approximation

a

27

Al associated with each angular momentum

Total

— —

31.59 0.044 0.55 43.52 0.29 0.24

76.23 430

(%) 41.44 0.05 0.72 57.08 0.37 0.31 — — 100 100

The process is dominated by the imaginary part of the amplitude.

In order to compare the branching ratio originating from the g:s: → g:s: transition with that associated with the transition to all Fnal states, we deFne, for convenience, the ratio Rgs (94:5)2 |Mgs→gs |2 ≡ : (130) = 2 2 R f (94:5 − Ex ) |Mgs→f | For the g:s: → g:s: transition pe = 94:5 MeV according to Eq. (129). Since me c2 pe we can consider the approximation pe ≈ Ee , which is equivalent to neglecting the electron mass (me ) in the kinematics of reaction (105). According to our calculation the ratio takes the value 0.41. This means that the ground state transition exhausts a large portion (41%) of the contribution of all Fnal states. As can be seen from Table 8, the main contribution to the total rate comes from the (5=2)+ states which contribute about 57% of the total rate. The rest of the portion originates mainly from the (3=2)+ ; (7=2)+ and (9=2)+ states. The spreading of the above contributions over the excitation spectrum of the Fnal nucleus is studied in detail by Divari et al. [115]. The common feature of these distributions is the fact that for each multipolarity the main contribution comes from low-lying states and that excitations lying above 12–15 MeV contribute negligibly.

J.D. Vergados / Physics Reports 361 (2002) 1–56

43

The total transition rate can also be calculated using some version of the closure approximation. In other words, the contribution of each individual state is e5ectively taken into account S by assuming a mean excitation energy E x = Ef − Egs , and using the completeness relation f |f f | = 1. Therefore, |f|?|i|2 = i|?+ ?|i : f

The matrix element i|?+ ?|i can be written as a sum of two pieces: a two-body term and a four-body one, that is i|?+ ?|i = i|(?+ ?)2b |i + i|(?+ ?)4b |i :

Thus

f2 A− (i)A+ (i)A− (j)A+ (j) V2 −
(131)

2

R0 rij

2

:

Assuming further that (i) the average excitation energy is 20 MeV, (ii) the main contribution comes from the spin singlet states and (iii) the two-body density is uniform (113), one Fnds i|?+ ?|i ≈ 430. In other words the results obtained in the above state-by-state calculation are much smaller than those resulting by employing the closure approximation. The disagreement between the closure approximation and the state-by-state shell model calculation can be attributed to the following reasons: (i) The closure approximation takes into account not only the contribution of 0˝! space but also includes excitations E ¿ 1˝!, as well as the continuum states. A possible extension of the s–d shell model space is quite diQcult. (ii) In closure approximation the second term in Eq. (131), which includes the very complicated four-body forces, was not taken into account in the previous calculations. Of course, the obvious question is: how important the contribution of four-body forces? (iii) On the other hand, from (− ; e− ) conversion, it is known that the results of the simple closure approximation are quite sensitive to the assumed mean excitation energy ES x . It is quite likely that the same is true for the (− ; e+ ) conversion as well. 6.5. Summary and conclusions In this section we have investigated the exotic double-charge exchange neutrinoless muon-topositron conversion, − → e+ , in the presence of nuclei. The appropriate operators in the form of spherical tensors have been constructed for a wide range of mechanisms. Since there are no restrictions imposed (like e.g. in the 0-decay) for the nuclear target to be used, the nucleus 27 Al has been chosen as an example, since it is going to be used as a stopping target in the Brookhaven experiment. This nucleus is a good prototype, since the s–d shell is perhaps the best understood and the s–d interaction well tested. From the study of reaction 27 Al(− ; e+ )27 Na within the s–d shell, the following conclusions can be drawn for the light intermediate neutrino mechanism: (i) The contribution coming from the Gamow–Teller component of the − → e+ operator is dominant.

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J.D. Vergados / Physics Reports 361 (2002) 1–56

(ii) The portion of the total strength, which goes to g:s: → g:s: transition is 41%. This is good news since eventually the experiments will focus on the ground state. (iii) The total strength, resulting by summing over all partial transition matrix elements in the s–d shell, is much smaller than that found previously by using closure approximation. If this is true, the light neutrino contribution to muon–positron conversion is unobservable, in spite of the 25-fold enhancement due to the imaginary part. 7. Extraction of the lepton violating parameters The limits on the lepton-number violating parameters extracted depend on the values of nuclear matrix elements, the kinematical factors and the current experimental limit on the half-life for a given isotope [see Eq. (10)]. 7.1. Traditional lepton violating parameters In some cases the available energy allows transitions not only to the ground state, but transitions to some low lying states as well. Transitions to the ground state are favored, since they are kinematically enhanced by large factors. Furthermore, the Q-value is smaller, which as we have seen in Section 3 is a disadvantage. Furthermore, the nuclear matrix elements are not expected to be larger than those involving the ground state, since the collective states are above threshold [117]. Furthermore to leading order only the 0+ states can be populated in the presence of standard V–A theories. Other excited states can be populated only either in the presence of right-handed currents (see Section 2.3) or in the R-parity violating SUSY with mixing between the isosinglet and isodoublet s-fermions or via the mass mechanism due to the nucleon recoil term [95]. In any case it has, however, been recognized a long time ago [118] that one should pay attention to such transitions to excited states, since the photon emitted after de-excitation can provide an additional signal against background. It is, therefore, seriously being considered in current experiments [81]. It may very well happen that the relevant nuclear matrix elements are enhanced in some special cases, so that the unfavorable kinematics is overcome. We will begin our discussion with transitions to the ground state. 7.1.1. Transitions to the ground state Even though, we expect the nuclear matrix elements entering the light neutrino mass mechanism to be decreased by about 30%, independent of the nuclear model, with the exception of our calculation marked P in the tables, we will stick to the calculations as reported. The results thus obtained are given in Table 9. As we have seen, given the nuclear matrix elements, the lifetime is still a function of the various lepton violating parameters (see Section 2). One may draw exclusion plots assuming that the lepton violating parameters are real and taking all the lepton violating parameters except two as known (set, e.g., equal to zero) [7]. In the present review we will adopt the simplest point of view assuming that only one mechanism at a time is dominant. In comparing the various theoretical results it has become customary [9,10] to introduce a standard value for a given lepton violating parameter and for each set of nuclear matrix elements

J.D. Vergados / Physics Reports 361 (2002) 1–56

45

Table 9 exp−0 The present state of the Majorana neutrino mass searches in -decay experiments. T1=2 (present) is the best a presently available lower limit on the half-life of the 0-decay for a given isotope Nucleus

76

Ge

exp−0 T1=2 (present) (yr ) 1:9 × 1025 Ref. (78) m (eV) 0.53 exp−0 (yr ) T1=2 (m best ) 1:9 × 1025 N 0:75 × 10−7 exp−0 best T1=2 (N ) (yr ) 1:9 × 1025

Nucleus

128

Te

exp−0 T1=2 (present) (yr ) 7:7 × 1024 Ref. (132) m (eV) 1.8 exp−0 T1=2 (m best ) (yr ) 1:1 × 1026 N 2:9 × 10−7 exp−0 best T1=2 [y](N ) (yr ) 1:0 × 1026

82

96

Se

2:7 × 1022 (128) 6.3

3:9 × 1019 (129) 203.0

4:8 × 1024 1:1 × 10−6 5:0 × 1024 130

Te

8:2 × 1021 (133) 13.0 6:6 × 1024 2:0 × 10−6 5:4 × 1024

100

Zr

5:2 × 1022 (130) 2.9

5:6 × 1024 4:0 × 10−5 2:0 × 1025 136

Xe

4:2 × 1023 (134) 4.9 4:7 × 1025 4:5 × 10−7 1:4 × 1025

Mo

2:1 × 1024 6:2 × 10−7 3:1 × 1024 150

116

Cd

2:9 × 1022 (131) 5.9 4:5 × 1024 1:1 × 10−6 5:5 × 1024

Nd

1:2 × 1021 (135) 8.5 4:0 × 1023 1:6 × 10−6 4:7 × 1023

a

The corresponding upper limits on lepton number non-conserving parameters m and N are presented. For the deFnition of the references and “best” see text.

predicting a lifetime of a given nucleus and comparing it with the present experimental limit. As one can see from Tables 10 and 11 there is a spread in the lepton violating parameters extracted from the data depending on the nuclear matrix elements. Thus, the average neutrino mass extracted from the Ge data ranges from approximately 0.5 –1:0 eV. Thus, the most restrictive limits extracted (see Refs. [29,78]) are as follows: m best ¡ 0:5 eV ;

(132)

N best ¡ 0:8 × 10−7 :

(133)

For the earlier data on 76 Ge see Baudis et al. [127] and for other experiments see Refs. [77– 100,128–137]. By assuming m = m best and N = best [10] we calculated the half-lives of N exp−0 exp−0 best best the 0-decay T1=2 (m ), T1=2 (N ) for nuclear systems of interest. The results thus obtained are given in Table 9. Since the quantities m , N depend only on particle theory parameters, these quantities indicate the experimental half-life limit for a given isotope, which the relevant experiments should reach in order to extract the best present bound on the corresponding lepton number violating parameter from their data. Some of them have a long way to go to reach the Ge target limit. A summary involving most of the available nuclear matrix elements and taking into account what, at present, is a good guess as canonical values of the lepton violating parameters is

46

J.D. Vergados / Physics Reports 361 (2002) 1–56

Table 10 The lifetimes predicted for 0+ → 0+ 0-decay in various mechanisms (light neutrino, heavy neutrino, and terms and SUSY contribution) for the suitable input of lepton violating parameters and available nuclear calculationsa ()0 -decay: 0+ → 0+ transition 0−theor T1=2 (m ; ; ; N ; SUSY ) Ref.

48

Ca

1024

76

Ge

1024

82

Se

1024

96

Zr

1024

m = 1 eV; = 0; = 0; N = 0; SUSY = 0 [42] 6.40 17.4 2.40 [36] 2.17 1.68 0.58 [51] 2.30 0.92 [48] 14.0 5.60 [34] 4.06 1.43 [50] 2.33 0.60 [8] 2.16 0.61 P1 2.50 3.60 1.50 0.61 P2 28.0 18.0 2.80 27.0 [57] 8.90 P 2.11 0.54 0.80

100

Mo

1024

1.27 0.26 3.90 0.25 0.23

116

Cd

1024

4.70 4.90 0.72 0.48

128

Te

1025

0.40 0.45 1.50 1.80 0.77 0.98 1.90 15.0 1.09 1.26

130

Te

1024

0.16 0.24 0.61 0.83 0.49 0.54 0.86 2.10 0.73

m = 0; = 10−6 ; = 0; N = 0; SUSY = 0

[42] [34] [50] [8] P1 P2

7.45

2.71 27.9

50.2 7.75 7.35 8.02 8.90 41.2

3.25 1.14 0.99 1.07 2.08 4.39

0.94 27.7

0.95 0.55 30.6 10.3

39.1 10.8

14.8 13.5 21.1 22.7 165

0.89 0.95 1.18 1.34 2.22

136

Xe

1024

150

Nd

1022

12.1 3.30 2.21 1.40 3.30 2.80 8.80 5.05

3.37 4.45

4.39

22.2 4.90 3.47 2.73 4.42

3.73 6.71

a

The unspeciFed references are deFned as follows: P1 = Pantis et al. [14], P2 = Pantis et al. [14] (p–n pairing), F = Faessler et al. [21,22], P = present calculation (see [29] for the nuclear Matrix elements).

provided in Tables 10 and 11. Notice that the present calculation, marked P in the table, involves not only renormalized QRPA [30,61], but takes into account the corrections in the hadronic current (29) discussed above (see Table 10), which decreases the nuclear matrix elements. 7.1.2. Transitions to excited states As it has been mentioned above transitions to excited states have some advantages and disadvantages. In general, the calculations of the relevant nuclear matrix elements for 0+ Fnal states in the context of the shell model can be carried out simultaneously with those for the ground state transitions. For the simplest nuclear system, 48 Ca, one Fnds that the nuclear matrix elements to the Frst excited 0+ 1 are even more suppressed than those to the ground state [26,5]. Most of the strength goes to the second excited 0+ 2 state, which is not kinematically allowed. The same kind of suppression appears to operate for the low lying 2+ states. As we have already mentioned, exact shell model calculations are not possible for most of the double beta decaying systems. The alternative approach, QRPA, is actually tailored

J.D. Vergados / Physics Reports 361 (2002) 1–56

47

Table 11 The previous table continued m = 0; = 0; = 10−8 ; N = 0; SUSY = 0

[42] [34] [50] [8] P1 P2

6.42

15.11 43.2

27.2 36.7 7.35 2.25 3.10 22.8

6.24 11.1 0.99 0.65 6.51 5.16

1.48 7.95

m = 0; = 0; = 0; N = 10−7 ; SUSY = 0

P1 4.95 0.25 3.35 67.1 P2 124 0.59 7.23 671 P 15.4 4.10 8.10 m = 0; = 0; = 0; N = 0; SUSY = 10−8 F 3.3 0.86 0.71 P 4.5 1.0 1.4

22.2

19.2 83.2

10.7 13.5 0.67 1.20 1.90

5.92 0.95 0.44 0.62 1.05

4.70 1.47 0.97

23.5 33.6 8.40

0.78 1.27 8.51

3.03 1.31 4.54

1.42 1.01 3.94

40.6

0.30 0.59

0.85 1.4

0.93 1.5

0.45 0.71

1.2 2.0

3.2 5.2

0.95 0.28 3.44 102

4.90 1.21 1.23 0.96

3.73 3.39

for transitions to the ground states. It has, however, recently been modiFed to give a good description of some excited states of the Fnal system, which can be described in terms of two-phonon excitations. In fact, the Frst excited 0+ state can be written as 1 + + 0 |0+ (134) 1 = √ [21 ⊗ 21 ] ; 2 where 2+ 1 is the low lying RPA phonon of the Fnal nucleus. Thus Suhonen [119] studied the 0-decay to the Frst excited 0+ state for 76 Ge and 82 Se in the mass mechanism for light intermediate neutrinos. In a recent calculation, using two methods, a boson expansion method ^ (BEM) as well as an operator recoupling method (RCE), Simkovic et al. [120] extended the + calculation to include transitions to the excited 0 states of the A = 100 and 136 systems as well. This last calculation does not include the and terms. It takes into account, however, not only the light neutrino but the heavy majorana neutrino as well as parity violating SUSY contributions. Both calculations conFrm that such transitions are suppressed, both kinematically and from the point of view of the nuclear matrix elements leading to lifetimes, which are about 100 times longer than those associated with the ground state transition [119,120]. It was found, however, that the nuclear matrix elements of 0-decay of 96 Zr to the Frst excited 0+ state of 96 Mo, described as monopole-vibrational state, are enhanced, i.e. comparable to those of the transition ^ to the ground state [121]. Similarly, Simkovic et al. [120] Fnd that with the RCM method there appears an anomaly in the case of the A = 136 system, in the sense that the nuclear matrix element for the transition to the excited state is about four times larger than that to the ground state. This is not conFrmed by their second method, i.e. the boson expansion method (BEM), which predicts a matrix element 10 times smaller than the previous method. Thus the RCM predicts a lifetime, which is an order of magnitude shorter than that to the gs transition. Such a disagreement between the two methods is rather surprising, since the absolute value of the matrix elements takes values, which are typical, not the result of unusual cancellations. Therefore, the

48

J.D. Vergados / Physics Reports 361 (2002) 1–56

accuracy of the method should be veriFed further. They also Fnd that the two methods di5er substantially in the case of the short ranged operators, which is not very surprising. Once the nuclear matrix elements are calculated one can use the lepton violating parameters m ; and etc to predict the lifetimes of the transitions to the excited states. Thus, using m = 0:72 eV; = 9:4 × 10−7 and = 7:6 × 10−9 , they Fnd a lifetime of T1=2 = 4:5 × 1024 yr for the A = 96 system, i.e. only one order of magnitude longer than that to the ground state [121,122]. The appearance of such unusual enhancements is obviously good news. One would like, however, to see this conFrmed by a simple soluble model or a shell model calculation, whereby the e5ects of Pauli principle are better treated. Unfortunately one does not Fnd similar enhancement in other systems, e.g. for the decay of 116 Cd [121,122], which is more important from an experimental point of view. 7.2. R-parity induced lepton violating parameters With the above ingredients and using the available nuclear matrix elements one can extract from the data values of SUSY . Then one can use these values of SUSY in order to extract . values for the R-parity violating parameters 111 As we have already mentioned one must start with Fve parameters in the allowed SUSY parameter space and solve the RGE equations to obtain the values of the needed parameters at low energies [138–140]. For our purposes it is adequate to utilize typical parameters, which have already appeared in the literature [138,139]. One then Fnds 111 = C'˜0 (SUSY )1=2 111 = Cg˜(SUSY )1=2

(neutralinos only) ;

(135)

(gluino only) :

(136)

When both neutralinos and gluinos are included we write 111 = C'˜0 ; g˜(SUSY )1=2 :

(137)

The values of these coeQcients C are given in Table 12 for the nine SUSY models mentioned above. From Table 12 we see that there is quite spread in the quantities C'˜0 ; Cg˜ and C'˜0 ; g˜, depending on the SUSY parameter space. We see that this is the largest uncertainty in estimating the SUSY contribution to 0-decay. In all of these cases the intermediate s-electron–neutralino mechanism appears to be the most dominant.The most favorable situation occurs in the case . Combining the above # 7 of Table 12, which we will utilize in extracting the limits on 111 values of the couplings .k ; k = 1; 2, with the corresponding nuclear matrix elements of Faessler et al. [21] and the two nucleon ME of Wodecki et al. [140] we obtain the limits listed as Pr in Table 13. From the data of Table 13 we see that the most stringent limit is 111 6 0:68 × 10−3 (for case # 7 Table 12) :

The above quantities were assumed positive. If not, the absolute value is understood.

(138)

J.D. Vergados / Physics Reports 361 (2002) 1–56

49

Table 12 A sample of relevant parameters obtained by some choices in the allowed SUSY parameter space. It is clear that in all cases the neutralino mediated mechanism is dominant (for deFnitions see text). The parameters C shown have been multiplied by 10−3 Input

tan m' 0 1 m' 0 2 m' 0 3 m' 0 4 me˜L mu˜ L md˜R mg˜ C'˜0 × 10−3 Cg˜ × 10−3 C'˜0 ; g˜ × 10−3

Kane et al. [138] (# 1–3)

Ramond et al. [139] (# 4 –9)

#1

#2

#3

#4

#5

#6

#7

#8

#9

10. 124. 237. 455. 471. 328. 700. 676. 718. 3.3 14 3.2

1.5 26 65 219 263 124 283 276 292 0.023 1.6 0.023

5.0 96 173 310 342 211 570 550 610 0.46 54 0.45

5.4 83 150 391 409 426 590 577 483 5.9 56 5.3

2.7 124 204 445 472 472 664 638 706 14 110 12

2.7 58 108 336 361 310 449 441 371 1.4 13 1.3

5.2 34 66 170 208 90 251 246 280 0.0068 0.97 0.0068

2.6 34 74 191 236 94 275 268 304 0.0089 1.5 0.0089

6.3 50 92 208 244 109 319 310 350 0.019 3.1 0.019

Table 13 obtained The limits for SUSY and 111 Pion modea (A; Z) 76

Ge Mo 116 Cd 128 Te 130 Te 136 Xe 150 Nd 100

SUSY (Pr ) 1:6 × 10−8 3:2 × 10−8 7:6 × 10−8 1:6 × 10−8 1:0 × 10−7 2:4 × 10−8 7:3 × 10−8

111 (Pr ) 1:0 × 10−3 1:8 × 10−3 1:8 × 10−3 8:1 × 10−4 1:8 × 10−3 9:4 × 10−4 2:3 × 10−3

SUSY (F) 9:5 × 10−9 2:4 × 10−8 5:4 × 10−8 1:1 × 10−8 5:5 × 10−8 1:7 × 10−8 5:2 × 10−8

Only nucleonsb 111 ( F) 8:3 × 10−4 1:1 × 10−3 1:5 × 10−3 6:8 × 10−4 1:5 × 10−3 7:8 × 10−4 1:4 × 10−3

SUSY (Pr ) 1:2 × 10−7 1:1 × 10−7 2:6 × 10−7 5:6 × 10−8 1:3 × 10−7 8:7 × 10−7 2:4 × 10−7

111 (Pr ) 1:9 × 10−3 2:2 × 10−3 3:3 × 10−3 1:6 × 10−3 6:5 × 10−3 2:2 × 10−3 3:0 × 10−3

a

For the pion mechanism using the values of .1 and .2 computed in this work and the nuclear ME of Faessler et al. [21,22]. b Using the nuclear ME of the two nucleon mode of Wodecki et al. [140]. In extracting the values of 111 we used the SUSY data of #7 of Table 12. The experimental lifetimes employed are those of Table 10.

8. Conclusions We have seen that 0-decay pops up in almost any fashionable particle model. Thus, it can set useful limits not only on the light neutrino mass (132), but in addition on other lepton violating parameters like N (133) or the parameters and (see Section 2.2). Finally, we , Eq. (138). The not mention again the limit extracted on the R-parity violating parameter 111 so fashionable composite models, not discussed in the present paper, can also be tested in the scale of 1 TeV competitive with the LHC sensitivity.

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J.D. Vergados / Physics Reports 361 (2002) 1–56

We should stress that the nuclear matrix elements involve a long range (light intermediate neutrinos) and a short range (heavy intermediate particles) part. The nuclear matrix elements associated with the neutrino mass are the most accurate, since the spin-dependent and the spin-independent parts are additive. In fact, we found that the spread in the value m extracted is within a factor of three, i.e. between 0.35 and 0:96 eV. This is indeed good news, if we take into account the fact that very di5erent nuclear models have been involved, large basis shell model calculations in one extreme and various versions of QRPA in the other. The parameter is not expected to be equally accurate, since the vector, axial vector and tensor components do not always have the same sign. The extraction of the value of may be even less accurate since it is dominated by the recoil term, which vanishes for point-like nucleons, or by the spin antisymmetric term, which is somewhat more sensitive to details of nuclear structure. In any case the long range part is under control. The short term part, which is viewed as extremely important due to the recent developments in R-parity violating supersymmetry, cannot, unfortunately, be determined equally accurately. As we have emphasized in the introduction and seen in Section 5.1 we believe that we have −1 −1 solved the problem of going from the range of Mheavy to ranges of order m−1 nucleon or mpion by at least three ways: (1) considering the nucleons as a bag of quarks or describing the Fnite size of the nucleon by the experimentally determined form factor, (2) assuming that neutrinoless double beta decay is caused by the decay of pions in Eight between nucleons and (3) estimating the probability of two nucleons in the nucleus to be in a six quark cluster and, thus, assuming that the decay takes place in the same hadron (cluster). Clearly, one of the above mechanisms may be more important than the others in some instances. All these mechanisms, however, yield results of the same order with matrix elements not terribly suppressed. One, of course, cannot claim that the problems associated with short −1 ranges of the size m−1 nucleon or mpion , which have plagued nuclear physics for years, are completely under control. We believe, however, that for ground states the problems may not be so serious. We conclude, therefore, that the short ranged nuclear matrix elements entering 0-decay are reliable, albeit not so reliable as those entering the mass mechanism. Keeping the above points in mind, for a choice of nuclear matrix elements, a set of limits for the most interesting quantities and for various nuclear targets is given in Table 14. We see that the obtained limits are quite stringent. It is clear that during the last years the interest of most people is being focused again on the light neutrino mass mechanism. This is due to the experimental indications for neutrino oscillations of solar (Homestake [141], Kamiokande [142], Gallex [143] and SAGE [144]), atmospheric (Kamiokande [145], IMB [146], Soudan [147], Super-Kamiokande [148]) and terrestrial (LSND) [149] experiments. These experiments only yield constraints on the two Ym2 and the mixing angles. They cannot, of course, predict the scale of the masses or the two independent relative Majorana phases. For the determination of these quantities one needs additional experimental information resulting from (i) the decay experiments, e.g the triton decay, (ii) neutrinoless double beta decay, which yields m and (iii) muon to positron conversion, which yield m e+ . Bilenky et al. [150] and others [152], however, have shown that under quite reasonable assumptions in a general scheme with three light Majorana neutrinos, it is |m | is smaller than 10−2 eV. In another study outlined in Ref. [151] the authors end up with |m | ≈ 0:14 eV. In any case one can

J.D. Vergados / Physics Reports 361 (2002) 1–56

51

Table 14 Summary of the results presented in this work using the same experimental limits as in Table 9 (A; Z)

76

Ge Mo 116 Cd 128 Te 130 Te 136 Xe 150 Nd 100

m eV

×10−6

×10−8

N ×10−8

SUSY ×10−8

111 ×10−4

Pr

P1

P1

Pr

Pr

Pr

0.51 2.9 5.9 1.8 13 49 8.5

0.97 26 37 5.6 7.6 2.1 5.6

0.55 8.8 26 1.3 5.2 1.4 5.3

0.76 6.2 11 2.9 20 4.5 16

0.54 3.2 7.6 1.6 10 2.4 7.3

10 18 18 8.1 18 9.4 23

see that, the current limit on m in (132) is quite a bit higher than the neutrino oscillation data. As we have seen in Section 3, very innovative neutrinoless double beta decay projects are underway, e.g. GENIUS, CUORICINO NEMO2, planned for the near future, e.g. GENIUS II, CUORE, MAJORANA, CAMEO I and II, NEMO3, IGEX, ELEGANT GEM-I, or planned for the not so distant future, Like GEM-II, EXO (2000), Xe-CTF (2000), Mo-MOON, etc. (see the review talk by Zdesenko [72]). Thus, we have every reason to expect that the present limits will substantially improve. It is not clear as to which experiment will win the race, since essentially all groups are competent and have accumulated quite a lot of experience. Based on their claims we will single out two. The Frst is the new experimental proposal for measurement of the 0-decay of 76 Ge, which intents to use 1 ton (in an extended version 10 t) of enriched 76 Ge to reach the half-life limit 0−exp 0−exp T1=2 ¿ 5:8 × 1027 and T1=2 ¿ 6:4 × 1028 after one and 10 yr of measurements, respectively. From these half-life values one can deduce [see Eq. (10) and Table 6] the possible future limits on the e5ective light neutrino mass 2:7 × 10−2 and 8:1 × 10−3 eV, respectively [78,127,153]. The second approach [97] is even more ambitious. Using Laser Tagging Techniques it claims to reach the level of m of order of 10−3 eV. The above goals, if accomplished, will yield results which will be in a position to compete with those advocated by the neutrino oscillation experiments. This means that they will reach detection levels, which will permit the extraction of very small lepton violating parameters, provided, of course, that the neutrinos are Majorana particles. Current fashionable theories in extra dimensions [156] have implications on neutrinoless double beta decay not just due to the neutrino induced neutrino masses, but also through the heavy neutrino contribution [157]. We must emphasize that the plethora of other 0-decay mechanisms predicted by GUTs and SUSY models do not diminish the importance of this reaction in settling the outstanding neutrino properties. One can show that the presence of these exotic mechanisms implies that the neutrinos are massive Majorana particles, even if the mass mechanism is not the dominant one [12,154,155].

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J.D. Vergados / Physics Reports 361 (2002) 1–56

Thus one can say with certainty that the experimental detection of the 0-decay process would be a major achievement with important implications on the Feld of particle and nuclear physics as well as on cosmology, complementing, of course, the LHC searches. Acknowledgements I would like to express my sincere thanks to the Physics Department of the University of Cyprus for its hospitality. I would also like to thank Dr H. Tsertos and Mr Symintzis for ^ their help in preparing some of the Fgures and Dr Simkovic for useful discussions and a careful reading of the manuscript. Finally, I would like to thank Professor Faessler for his hospitality in Tuebingen during the Frst stages of this work. This work was partially supported by a Humboldt Research Award and the EU contract Nos ERBFMRXCT-96-090 and HPRN-CT-2000-00152. References [1] W. Furry, Phys. Rev. 56 (1939) 1184. [2] H. Primako5, Phys. Rev. 85 (1925) 888; H. Primako5, S.P Rosen, Phys. Rev. 184 (1969) 1925; Proc. Roy. Soc. (London) 78 (1961) 464. [3] M. Doi, T. Kotani, N. Nishiura, K. Okuda, E. Takasugi, Phys. Lett. B 103 (1981) 219; 113 (1982) 513(E). [4] J.D. Vergados, Nucl. Phys. B 218 (1983) 109. [5] W.C. Haxton, G.S. Stephenson, Prog. Part. Nucl. Phys. 12 (1984) 409. [6] M. Doi, T. Kotani, E. Takasugi, Prog. Theor. Phys. (Supp.) 83 (1985) 1. [7] J.D. Vergados, Phys. Rep. 133 (1986) 1. [8] T. Tomoda, Rep. Prog. Phys. 54 (1991) 53. [9] J. Suhonen, O. Civitarese, Phys. Rep. 300 (1998) 123. ^ [10] A. Faessler, F. Simkovic, J. Phys. G 24 (1998) 2139. [11] P. Vogel, Double Beta Decay: Theory and Experiment, nucl-th=0005020; nucl-th=9904065. [12] J. Schechter, J.W.F. Valle, Phys. Rev. D 25 (1982) 2951. [13] R.N. Mohapatra, hep-ph=9808284. ^ [14] G. Pantis, F. Simkovic, J.D. Vergados, A. Faessler, Phys. Rev. C 53 (1996) 695. [15] R. Mohapatra, Phys. Rev. D 34 (1986) 3457. [16] J.D. Vergados, Phys. Lett. B 184 (1987) 55. [17] M. Hirsch, H.V. Klapdor-Kleingrothaus, S.G Kovalenko, Phys. Rev. Lett. 75 (1995) 17; Phys. Rev. D 53 (1996) 13297; Phys. Lett. B 398 (1999) 311. [18] H. Paes, M. Hirsch, H.V. Klapdor-Kleingrothaus, Phys. Lett. B 459 (1999) 450. [19] A. Wodecki, W. KamiVnski, S. Pagerka, Phys. Lett. B 413 (1997) 342. ^ [20] A. Faessler, S. Kovalenko, F. Simkovic, J. Schwieger, Phys. Rev. Lett. 78 (1997) 183; Phys. Atom. Nucl. ^ 61 (1998) 1229. A. Faessler, F. Simkovic, Phys. Atom. Nucl. 63 (2000) 1165. ^ [21] A. Faessler, S. Kovalenko, F. Simkovic, Phys. Rev. D 58 (1998) 115 004. ^ [22] A. Faessler, S. Kovalenko, F. Simkovic, Phys. Rev. D 58 (1998) 055 004. [23] M. Hirsch, J.W.F. Valle, Nucl. Phys. B 557 (2001) 355. [24] J.D. Vergados, Phys. Rev. C 24 (1981) 640. [25] B. Pontecorvo, Phys. Lett. B 26 (1968) 630. [26] J.D. Vergados, Phys. Rev. D 25 (1982) 914. [27] J.D. Vergados, Nucl. Phys. B 250 (1985) 618. [28] T. Tomoda, A. Faessler, K.W. Schmidt, F. Gr`ummer, Phys. Lett. B 157 (1985) 4. ^ [29] F. Simkovic, G. Pantis, J.D. Vergados, A. Faessler, Phys. Rev. C 60 (1999) 055 502. ^ [30] F. Simkovic, G.V. EFmov, M.A. Ivanov, V.E. Lyubovitskij, Z. Phys. A 341 (1992) 193.

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Physics Reports 361 (2002) 57 – 265 www.elsevier.com/locate/physrep

Brownian motors: noisy transport far from equilibrium Peter Reimann Institut fur Physik, Universitat Augsburg, Universitatsstr. 1, 86135 Augsburg, Germany Received August 2001; Editor: I: Procaccia

Contents 1. Introduction 1.1. Outline and scope 1.2. Historical landmarks 1.3. Organization of the paper 2. Basic concepts and phenomena 2.1. Smoluchowski–Feynman ratchet 2.2. Fokker–Planck equation 2.3. Particle current 2.4. Solution and discussion 2.5. Tilted Smoluchowski–Feynman ratchet 2.6. Temperature ratchet and ratchet e5ect 2.7. Mechanochemical coupling 2.8. Curie’s principle 2.9. Brillouin’s paradox 2.10. Asymptotic analysis 2.11. Current inversions 3. General framework 3.1. Working model 3.2. Symmetry 3.3. Main ratchet types 3.4. Physical basis 3.5. Supersymmetry 3.6. Tailoring current inversions 3.7. Linear response and high temperature limit 3.8. Activated barrier crossing limit 4. Pulsating ratchets 4.1. Fast and slow pulsating limits 4.2. On–o5 ratchets 4.3. Fluctuating potential ratchets 4.4. Traveling potential ratchets 4.5. Hybrids and further generalizations

59 59 60 61 63 63 67 68 69 72 75 79 80 81 82 83 86 86 90 91 93 100 107 108 109 111 111 113 115 121 127

4.6. Biological applications: molecular pumps and motors 5. Tilting ratchets 5.1. Model 5.2. Adiabatic approximation 5.3. Fast tilting 5.4. Comparison with pulsating ratchets 5.5. Fluctuating force ratchets 5.6. Photovoltaic e5ects 5.7. Rocking ratchets 5.8. In=uence of inertia and Hamiltonian ratchets 5.9. Two-dimensional systems and entropic ratchets 5.10. Rocking ratchets in SQUIDs 5.11. Giant enhancement of di5usion 5.12. Asymmetrically tilting ratchets 6. Sundry extensions 6.1. Seebeck ratchets 6.2. Feynman ratchets 6.3. Temperature ratchets 6.4. Inhomogeneous, pulsating, and memory friction 6.5. Ratchet models with an internal degree of freedom 6.6. Drift ratchet 6.7. Spatially discrete models and Parrondo’s game 6.8. In=uence of disorder 6.9. ECciency 7. Molecular motors 7.1. Biological setup

c 2001 Elsevier Science B.V. All rights reserved. 0370-1573/01/$ - see front matter PII: S 0 3 7 0 - 1 5 7 3 ( 0 1 ) 0 0 0 8 1 - 3

130 133 133 133 135 136 137 143 144 148 150 152 154 156 159 160 162 164 165 168 169 173 175 176 178 179

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P. Reimann / Physics Reports 361 (2002) 57 – 265 7.2. 7.3. 7.4. 7.5. 7.6.

Basic modeling-steps SimpliFed stochastic model Collective one-head models Coordinated two-head model Further models for a single motor enzyme 7.7. Summary and discussion 8. Quantum ratchets 8.1. Model 8.2. Adiabatically tilting quantum ratchet 8.3. Beyond the adiabatic limit 8.4. Experimental quantum ratchet systems 9. Collective e5ects 9.1. Coupled ratchets

181 184 190 199 200 203 205 206 209 215 217 220 222

9.2. Genuine collective e5ects 10. Conclusions Acknowledgements Appendix A. Supplementary material regarding Section 2:1:1 A.1. Gaussian white noise A.2. Fluctuation–dissipation relation A.3. Einstein relation A.4. Dimensionless units and overdamped dynamics Appendix B. Alternative derivation of the Fokker–Planck equation Appendix C. Perturbation analysis References

223 232 234 234 234 235 236 237 239 239 241

Abstract Transport phenomena in spatially periodic systems far from thermal equilibrium are considered. The main emphasis is put on directed transport in so-called Brownian motors (ratchets), i.e. a dissipative dynamics in the presence of thermal noise and some prototypical perturbation that drives the system out of equilibrium without introducing a priori an obvious bias into one or the other direction of motion. Symmetry conditions for the appearance (or not) of directed current, its inversion upon variation of certain parameters, and quantitative theoretical predictions for speciFc models are reviewed as well as a wide variety of experimental realizations and biological applications, especially the modeling of molecular motors. Extensions include quantum mechanical and collective e5ects, Hamiltonian ratchets, the in=uence of spatial disorder, and di5usive transport. c 2001 Elsevier Science B.V. All rights reserved. PACS: 5.40.−a; 5.60.−k; 87.16.Nn Keywords: Ratchet; Transport; Nonequilibrium process; Brownian motion; Molecular motor; Periodic potential; Symmetry breaking

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59

1. Introduction 1.1. Outline and scope The subject of the present review are transport phenomena in spatially periodic systems out of thermal equilibrium. While the main emphasis is put on directed transport, also some aspects of di1usive transport will be addressed. We furthermore focus mostly on small-scale systems for which thermal noise plays a non-negligible or even dominating role. Physically, the thermal noise has its origin in the thermal environment of the actual system of interest. As an unavoidable consequence, the system dynamics is then always subjected to dissipative e5ects as well. Apart from transients, directed transport in a spatially periodic system in contact with a single dissipation- and noise-generating thermal heat bath is ruled out by the second law of thermodynamics. The system has therefore to be driven away from thermal equilibrium by an additional deterministic or stochastic perturbation. Out of the inFnitely many options, we will mainly focus on either a periodic driving or a restricted selection of stochastic processes of prototypal simplicity. In the most interesting case, these perturbations are furthermore unbiased, i.e. the time-, space-, and ensemble-averaged forces which they entail are required to vanish. Physically, they may be either externally imposed (e.g. by the experimentalist) or of system-intrinsic origin, e.g. due to a second thermal heat reservoir at a di5erent temperature or a non-thermal bath. Besides the breaking of thermal equilibrium, a further indispensable requirement for directed transport in spatially periodic systems is clearly the breaking of the spatial inversion symmetry. There are essentially three di5erent ways to do this, and we will speak of a Brownian motor, or equivalently, a ratchet system whenever a single one or a combination of them is realized: First, the spatial inversion symmetry of the periodic system itself may be broken intrinsically, that is, already in the absence of the above mentioned non-equilibrium perturbations. This is the most common situation and typically involves some kind of periodic and asymmetric, so-called ratchet potential. A second option is that the non-equilibrium perturbations, notwithstanding the requirement that they must be unbiased, bring about a spatial asymmetry of the dynamics. A third possibility arises as a collective e5ect in coupled, perfectly symmetric non-equilibrium systems, namely in the form of spontaneous symmetry breaking. As it turns out, these two conditions (breaking of thermal equilibrium and of spatial inversion symmetry) are generically suCcient for the occurrence of the so-called ratchet e1ect, i.e. the emergence of directed transport in a spatially periodic system. Elucidating this basic phenomenon in all its facets is the central theme of our present review. We will mainly focus on two basic classes of ratchet systems, which may be roughly characterized as follows (for a more detailed discussion see Section 3.3): The Frst class, called pulsating ratchets, are those for which the above-mentioned periodic or stochastic non-equilibrium perturbation gives rise to a time-dependent variation of the potential shape without a5ecting its spatial periodicity. The second class, called tilting ratchets, are those for which these non-equilibrium perturbations act as an additive driving force, which is unbiased on the average. In full generality, also combinations of pulsating and tilting ratchet schemes are possible, but they exhibit hardly any fundamentally new basic features (see Section 3.4.2). Even within those two classes, the possibilities of breaking thermal equilibrium and symmetry in a ratchet system are still numerous and in many cases, predicting the actual direction of the transport is already far from obvious, not to speak of its quantitative value.

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In particular, while the occurrence of a ratchet e5ect is the rule, exceptions with zero current are still possible. For instance, such a non-generic situation may be created by Fne-tuning of some parameter. Usually, the direction of transport then exhibits a change of sign upon variation of this parameter, called current inversions. Another type of exception can be traced back to symmetry reasons with the characteristic signature of zero current without Fne-tuning of parameters. The understanding and control of such exceptional cases is clearly another issue of considerable theoretical and practical interest that we will discuss in detail (especially in Sections 3.5 and 3.6). 1.2. Historical landmarks Progress in the Feld of Brownian motors has evolved through contributions from rather di5erent directions and re-discoveries of the same basic principles in di5erent contexts have been made repeatedly. Moreover, the organization of the much more detailed subsequent chapters will not always admit it to keep the proper historical order. For these reasons, a brief historical tour d’horizon seems worthwhile at this place. At the same time, this gives a Frst =avor of the wide variety of applications of Brownian motor concepts. Though certain aspects of the ratchet e5ect are contained implicitly already in the works of Archimedes, Seebeck, Maxwell, Curie, and others, Smoluchowski’s Gedankenexperiment from 1912 [1] regarding the prima facie quite astonishing absence of directed transport in spatially asymmetric systems in contact with a single heat bath, may be considered as the Frst seizable major contribution (discussed in detail in Section 2.1). The next important step forward represents Feynman’s famous recapitulation and extension [2] to the case of two thermal heat baths at di5erent temperatures (see Section 6.2). Brillouins paradox [3] from 1950 (see Section 2.9) may be viewed as a variation of Smoluchowski’s counterintuitive observation. In turn, Feynman’s prediction that in the presence of a second heat bath a ratchet e5ect will manifest itself, has its Brillouin-type correspondence in the Seebeck e5ect (see Section 6.1), discovered by Seebeck in 1822 of course without any idea about the underlying microscopic ratchet e5ect. Another root of Brownian motor theory leads us into the realm of intracellular transport research, speciFcally the biochemistry of molecular motors and molecular pumps. In the case of molecular motors, the concepts which we have in mind here have been unraveled in several steps, starting with A. Huxley’s ground-breaking 1957 work on muscle contraction [4], and continued in the late 1980s by Braxton and Yount [5,6] and in the 1990s by Vale and Oosawa [7], Leibler and Huse [8,9], Cordova, Ermentrout, and Oster [10], Magnasco [11,12], Prost’s group [13,14], Astumian and Bier [15,16], Peskin et al. [17,18] and many others, see Section 7. In the case of molecular pumps, the breakthrough came with the theoretical interpretation of previously known experimental Fndings [19,20] as a ratchet e5ect in 1986 by Tsong, Astumian and coworkers [21,22], see Section 4.6. While the general importance of asymmetry induced rectiFcation, thermal =uctuations, and the coupling of non-equilibrium enzymatic reactions to mechanical currents according to Curie’s principle for numerous cellular transport processes is long known [23,24], the above works introduced for the Frst time a quantitative microscopic modeling beyond the linear response regime close to thermal equilibrium. On the physical side, a ratchet e5ect in the form of voltage rectiFcation by a DC-SQUID in the presence of a magnetic Feld and an unbiased AC-current (i.e. a tilting ratchet scheme) has been

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experimentally observed and theoretically interpreted as early as in 1967 by De Waele et al. [25,26]. Further, directed transport induced by unbiased, time-periodic driving forces in spatially periodic structures with broken symmetry has been the subject of several hundred experimental and theoretical papers since the mid-1970s. In this context of the so-called photovoltaic and photorefractive e5ects in non-centrosymmetric materials, a ground breaking experimental contribution represents the 1974 paper by Glass et al. [27]. The general theoretical framework was elaborated a few years later by Belinicher, Sturman and coworkers, as reviewed—together with the above mentioned numerous experiments—in their capital works [28,29]. They identiFed as the two main ingredients for the occurrence of the ratchet e5ect in periodic systems the breaking of thermal equilibrium (detailed balance symmetry) and of the spatial symmetry, and they pointed out the much more general validity of such a tilting ratchet scheme beyond the speciFc experimental systems at hand, see Section 5.6. The possibility of producing a DC-output by two superimposed sinusoidal AC-inputs at frequencies ! and 2! in a spatially periodic, symmetric system, exemplifying a so-called asymmetrically tilting ratchet mechanism, has been observed experimentally 1978 by Seeger and Maurer [30] and analyzed theoretically 1979 by Wonneberger [31], see Section 5.12.1. The occurrence of a ratchet e5ect has been theoretically predicted 1987 by Bug and Berne [32] for the simplest variant of a pulsating ratchet scheme, termed on–o5 ratchet (see Section 4.2). A ratchet model with a symmetric periodic potential and a state-dependent temperature (multiplicative noise) with the same periodicity but out of phase, i.e. a simpliFed microscopic model for the Seebeck e5ect (see Section 6.1), has been analyzed 1987 by BMuttiker [33] and independently by van Kampen [463]. The independent re-inventions of the on–o5 ratchet scheme 1992 by Ajdari and Prost [34] and of the tilting ratchet scheme 1993 by Magnasco [11] together with the seminal 1994 works (ordered by date of receipt) [12,13,15,17,35–42] provided the inspiration for a whole new wave of great theoretical and experimental activity and progress within the statistical physics community as detailed in the subsequent chapters and reviewed e.g. in [14,43– 61]. While initially the modeling of molecular motors has served as one of the main motivations, the scope of Brownian motor studies has subsequently been extended to an ever increasing number of physical and technological applications, along with the re-discovery of the numerous pertinent works from before 1992. As a result, a much broader and uniFed conceptual basis has been achieved, new theoretical tools have been developed which lead to the discovery of many interesting and quite astonishing e5ects, and a large variety of exciting new experimental realizations have become available. Within the realm of noise-induced or -assisted non-equilibrium phenomena, an entire family of well-established major Felds are known under the labels of stochastic resonance [62], noise induced transitions [63] and phase transitions [64,65], reaction rate theory [66 – 68], and driven di5usive systems [69,70], to name but a few examples. One objective of our present review is to show that the important recent contributions of many workers to the theory and application of Brownian motors has given rise to another full-=edged member of this family. 1.3. Organization of the paper This review addresses two readerships: It may serve as an introduction to the Feld without requiring any specialized preknowledge. On the other hand, it o5ers to the active researcher a unifying view and guideline through the very rapidly growing literature. For this reason, not everything will be of

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equal interest for everybody. The following outline together with the table of contents may help to make one’s selection. Essentially, the subsequent eight sections (or chapters) can be divided into three units of rather di5erent characters: The Frst unit (Section 2) is predominantly of introductory and pedagogical nature, illustrating the basic phenomena, concepts, and applications by way of examples. Technically, the discussion is conducted on a rather elementary level and the calculations are to a far extent self-contained. “Standard” lines of reasoning and the derivation of basic working tools are discussed rather detailed in mathematically heuristic but physically suggestive terms. While these parts of Section 2 are not meant to replace a systematic introduction to the Feld of stochastic processes, they may hopefully serve as a minimal basis for the technically less detailed subsequent sections. Section 3 is devoted to general and systematic considerations which are relevant for the entire subsequent parts of the paper. The main classes of ratchet models and their physical origin are discussed with particular emphasis on symmetries, current inversions, and asymptotic regimes. Sections 4 – 6 represent the main body of the present work and are to a large extent of review character. It was only during the completion of these chapters that the amount of pertinent literature in this context became clear. As a consequence, speciFc new aspects of the considered ratchet systems and of the obtained results could only be included for a selection of particularly signiFcant such studies. Even then, the technical procedures and the detailed quantitative Fndings had to be mostly omitted. Besides the conceptual theoretical considerations and the systematic discussion of various speciFc model classes, a substantial part of Sections 4 – 6 has also been reserved for the manifold experimental applications of those ideas. Sections 7–9 represent the third main unit of our work, elaborating in somewhat more detail three major instances of applications and extensions. Of methodic rather than review character are the Frst three subsections of Section 7, illustrating a typical stochastic modeling procedure for the particularly important example of intracellular transport processes by molecular motors. The remainder of Section 7 presents a survey of the Feld with particular emphasis on cooperative molecular motors and the character of the mechanochemical coupling. Section 8 is devoted to the discussion of theoretically predicted new characteristic quantum mechanical signatures of Brownian motors and their experimental veriFcation on the basis of a quantum dot array with broken spatial symmetry. Finally, Section 9 deals with collective e5ects of interacting ratchet systems. On the one hand, we review modiFcations of the directed transport properties of single ratchets caused by their interaction (Section 9.1). On the other hand (Section 9.2) we exemplify genuine collective transport phenomena by a somewhat more detailed discussion of one speciFc model of paradigmatic simplicity—meant as a kind of “normal form” description which still captures the essence of more realistic models but omits all unnecessary details, in close analogy to the philosophy usually adopted in the theory of equilibrium phase transitions. Concluding remarks and future perspectives are presented in Section 10. Some technical details from the introductory Section 2 are contained in the appendices. Previously unpublished research represent the considerations about supersymmetry in Section 3.5, the method of tailoring current inversions in Section 3.6, the general treatment of the linear response regime in Section 3.7, the approximative molecular motor model with two highly cooperative “heads” in Section 7.5, as well as a number of additional minor new results which are indicated as such throughout the text, e.g. various exact mappings between di5erent classes of ratchet systems. New, mainly by the way of presentation but to some degree also by their content, are parts of

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Sections 2.1–2.4 and 6.1– 6.4, the systematic ratchet classiFcation scheme and its physical basis in Sections 3.3 and 3.4, the uniFed fast =uctuating force asymptotics in Section 5.5.1, as well as the coherent historical review in the preceding Section 1.2. A kind of red thread through the entire review consists in the asymptotic analysis of the so-called fast-driving limit. By collecting and rewriting the various results spread out in the literature and completing the missing pieces, a uniFed picture of this asymptotic regime emerges for the Frst time. The structural similarity of these analytical results in view of the rather di5erent underlying models is remarkable. For instance, within our standard working model—the overdamped Brownian motion in a periodic non-equilibrium system involving some ratchet-potential V (x) of period L—the direction of particle current is governed under very general circumstances by a factor of the form L the average n V (x)[d V (x)=d xn ]2 d x with a model-dependent n-value between 1 and 3. Especially, already 0 within this asymptotic regime, the intriguingly complicated dependence of the directed transport, e.g. on the detailed potential shape V (x), becomes apparent—a typical feature of systems far from thermal equilibrium. Basically, the review is organized in three levels (chapters, sections, subsections). While from the logical viewpoint, additional levels would have been desirable, the present rather “=at” structure simpliFes a quick orientation on the basis of the table of contents. Throughout the main text, cross-referencing to related subsections is used rather extensively. It may be ignored in case of a systematic reading, but is hopefully of use otherwise. 2. Basic concepts and phenomena This chapter serves as a motivation and Frst exposition of the main themes of our review, such as the absence of directed transport in ratchet systems at thermal equilibrium, its generic occurrence away from equilibrium, and the possibility of current inversions upon variation of some parameter. These fundamental phenomena are exempliFed in their simplest form in Section 2.1, Sections 2.6–2.9, and Section 2.11, respectively, and will then be elaborated in more generality and depth in the subsequent chapters. At the same time, this chapter also introduces the basic stochastic modeling concepts as well as the mathematical methods and “standard arguments” in this context. These issues are mainly contained in Sections 2.2–2.5 and 2.10, complemented by further details in the respective appendices. Readers who are already familiar with these basic physical phenomena and mathematical concepts may immediately proceed to Section 3. 2.1. Smoluchowski–Feynman ratchet Is it possible, and how is it possible to gain useful work out of unbiased random =uctuations? In the case of macroscopic =uctuations, the task can indeed be accomplished by various well-known types of mechanical and electrical rectiFers. Obvious daily-life examples are the wind-mill or the self-winding wristwatch. More subtle is the case of microscopic =uctuations, as demonstrated by the following Gedankenexperiment about converting Brownian motion into useful work. The basic idea can be traced back to a conference talk by Smoluchowski in MMunster 1912 (published as proceedings-article in Ref. [1]) and was later popularized and extended in Feynman’s Lectures on Physics [2].

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Fig. 2.1. Ratchet and pawl. The ratchet is connected by an axle with the paddles and with a spool, which may lift a load. In the absence of the pawl (leftmost object) and the load, the random collisions of the surrounding gas molecules (not shown) with the paddles cause an unbiased rotatory Brownian motion. The pawl is supposed to rectify this motion so as to lift the load.

2.1.1. Ratchet and pawl The main ingredient of Smoluchowski and Feynman’s Gedankenexperiment is an axle with at one end paddles and at the other end a so-called ratchet, reminiscent of a circular saw with asymmetric saw-teeth (see Fig. 2.1). The whole device is surrounded by a gas at thermal equilibrium. So, if it could freely turn around, it would perform a rotatory Brownian motion due to random impacts of gas molecules on the paddles. The idea is now to rectify this unbiased random motion with the help of a pawl (see Fig. 2.1). It is indeed quite suggestive that the pawl will admit the saw-teeth to proceed without much e5ort into one direction (henceforth called “forward”) but practically exclude a rotation in the opposite (“backward”) direction. In other words, it seems quite convincing that the whole gadget will perform on the average a systematic rotation in one direction, and this in fact even if a small load in the opposite direction is applied. Astonishingly enough, this naive expectation is wrong: In spite of the built in asymmetry, no preferential direction of motion is possible. Otherwise, such a gadget would represent a perpetuum mobile of the second kind, in contradiction to the second law of thermodynamics. The culprit must be our assumption about the working of the pawl, which is indeed closely resembling Maxwell’s demon. 1 Since the impacts of the gas molecules take place on a microscopic scale, the pawl needs to be extremely small and soft in order to admit a rotation even in the forward direction. As Smoluchowski points out, the pawl itself is therefore also subjected to non-negligible random thermal =uctuations. So, every once in a while the pawl lifts itself up and the saw-teeth can freely travel underneath. Such an event clearly favors on the average a rotation in the “backward” direction in Fig. 2.1. At overall thermal equilibrium (the gas surrounding the paddles and the pawl being at the same temperature) the detailed quantitative analysis [2] indeed results in the subtle probabilistic balance which just rules out the functioning of such a perpetuum mobile. 1

Both Smoluchowski and Feynman have pointed out the similarity between the working principle of the pawl and that of a valve. A valve, acting between two boxes of gas, is in turn one of the simplest realizations of a Maxwell demon [71]. For more details on Maxwell’s demon, especially the history of this apparent paradox and its resolution, we refer to the commented collection of reprints in [72].

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A physical system as described above will be called after Smoluchowski and Feynman. We will later go one step further and consider the case that the gas surrounding the paddles and the pawl are not at the same temperature (see Section 6.2). Such an extension of the original Gedankenexperiment appears in Feynman’s lectures, but has not been discussed by Smoluchowski, and will therefore be named after Feynman only. Smoluchowski and Feynman’s ratchet and pawl has been experimentally realized on a molecular scale by Kelly et al. [73–76]. Their synthesis of triptycene[4]helicene incorporates into a single molecule all essential components: The triptycene “paddlewheel” functions simultaneously as circular ratchet and as paddles, the helicene serves as pawl and provides the necessary asymmetry of the system. Both components are connected by a single chemical bond, giving rise to one degree of internal rotational freedom. By means of sophisticated nuclear magnetic resonance (NMR) techniques, the predicted absence of a preferential direction of rotation at thermal equilibrium has been conFrmed experimentally. The behavior of similar experimental systems beyond the realm of thermal equilibrium will be discussed at the end of Section 4.5.2. 2.1.2. Simpli8ed stochastic model In the sense that we are dealing merely with a speciFc instance of the second law of thermodynamics, the situation with respect to Smoluchowski–Feynman’s ratchet and pawl is satisfactorily clariFed. On the other hand, the obvious intention of Smoluchowski and Feynman is to draw our attention to the amazing content and implications of this very law, calling for a more detailed explanation of what is going on. A satisfactory modeling and analysis of the relatively complicated ratchet and pawl gadget as it stands is possible but rather involved, see Section 6.2. Therefore, we focus on a considerably simpliFed model which, however, still retains the basic qualitative features: We consider a Brownian particle in one dimension with coordinate x(t) and mass m, which is governed by Newton’s equation of motion 2 m x(t) M + V (x(t)) = − x(t) ˙ + (t) :

(2.1)

Here V (x) is a periodic potential with period L, V (x + L) = V (x)

(2.2)

and broken spatial symmetry, 3 thus playing the role of the ratchet in Fig. 2.1. A typical example is V (x) = V0 [sin(2 x=L) + 0:25 sin(4 x=L)] ;

(2.3)

see Fig. 2.2. The left-hand side in (2.1) represents the deterministic, conservative part of the particle dynamics, while the right-hand side accounts for the e5ects of the thermal environment. These are energy dissipation, modeled in (2.1) as viscous friction with friction coeCcient , and randomly =uctuating forces in the form of the thermal noise (t). These two e5ects are not independent of each other since they have both the same origin, namely the interaction of the particle x(t) with a huge number of 2 3

Dot and prime indicate di5erentiations with respect to time and space, respectively. Broken spatial symmetry means that there is no x such that V (−x) = V (x + Ux) for all x.

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V(x)/V

0

1

0

-1

-2

-1

- 0.5

0

0.5

1

x /L

Fig. 2.2. Typical example of a ratchet-potential V (x), periodic in space with period L and with broken spatial symmetry. Plotted is the example from (2.3) in dimensionless units.

microscopic degrees of freedom of the environment. As discussed in detail in Sections A.1 and A.2 of Appendix A, our assumption that the environment is an equilibrium heat bath with temperature T and that its e5ect on the system can be modeled by means of the phenomenological ansatz appearing on the right-hand side of (2.1) completely Fxes [66,77–97] all statistical properties of the =uctuations (t) without referring to any microscopic details of the environment (see also Sections 2.9, 3.4.1 and 8.1). Namely, (t) is a Gaussian white noise of zero mean, (t) = 0 ;

(2.4)

satisfying the 9uctuation–dissipation relation [79 –81] (t)(s) = 2 kB T(t − s) ;

(2.5)

where kB is Boltzmann’s constant, 2 kB T is the noise intesity or noise strength, and (t) is Dirac’s delta function. Note that the only particle property which enters the characteristics of the noise is the friction coeCcient , which may thus be viewed as the coupling strength to the environment. For the typically very small systems one has in mind, and for which thermal =uctuations play any notable role at all, the dynamics (2.1) is overdamped, that is, the inertia term m x(t) M is negligible (see also the more detailed discussion of this point in Section A.4 of Appendix A). We thus arrive at our “minimal” Smoluchowski–Feynman ratchet model

x(t) ˙ = −V (x(t)) + (t) :

(2.6)

According to (2.5), the Gaussian white noise (t) is uncorrelated in time, i.e. it is given by independently sampled Gaussian random numbers at any time t. This feature and the concomitant inFnitely large second moment 2 (t) are mathematical idealizations. In physical reality, the correlation time is meant to be Fnite, but negligibly small in comparison with all other relevant time scales of the system. In this spirit, we may introduce a small time step Ut and consider a time-discretized

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version of the stochastic dynamics (2.6) of the form x(tn+1 ) = x(tn ) − Ut[V (x(tn )) + n ]= ;

(2.7)

where tn := nUt and where the n are independently sampled, unbiased Gaussian random numbers with second moment 2n = 2 kB T=Ut :

(2.8)

The continuous dynamics (2.6) with uncorrelated noise is then to be understood [98–100] as the mathematical limit of (2.7) for Ut → 0. Moreover, this discretized dynamics (2.7) is a suitable starting point for a numerical simulation of the problem. Finally, a derivation of the so-called Fokker– Planck equation (see Eq. (2.14) below) based on (2.7) is given in Appendix B. 2.2. Fokker–Planck equation The following four sections are mainly of methodological nature without much new physics. After introducing the Fokker–Planck equation in the present section, we turn in Sections 2.3 and 2.4 to the evaluation of the particle current x, ˙ with the result that even when the spatial symmetry is broken by the ratchet potential V (x), there arises no systematic preferential motion of the random dynamics in one or the other direction. Finally, in Section 2.5 the e5ect of an additional static “tilting” force F in the Smoluchowski–Feynman ratchet dynamics (2.6) is considered, with the expected result of a Fnite particle current x ˙ with the same sign as the applied force F. Readers who are already familiar with or not interested in these standard techniques are recommended to continue with Section 2.6. Returning to (2.6), a quite natural next step is to consider a statistical ensemble of these stochastic processes belonging to independent realizations of the random =uctuations (t). The corresponding probability density P(x; t) in space x at time t describes the distribution of the Brownian particles and follows as an ensemble average 4 of the form P(x; t) := (x − x(t)) : An immediate consequence of this deFnition is the normalization ∞ d x P(x; t) = 1 : −∞

(2.9)

(2.10)

Another trivial consequence is that P(x; t) ¿ 0 for all x and t. In order to determine the time-evolution of P(x; t), we Frst consider in (2.6) the special case V (x) ≡ 0. As discussed in detail in Section A.3 of Appendix A, we are thus dealing with the force-free thermal di5usion of a Brownian particle with a di5usion coeCcient D that satisFes Einstein’s relation [77] D = kB T= : 4

(2.11)

To be precise, an average over the initial conditions x(t0 ) according to some prescribed statistical weight P(x; t0 ) together with an average over the noise is understood on the right-hand side of (2.9).

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Consequently, P(x; t) is governed by the di5usion equation kB T 92 9 P(x; t) = P(x; t) 9t

9x2

if V (x) ≡ 0 :

(2.12)

Next, we address the deterministic dynamics (t) ≡ 0 in (2.6). In complete analogy to classical Hamiltonian mechanics, one then Fnds that the probability density P(x; t) evolves according to a Liouville-equation of the form 5 9 9 V (x) P(x; t) = P(x; t) if (t) ≡ 0 : (2.13) 9t 9x

Since both (2.12) and (2.13) are linear in P(x; t) it is quite obvious that the general case follows by combination of both contributions, i.e. one obtains the so-called Fokker-Planck equation [99,101] 9 V (x) k B T 92 9 P(x; t) = P(x; t) + P(x; t) ; (2.14) 9t 9x

9x2 where the Frst term on the right-hand side is called “drift term” and the second “di5usion term”. While our above derivation of the Fokker–Planck equation is admittedly of a rather heuristic nature, it is appealing due to its extreme simplicity and the intuitive physical way of reasoning. A more rigorous calculation, based on the discretized dynamics (2.7) in the limit Ut → 0 is provided in Appendix B. Numerous alternative derivations can be found, e.g. in [98–105] and further references therein. A brief historical account of the Fokker–Planck equation has been compiled in [106], see also [107]. 2.3. Particle current The quantity of foremost interest in the context of transport in periodic systems is the particle current x, ˙ deFned as the time-dependent ensemble average over the velocities x ˙ := x(t) ˙ :

(2.15)

For later convenience, the argument t in x ˙ is omitted. Obviously, the probability density P(x; t) contains the entire information about the system; in this section we treat the question of how to extract the current x ˙ out of it. The simplest way to establish such a connection between x ˙ and P(x; t) follows by averaging in (2.6) and taking into account (2.4), i.e. x ˙ = −V (x(t))= . Since the ensemble average means

5

Proof. Let x(t) be a solution of x(t) ˙ = f(x(t)) and deFne P(x; t) := (x − x(t)). Note that the variable x and the function x(t) are mathematically completely unrelated objects. Then (9=9t)P(x; t) = −x(t) ˙ (9=9x)(x − x(t)) = −f(x(t)) (9=9x)(x −x(t))=−(9=9x) {f(x(t))(x −x(t))}=−(9=9x){f(x)(x −x(t))} (the last identity can be veriFed by operating with d x h(x) on both sides, where h(x) is an arbitrary test function with h(x → ± ∞) = 0, and then performing a partial integration). Thus (2.13) is recovered for a -distributed initial condition. Since this Eq. (2.13) is linear in P(x; t), the case of a general initial distribution follows by linear superposition.

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by deFnition an average with respect to the probability density P(x; t) we arrive at our Frst basic observation, namely the connection between x ˙ and P(x; t): ∞ V (x) x ˙ =− P(x; t) : (2.16) dx

−∞ The above derivation of (2.16) has the disadvantage that the speciFc form (2.6) of the stochastic dynamics has been exploited. For later use, we next sketch an alternative, more general derivation: From the deFnition (2.9) one obtains, independently of any details of the dynamics governing x(t), a so-called master equation [99 –101] 9 9 P(x; t) + J (x; t) = 0 ; 9t 9x

(2.17)

J (x; t) := x(t) ˙ (x − x(t)) :

(2.18)

Note that the symbols x and x(t) denote here completely unrelated mathematical objects. The master equation (2.17) has the form of a continuity equation for the probability density associated with the conservation of particles, hence J (x; t) is called the probability current. Upon integrating (2.18), the following completely general connection between the probability current and the particle current is obtained: ∞ x ˙ = d x J (x; t) : (2.19) −∞

By means of a partial integration, the current in (2.19) can be rewritten as − d x x 9J (x; t)=9x and by exploiting (2.17) one recovers the relation ∞ d x ˙ = d x x P(x; t) ; (2.20) dt −∞ which may thus be considered as an alternative deFnition of the particle current x. ˙ For the speciFc stochastic dynamics (2.6), we Fnd by comparison of the Fokker–Planck equation (2.14) with the general master equation (2.17) the explicit expression for the probability current V (x) kB T 9 + P(x; t) ; (2.21) J (x; t) = −

9x up to an additive, x-independent function. Since both, J (x; t) and P(x; t) approach zero for x → ± ∞, it follows that this function must be identically zero. By introducing (2.21) into (2.19) we Fnally recover (2.16). 2.4. Solution and discussion Having established the evolution equation (2.14) governing the probability density P(x; t) our next goal is to actually solve it and determine the current x ˙ according to (2.19). Such a calculation is illustrated in detail in this section.

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We start with introducing the so-called reduced probability density and reduced probability current ˆ t) := P(x;

∞

P(x + nL; t) ;

(2.22)

J (x + nL; t) :

(2.23)

n=−∞

Jˆ(x; t) :=

∞

n=−∞

Taking into account (2.10), (2.19) it follows that ˆ + L; t) = P(x; ˆ t) ; P(x 0

L

ˆ t) = 1 ; d x P(x;

x ˙ =

0

L

d x Jˆ(x; t) :

(2.24) (2.25) (2.26)

With P(x; t) being a solution of the Fokker–Planck equation (2.14) it follows with (2.2) that also P(x + nL; t) is a solution for any integer n. Since the Fokker–Planck equation is linear, it is also satisFed by the reduced density (2.22). With (2.21) it can furthermore be recast into the form of a continuity equation 9 ˆ 9 P(x; t) + Jˆ(x; t) = 0 9t 9x with the explicit form of the reduced probability current k 9 V (x) T B ˆ t) : + P(x; Jˆ(x; t) = −

9x

(2.27)

(2.28)

In other words, as far as the particle current x ˙ is concerned, it su
(2.30)

We recall that in general the current x ˙ is time dependent but the argument t is omitted (cf. (2.15)). However, the most interesting case is usually its behavior in the long-time limit,

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corresponding to a steady state in the reduced description (unless an external driving prohibits its existence, see e.g. Section 2.6.1 below). In this case, no implicit t-dependent of x ˙ is present any more, see (2.30). We have tacitly assumed here that the original problem (2.6) extends over the entire real x-axis. In some cases, a periodicity condition after one or several periods L of the potential V (x) may represent a more natural modeling, for instance in the original Smoluchowski–Feynman ratchet of circular shape (Fig. 2.1). One readily sees, that in such a case (2.24) – (2.30) remain valid without any change. We furthermore remark that the speciFc form of the stochastic dynamics (2.6) or of the equivalent master equation (2.17), (2.21) has only been used in (2.28), while equations (2.22) – (2.27), (2.29), (2.30) remain valid for more general stochastic dynamics. ˆ t) indeed approaches For physical reasons we expect that the reduced probability density P(x; st st a steady state Pˆ (x) in the long-time limit t → ∞ and hence Jˆ(x0 ; t) → Jˆ . From the remaining ordinary Frst order di5erential equation (2.28) for P st (x) in combination with (2.24) it follows that st Jˆ must be zero and therefore the solution is st Pˆ (x) = Z −1 e−V (x)=kB T ;

Z :=

L

0

d x e−V (x)=kB T ;

(2.31) (2.32)

while (2.26) implies for the steady state particle current the result x ˙ = 0:

(2.33)

It can be shown that the long-time asymptotics of a Fokker–Planck equation like in (2.27), (2.28) is unique [82,83,100,108,109]. Consequently, this unique solution must be (2.31), independent of the initial conditions. Furthermore, our assumption that a steady state is approached for t → ∞ is self-consistently conFrmed. The above results justify a posteriori our proposition that (2.6) models an overdamped Brownian motion under the in=uence of a thermal equilibrium heat bath at temperature T : indeed, in the long-time limit (steady state), Eq. (2.31) correctly reproduces the expected Boltzmann distribution and the average particle current vanishes (2.33), as required by the second law of thermodynamics. The importance of such consistency checks when modeling thermal noise is further discussed in Section 2.9. Much like in the original ratchet and pawl gadget (Fig. 2.1), the absence of an average current (2.33) is on the one hand, a simple consequence of the second law of thermodynamics. On the other hand, when looking at the stochastic motion in a ratchet-shaped potential like in Fig. 2.2, it is nevertheless quite astonishing that in spite of the broken spatial symmetry there arises no systematic preferential motion of the random dynamics in one or the other direction. Note that if the original problem (2.6) extends over the entire real axis (bringing along natural boundary conditions), then the probability density P(x; t) will never approach a meaningful 6 steady 6

The trivial long time behavior P(x; t) → 0 does not admit any further conclusions and is therefore not considered as a meaningful steady state.

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ˆ t), associated with periodic boundary conditions, which state. It is only the reduced density P(x; tends toward a meaningful t-independent long-time limit. In particular, only after this mapping, which leaves the particle current unchanged, are the concepts of equilibrium statistical mechanics applicable. Conceptually, the simpliFed Smoluchowski–Feynman ratchet model (2.6) has one crucial advantage in comparison with the original full-blown ratchet and pawl gadget from Fig. 2.1: The second law of thermodynamics has not to be invoked as a kind of deus ex machina, rather the absence of a current (2.33) now follows directly from the basic model (2.6), without any additional assumptions. As a consequence, modiFcations of the original situation, for which the second law of thermodynamics no longer applies, are relatively straightforward to treat within a correspondingly modiFed Smoluchowski–Feynman ratchet model (2.6), but become very cumbersome [110,111] for the more complicated original ratchet and pawl gadget from Fig. 2.1. A Frst, very simple such modiFcation of the Smoluchowski–Feynman ratchet model will be addressed next. 2.5. Tilted Smoluchowski–Feynman ratchet In this section we consider the generalization of the overdamped Smoluchowski–Feynman ratchet model (2.6) in the presence of an additional homogeneous, static force F:

x(t) ˙ = −V (x(t)) + F + (t) :

(2.34)

This scenario represents a kind of “hydrogen atom” in that it is one of the few exactly solvable cases and will furthermore turn out to be equivalent to the archetypal ratchet models considered later in Sections 4.3.2, 4.4.1, 5.2, 6.1 and 9.2. For instance, in the original ratchet and pawl gadget (Fig. 2.1) such a force F in (2.34) models the e5ect of a constant external torque due to a load. We may incorporate the ratchet potential V (x) and the force F into a single e5ective potential Ve5 (x) := V (x) − x F ;

(2.35)

which the Brownian particle (2.34) experiences. E.g. for a negative force F¡0, pulling the particles to the left, the e5ective potential will be tilted to the left as well, see Fig. 2.3. In view of x ˙ =0 for F = 0 (see previous section) it is plausible that in such a potential the particles will move on the average “downhill”, i.e. x¡0 ˙ for F¡0 and similarly x¿0 ˙ for F¿0. This surmise is conFrmed by a detailed calculation along the very same lines as for F = 0 (see Section 2.4), with the result (see [112–114] and also Vol. 2, Chapter 9 in [115]) that in the steady state (long-time limit)

−Ve5 (x)=kB T x+L st e Pˆ (x) = N dy eVe5 (y)=kB T ; (2.36) kB T x x ˙ = L N [1 − e[Ve5 (L)−Ve5 (0)]=kB T ] ; kB T N :=

0

L

dx

x

x+L

dy e

[Ve5 (y)−Ve5 (x)]=kB T

(2.37) −1 :

(2.38)

Note that for the speciFc form (2.35) of the e5ective potential we can further simplify (2.37) by exploiting that Ve5 (L) − Ve5 (0) = −LF. However, the result (2.36) – (2.38) is valid for completely (x) provided V (x + L) = V (x). general e5ective potentials Ve5 e5 e5

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73

4

2

3 2

<x>

1

.

0

V

eff

(x)

1

0 -1 -2

-1

-3 -2 -1

-0.5

0

x

0.5

1

-4 -6

-4

-2

0

2

4

6

F

Fig. 2.3. Typical example of an e5ective potential from (2.35) “tilted to the left”, i.e. F¡0. Plotted is the example from (2.3) in dimensionless units (see Section A.4 in Appendix A) with L = V0 = 1 and F = −1, i.e. Ve5 (x) = sin(2 x) + 0:25 sin(4 x) + x. Fig. 2.4. Steady state current x ˙ from (2.37) versus force F for the tilted Smoluchowski–Feynman ratchet dynamics (2.5), (2.34) with the potential (2.3) in dimensionless units (see Section A.4 in Appendix A) with = L = V0 = kB = 1 and T = 0:5. Note the broken point-symmetry. st Our Frst observation is that a time-independent probability density Pˆ (x) does not exclude a non-vanishing particle current x. ˙ Exploiting (2.35), one readily sees that—as expected—the sign of this current (2.37) agrees with the sign of F. Furthermore one can prove that the current is a monotonically increasing function of F [116] and that for any Fxed F-value, the current is maximal (in modulus) when V (x) = const: (see Section 4.4.1). The typical quantitative behavior of the steady state current (2.37) as a function of the applied force F (called “response curve”, “load curve”, or (current-force-) “characteristics”) is exempliFed in Fig. 2.4. Note that the leading-order (“linear response”) behavior is symmetric about the origin, but not the higher order contributions. The occurrence of a non-vanishing particle current in (2.37) signals that (2.36) describes a steady state which is not in thermal equilibrium, and actually far from equilibrium unless F is very small. 7 As mentioned already at the end of the previous section, while at (and near) equilibrium one may question the need of a microscopic model like in (2.34) in view of the powerful principles of equilibrium statistical mechanics, such an approach has the advantage of remaining valid far from equilibrium, 8 where no such general statistical mechanical principles are available. ˆ t) As pointed out at the end of the preceding section, only the reduced probability density P(x; approaches a meaningful steady state, but not the original dynamics (2.34), extending over the entire x-axis. Thus, stability criteria for steady states, both mechanical and thermodynamical, can only be 7

In particular, the e5ective di5usion coeCcient is no longer related to the mobility via a generalized Einstein relation (2.11), i.e. De5 = kB T 9x =9F ˙ only holds for F = 0 [117]. 8 Note that there is no inconsistency of a thermal (white) noise (t) appearing in a system far from thermal equilibrium: any system (equilibrium or not) can be in contact with a thermal heat bath.

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discussed in the former, reduced setup. As compared to the usual re=ecting boundary conditions in this context, the present periodic boundary conditions entail some quite unusual consequences: With x +L st the deFnition (F; x0 ) := x00 d x x Pˆ (x) for the “center of mass” in the steady state (cf. (2.29)), L st st st one can infer from the periodicity Pˆ (x + L) = Pˆ (x) and the normalization 0 d x 9Pˆ (x)=9F = 0 that 9(F; x0 + L)=9F = 9(F; x0 )=9F, where x0 is an arbitrary reference position. Furthermore, one Fnds that L L L st 9Pˆ (x + x0 ) 9(F; x0 ) = = 0: (2.39) d x0 d x0 d x (x + x0 ) 9F 9F 0 0 0 Excluding the non-generic case that 9(F; x0 )=9F is identically zero, it follows 9 that 9(F; x0 )=9F may be negative or positive, depending on the choice of x0 . In other words, the “center of mass” may move either in the same or in the opposite direction of the applied force F, and this even if the unperturbed system is at thermal equilibrium. Similarly, also with respect to the dependence of the steady state current x ˙ upon the applied force F, no general a priori restrictions due to certain “stability criteria” for steady states exist. 2.5.1. Weak noise limit In this section we work out the simpliFcation of the current-formula (2.37) for small thermal energies kB T —see Eq. (2.44) below—and its quite interesting physical interpretation, repeatedly re-appearing later on. Focusing on not too large F-values, such that Ve5 (x) in (2.35) still exhibits at least one local minimum and maximum within each period L, one readily sees that the function Ve5 (y) − Ve5 (x) has generically a unique global maximum within the two-dimensional integration domain in (2.38), say at the point (x; y) = (xmin ; xmax ), where xmin is a local minimum of Ve5 (x) and xmax a local maximum, sometimes called metastable and activated states, respectively. Within (xmin ; xmin + L) the point xmax is moreover a global maximum of Ve5 (x) and similarly xmin a global minimum within (xmax − L; xmax ), i.e. UVe5 := Ve5 (xmax ) − Ve5 (xmin )

(2.40)

is the e5ective potential barrier that the particle has to surmount in order to proceed from the metastable state xmin to xmin + L. Likewise, Ve5 (xmax − L) − Ve5 (xmin ) = UVe5 − [Ve5 (L) − Ve5 (0)]

(2.41)

is the barrier between xmin and xmin − L. For small thermal energies kB T {UVe5 ; UVe5 − [Ve5 (L) − Ve5 (0)] } ;

(2.42)

the main contribution in (2.38) stems from a small vicinity of the absolute maximum (xmin ; xmax ) and we thus can employ the so-called saddle point approximation Ve5 (y) − Ve5 (x) UVe5 − 9

(x |Ve5 |V (xmin )| max )| (y − xmax )2 − e5 (x − xmin )2 ; 2 2

Note that we did not exploit any speciFc property of the underlying stochastic dynamics.

(2.43)

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(x where we have used that Ve5 max ) = Ve5 (xmin ) = 0 and Ve5 (xmax )¡0, Ve5 (xmin )¿0. Within the same approximation, the two integrals in (2.38) can now be extended over the entire real x- and y-axis. Performing the two remaining Gaussian integrals in (2.38) yields for the current (2.37) the result

x ˙ = L [k+ − k− ] ; k+ :=

(x 1=2 |Ve5 max )Ve5 (xmin )| e−UVe5 =kB T ; 2

(2.44) (2.45)

k− := k+ e[Ve5 (L)−Ve5 (0)]=kB T =

(x 1=2 |Ve5 max − L)Ve5 (xmin )| e−[Ve5 (xmax −L)−Ve5 (xmin )]=kB T ; 2

(2.46)

(x) in the last relation in (2.46). where we have exploited (2.41) and the periodicity of Ve5 One readily sees that k+ is identical to the so-called Kramers–Smoluchowski rate [66] for transitions from xmin to xmin + L, and similarly k− is the escape rate from xmin to xmin − L. For weak thermal noise (2.42) these rates are small and the current (2.44) takes the suggestive form of a net transition frequency (rate to the right minus rate to the left) between adjacent local minima of Ve5 (x) times the step size L of one such transition.

2.6. Temperature ratchet and ratchet e1ect We now come to the central issue of the present chapter, namely the phenomenon of directed transport in a spatially periodic, asymmetric system away from equilibrium. This so-called ratchet e5ect is very often illustrated by invoking as an example the on–o5 ratchet model, as introduced by Bug and Berne [32] and by Ajdari and Prost [34], see Section 4.2. Here, we will employ a di5erent example, the so-called temperature ratchet, which in the end will however turn out to be actually very closely related to the on–o5 ratchet model (see Section 6.3). We emphasize that the choice of this example is not primarily based on its objective or historical signiFcance but rather on the author’s personal taste and research activities. Moreover, this example appears to be particularly suitable for the purpose of illustrating besides the ratchet e5ect per se also many other important concepts (see Sections 2.6.3–2.11) that we will encounter again in much more generality in later sections. 2.6.1. Model As an obvious generalization of the tilted Smoluchowski–Feynman ratchet model (2.34) we consider the case that the temperature of the Gaussian white noise (t) in (2.5) is subjected to periodic temporal variations with period T [118], i.e. (t)(s) = 2 kB T (t) (t − s) ;

(2.47)

T (t) = T (t + T) ;

(2.48)

where T (t) ¿ 0 for all t is taken for granted. Note that due to the time-dependent temperature in (2.47) the noise (t) is strictly speaking no longer stationary. A stationary noise is, however, readily

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recovered by rewriting (2.34), (2.47) as ˆ ;

x(t) ˙ = −V (x(t)) + F + g(t) (t)

(2.49)

ˆ is a Gaussian white noise with (t) ˆ (s)=2(t ˆ where (t) −s) and g(t) := [ kB T (t)]1=2 . Two typical examples which we will adopt for our numerical explorations below are T (t) = TW [1 + A sign{sin(2 t=T)}] ;

(2.50)

T (t) = TW [1 + A sin(2 t=T)]2 ;

(2.51)

where sign(x) denotes the signum function and |A|¡1. The Frst example (2.50) thus jumps between T (t) = TW [1 + A] and T (t) = TW [1 − A] at every half-period T=2. The motivation for the square on the right-hand side of (2.51) becomes apparent when rewriting the dynamics in the form (2.49). Similarly as in Section 2.2, one Fnds that the reduced particle density (2.22) for this so-called temperature ratchet model (2.34), (2.47), (2.48) is governed by the Fokker–Planck equation 9 ˆ 9 V (x) − F ˆ kB T (t) 92 ˆ P(x; t) : (2.52) P(x; t) = P(x; t) + 9t 9x

9x2 Due to the permanent oscillations of T (t), this equation does not admit a time-independent solution. ˆ t) will not approach a steady state but rather a unique periodic Hence, the reduced density P(x; behavior in the long-time limit. 10 It is therefore natural to include a time average into the deFnition (2.15) of the particle current. Keeping for convenience the same symbol x, ˙ the generalized expression (2.26), (2.28) for this current becomes t+T L 1 F − V (x) ˆ x ˙ = P(x; t) : (2.53) dt dx T t

0 Note that in general, the current x ˙ in (2.53) is still t-dependent. Only in the long time limit, corˆ t), this t-dependence disappears. Usually, responding in the reduced description to a T-periodic P(x; this latter long-time limit is of foremost interest. 2.6.2. Ratchet e1ect After these technical preliminaries, we return to the physics of our model (2.34), (2.47), (2.48): In the case of the tilted Smoluchowski–Feynman ratchet (time-independent temperature T ), Eq. (2.37) tells us that for a given force, say F¡0, the particles will move “downhill” on the average, i.e. x¡0, ˙ and this for any Fxed (positive) value of the temperature T . Turning to the temperature ratchet with T being now subjected to periodic temporal variations, one therefore should expect that the particles still move “downhill” on the average. The numerically calculated “load curve” in Fig. 2.5 demonstrates that the opposite is true within an entire interval of negative F-values. Surprisingly indeed, the particles are climbing “uphill” on the average, thereby performing work against the load force F, which apparently can have no other origin than the white thermal noise (t). ˆ t) also P(x; ˆ t + T) solves (2.52). Moreover, for the long time Proof. Since T (t + T) = T (t) we see that with P(x; ˆ t + T) must converge asymptotics of (2.52) the general proof of uniqueness from [83,109] applies. Consequently, P(x; ˆ ˆ towards P(x; t), i.e. P(x; t) is periodic and unique for t → ∞. 10

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77

0.04

<x>

0.02

.

0

-0.02 -0.04

-0.02

0

0.02

F

Fig. 2.5. Average particle current x ˙ versus force F for the temperature ratchet dynamics (2.3), (2.34), (2.47), (2.50) in dimensionless units (see Section A.4 in Appendix A). Parameter values are = L = T = kB = 1, V0 = 1=2 , TW = 0:5, A = 0:8. The time- and ensemble-averaged current (2.53) has been obtained by numerically evolving the Fokker–Planck equation (2.52) until transients have died out. Fig. 2.6. The basic working mechanism of the temperature ratchet (2.34), (2.47), (2.50). The Fgure illustrates how Brownian particles, initially concentrated at x0 (lower panel), spread out when the temperature is switched to a very high value (upper panel). When the temperature jumps back to its initial low value, most particles get captured again in the basin of attraction of x0 , but also substantially in that of x0 + L (hatched area). A net current of particles to the right, i.e. x ¿0 ˙ results. Note that practically the same mechanism is at work when the temperature is kept Fxed and instead the potential is turned “on” and “o5 ” (on–o5 ratchet, see Section 4.2).

A conversion (rectiFcation) of random =uctuations into useful work as exempliFed above is called “ratchet e1ect”. For a setup of this type, the names thermal ratchet [7,10,11], Brownian motor [48,118], Brownian recti8er [51] (mechanical diode [11]), stochastic ratchet [119,120], or simply ratchet are in use. 11 Since the average particle current x ˙ usually depends continuously on the load force F, it is for a qualitative analysis suCcient to consider the case F = 0: the occurrence of the ratchet e1ect is then tantamount to a 8nite current x ˙ = 0

for F = 0 ;

(2.54)

i.e. the unbiased Brownian motor implements a “particle pump”. The necessary force F which leads to an exact cancellation of the ratchet e5ects, i.e x ˙ = 0, is called the “stopping force”. The property (2.54) is the distinguishing feature between the ratchet e5ect and the somewhat related so-called negative mobility e5ect, encountered later in Section 9.2.4. 11

The notion “molecular motor” should be reserved for models focusing speciFcally on intracellular transport processes, see Section 7. Similarly, the notion “Brownian ratchet” has been introduced in a rather di5eren context, namely as a possible operating principle for the translocation of proteins accross membranes [121–125].

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2.6.3. Discussion In order to understand the basic physical mechanism behind the ratchet e5ect at F = 0, we focus on the dichotomous periodic temperature modulations from (2.50). During a Frst time interval, say t ∈ [T=2; T], the thermal energy kB T (t) is kept at a constant value TW [1 − A] much smaller than the potential barrier UV between two neighboring local minima of V (x). Thus, all particles will have accumulated in a close vicinity of the potential minima at the end of this time interval, as sketched in the lower panel of Fig. 2.6. Then the thermal energy jumps to a value TW [1 + A] much larger than UV and remains there during another half-period, say t ∈ [T; 3T=2]. Since the particles then hardly feel the potential any more in comparison to the violent thermal noise, they spread out practically like in the case of free thermal di5usion (upper panel in Fig. 2.6). Finally, T (t) jumps back to its original low value TW [1 − A] and the particles slide downhill towards the respective closest local minima of V (x). Due to the asymmetry of the potential V (x), the original population of one given minimum is re-distributed asymmetrically and a net average displacement results after one time-period T. In the case that the potential V (x) has exactly one minimum and maximum per period L (as it is the case in Fig. 2.6) it is quite obvious that if the local minimum is closer to its adjacent maximum to the right (as in Fig. 2.6), a positive particle current x¿0 ˙ will arise, otherwise a negative current. For potentials with additional extrema, the determination of the current direction may be less obvious. As expected, a qualitatively similar behavior is observed for more general temperature modulations T (t) than in Fig. 2.6 provided they are suCciently slow. The e5ect is furthermore robust with respect to the potential shape [118] and persists even for (slow) random instead of deterministic changes of T (t) [126,127], e.g. (rare) random =ips between the two possible values in Fig. 2.6, as well as for a modiFed dynamics with a discretized state space [128,129]. The case of Fnite inertia and of various correlated (colored) Gaussian noises instead of the white noise in (2.34) or (2.49) has been addressed in [130] and [131], respectively. A somewhat more detailed quantitative analysis will be given in Sections 2.10 and 2.11 below. In practice, the required magnitudes and time scales of the temperature variations may be diCcult to realize experimentally by directly adding and extracting heat, but may well be feasible indirectly, e.g. by pressure variations. An exception are point contact devices with a defect which tunnels incoherently between two states and thereby changes the coupling strength of the device to its thermal environment [132–138]. In other words, when incorporated into an electrical circuit, such a device exhibits random dichotomous jumps both of its electrical resistance and of the intensity of the thermal =uctuations which it produces [139]. The latter may thus be exploited to drive a temperature ratchet system [126]. Further, it has been suggested [140,141] that microorganisms living in convective hot springs may be able to extract energy out of the permanent temperature variations they experience; the temperature ratchet is a particularly simple mechanism which could do the job. Moreover, a temperature ratchet-type modiFcation of the experiment by Kelly et al. [73–75] (cf. Section 2.1.1) has been proposed in [76]. Finally, it is known that certain enzymes (molecular motors) in living cells are able to travel along polymer Flaments by hydrolyzing ATP (adenosine triphosphate). The interaction (chemical “aCnity”) between molecular motor and Flament is spatially periodic and asymmetric, and thermal =uctuations play a signiFcant role on these small scales. On the crudest level, hydrolyzing an ATP molecule

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may be viewed as converting a certain amount of chemical energy into heat, thus we recover all the essential ingredients of a temperature ratchet. Such a temperature ratchet-type model for intracellular transport has been proposed in [7]. Admittedly, modeling the molecular motor as a Brownian particle without any relevant internal degree of freedom 12 and the ATP hydrolysis as a mere production of heat is a gross oversimpliFcation from the biochemical point of view, see Section 7, but may still be acceptable as a primitive sketch of the basic physics. Especially, quantitative estimates indicate [9,142,143] that the temperature variations (either their amplitude or their duration) within such a temperature ratchet model may not be suCcient to reproduce quantitatively the observed traveling speed of the molecular motor. 2.7. Mechanochemical coupling We begin with pointing out that the ratchet e5ect as exempliFed by the temperature ratchet model is not in contradiction with the second law of thermodynamics 13 since we may consider the changing temperature T (t) as caused by several heat baths at di5erent temperatures. 14 From this viewpoint, our system is nothing else than an extremely primitive and small heat engine [12]. SpeciFcally, the example from (2.50) and Fig. 2.5 represents the most common case with just two equilibrium heat baths at two di5erent temperatures. The fact that such a device can produce work is therefore not a miracle but still amazing. At this point it is crucial to recognize that there is also one fundamental di5erence between the usual types of heat engines and a Brownian motor as exempliFed by the temperature ratchet: To this end we Frst note that the two “relevant state variables” of our present system are x(t) and T (t). In the case of an ordinary heat engine, these state variables would always cycle through one and the same periodic sequence of events (“working strokes”). In other words, the evolutions of the state variables x(t) and T (t) would be tightly coupled together (interlocked, synchronized). As a consequence, a single suitably deFned e5ective state variable would actually be suCcient to describe the system. 15 In contrast to this standard scenario, the relevant state variables of a genuine Brownian motor are loosely coupled: Of course, some degree of interaction is indispensable for the functioning of the Brownian motor, but while T (t) completes one temperature cycle, x(t) may evolve in several essentially di5erent ways (it is not “slaved” by T (t)).

12

A molecular motor is a very complex enzyme with a huge number of degrees of freedom (see Section 7). Within the present temperature ratchet model, the ATP hydrolyzation energy is thought to be quickly converted into a very irregular vibrational motion of these degrees of freedom, i.e. a locally increased apparent temperature. As this excess heat spreads out, the temperature decreases again. Thus, the internal degrees of freedom play a crucial role but are irrelevant in so far as they do not give rise to any additional slow, collective state variable. 13 We also note that a current x ˙ opposite to the force F is not in contradiction with any kind of “stability criteria”, cf. the discussion below (2.39). 14 In passing we notice that the case F = 0 in conjunction with a time-dependent temperature T (t) is conceptually quite interesting: It describes a system which is at any given instant of time an equilibrium system in a non-equilibrium (typically far from equilibrium) state. 15 Note that a Fxed sequence of events does not necessarily imply a deterministic evolution in time. In particular, small (“microscopic”) =uctuations which can be described by some environmental (equilibrium or not) noise are still admissible.

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The loose coupling between state variables is a salient point which makes the Brownian motor concept more than just a cute new look at certain very small and primitive, but otherwise quite conventional thermo-mechanical or even purely mechanical engines. In most cases of practical relevance, the presence of some amount of (not necessarily thermal) random =uctuations is therefore an indispensable ingredient of a genuine Brownian motor; exceptionally, deterministic chaos may be a substitute (cf. Sections 5.8 and 5.12.2). We remark that most of the speciFc ratchet models that we will consider later on do have a second relevant state variable besides 16 x(t). One prominent exception are the so-called Seebeck ratches, treated in Section 6.1. In such a case the above condition of a loose coupling between state variables is clearly meaningless. This does, however, not imply that those are no genuine Brownian motors. The important issue of whether the coupling between relevant state variables is loose or tight has been mostly discussed in the context of molecular motors [12,16,144] and has been given the suggestive name mechanochemical coupling, see also Sections 7.4.3 and 7.7. The general fact that such couplings of non-equilibrium enzymatic reactions to mechanical currents play a crucial role for numerous cellular transport processes is long known [23,24]. 2.8. Curie’s principle The main, and a priori quite counterintuitive observation from Section 2.1 is the fact that no preferential direction of the random dynamics (2.5), (2.6) arises in spite of the broken spatial symmetry of the system. The next surprising observation from Section 2.6 is the appearance of the ratchet e5ect, i.e. of a Fnite current x, ˙ for the slightly modiFed temperature ratchet model (2.6), (2.48) in spite of the absence of any macroscopic static forces, gradients (of temperature, concentration, chemical potentials etc.), or biased time-dependent perturbations. Here the word “macroscopic” refers to “coarse grained” e5ects that manifest themselves over many spatial periods L. Of course, on the “microscopic” scale, a static gradient force −V (x) is acting in (2.6), but that averages out to zero for displacements by multiples of L. Similarly, at most time instants t, a non-vanishing thermal force (t) is acting in (2.6), but again that averages out to zero over long times or when an entire statistical ensemble is considered. The Frst observation, i.e. the absence of a current at thermal equilibrium, is a consequence of the second law of thermodynamics. In the second above-mentioned situation, giving rise to a ratchet e5ect, this law is no longer applicable, since the system is not in a thermal equilibrium state. So, in the absence of this and any other prohibitive a priori reason, and in view of the fact that, after all, the spatial symmetry of the system is broken, the manifestation of a preferential direction for the particle motion appears to be an almost unavoidable educated guess. This common sense hypothesis, namely that if a certain phenomenon is not ruled out by symmetries then it will occur, is called Curie’s principle 17 [147]. More precisely, the principle postulates 16 While this second state variable obviously in=uences x(t) in some or the other way, a corresponding back-reaction may or may not exist. The latter case is exempliFed by the temperature ratchet model. 17 In the biophysical literature [23,24] the notion of Curie’s principle (or Curie–Prigogine’s principle) is mostly used for its implications in the special case of linear response theory (transport close to equilibrium) in isotropic systems, stating that a force can couple only to currents of the same tensorial order, see also [145,146].

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the absence of accidental symmetries in the generic case. That is, an accidental symmetry may still occur as an exceptional coincidence or by Fne-tuning of parameters, but typically it will not. Accidental symmetries are structurally unstable, an arbitrarily small perturbation destroys them [12], while a broken symmetry is a structurally stable situation. In this context it may be worth noting that the absence of a ratchet e5ect at thermal equilibrium in spite of the spatial asymmetry is no contradiction to Curie’s principle: The very condition for a system to be at thermal equilibrium can also be expressed in the form of a symmetry condition, namely the so-called detailed balance symmetry 18 [98–101,148–152]. 2.9. Brillouin’s paradox As mentioned in Section 2.1.1, both Smoluchowski and Feynman have already pointed out the close similarity of the ratchet and pawl gadget from Fig. 2.1 with a Maxwell demon and also with the behavior of a mechanical valve. But also the analogy of such a ratchet device with an electrical rectiFer, especially the asymmetric response to an external static force Feld (cf. Fig. 2.4), has been pointed out in Feynman’s Lectures [2], see also Vol. III, Section 14-4 therein. In this modiFed context of an electrical rectiFer, the astonishing fact that random thermal =uctuations cannot be rectiFed into useful work is called Brillouin’s paradox [3] and has been extensively discussed, e.g. in [100,101,153–157]. The main point of this discussion can be most easily understood by comparison with the corresponding tilted Smoluchowski–Feynman ratchet model (2.34). Furthermore, we focus on the case of an electrical circuit with a semiconductor diode. 19 With the entire circuit being kept at thermal equilibrium, at any Fnite temperature and conductance, a random electrical noise arises and it is prima facie indeed quite surprising that its rectiFcation by the diode is impossible. The stepping stone becomes apparent in the corresponding Smoluchowski–Feynman ratchet model (2.34). While its response to an external force F in Eq. (2.37) and Fig. 2.4 shares the typical asymmetric shape with a diode, it is now clearly wrong to phenomenologically describe the e5ect of the thermal noise in such a system by simply averaging the current x ˙ from (2.37) with respect to F according to the probability with which the thermal noise takes these values F. Rather, the correct modeling, which in particular consistently incorporates the common microscopic origin of friction and thermal noise, is represented by (2.34) (with F = 0). In contrast, the response characteristics (2.37) is already the result of an averaging over the thermal noise under the additional assumption that F is practically constant on the typical transient time scales of the emerging current x. ˙ It is clear, that we do not recover the full-=edged noisy dynamics (2.34) by replacing phenomenologically F by (t) in (2.37), notwithstanding the fact that in (2.34) these two quantities indeed appear in the same way. The close analogy of this situation with that in a semiconductor diode becomes apparent by considering that also in the latter case the asymmetric response characteristics is the result of a thermal di5usion process of the electrons near the interface of the n–p junction under quasi-static conditions and after averaging out the thermal =uctuations. 18

To be precise, detailed balance is necessary but not suCcient for thermal equilibrium [101,148]. Conversely, detailed balance is suCcient but not necessary for a vanishing particle current x . ˙ 19 A tube diode requires permanent heating and it is not obvious how to reconcile this with the condition of thermal equilibrium.

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This example (see also [158] for another such example) demonstrates that the correct modeling of the thermal environment is not always obvious. Especially, taking the averaged macroscopic behavior of the system as a starting point for a phenomenological modeling of the noisy dynamics may be dangerous outside the linear response regime, as van Kampen and others are emphasizing since many years [100]. Much safer is a microscopic starting point in order to consistently capture the common origin of the dissipation and the =uctuations in the actual system of interest, as exempliFed in Sections 2.1.2, 3.4.1 and 8.1. Away from thermal equilibrium, the realization of the ratchet e5ect by diodes and other semiconductor heterostructures is further discussed in Sections 6.1 and 8.4. 2.10. Asymptotic analysis In the remainder of this section, we continue our exploration of the temperature ratchet model (2.34), (2.47), (2.48) with the objective to understand in somewhat more detail the behavior of the particle current x ˙ at zero load F = 0 as a function of various parameters of the model. Since a closed analytical solution of the Fokker–Planck equation (2.52) is not possible in general, we have to recourse to asymptotic expansions and qualitative physical arguments, complemented by accurate numerical results for a few typical cases. In the present, somewhat more technical section we analyze the behavior of the particle current for asymptotically slow and fast temperature oscillations. For asymptotically slow temporal oscillations in (2.48) the time- and ensemble-averaged particle current x ˙ approaches zero. 20 Considering that T → ∞ means a constant T (t) during any given, Fnite time interval, this conclusion x ˙ → 0 is physically quite obvious. It can also be formally conFrmed by the observation that ad Pˆ (x; t) := Z(t)−1 e−V (x)=kB T (t) (2.55) L with Z(t) := 0 d x e−V (x)=kB T (t) solves the Fokker–Planck equation (2.52) in arbitrarily good approximation for suCciently large T and F =0. Comparison with (2.31) shows that this so-called adiabatic approximation (2.55) represents an accompanying or instantaneous equilibrium solution in which the time t merely plays the role of a parameter. Introducing (2.55) into (2.53) with F =0 indeed conFrms the expected result x ˙ = 0. Turning to Fnite but still large T, one expects that x ˙ decreases proportional to T−1 in the general case. In the special case that T (t) is symmetric under time inversion 21 , as for instance in (2.50), (2.51), the current x ˙ must be an even function of T and thus typically decreases − 2 proportional to T for large T. Furthermore, our considerations along the lines of Fig. 2.6 suggest that, at least for potentials with only one maximum and minimum per period L, the current x ˙ approaches zero from above if the minimum is closer to the adjacent maximum to the right, and from below otherwise. Here and in the following, we tacitly assume that apart form the variation of

20

In the following we tacitly restrict ourselves to smooth T (t), like e.g. in (2.51). For discontinuous T (t), for instance (2.50), the conclusion x ˙ → 0 for T → ∞ remains valid, but the reasoning has to be modiFed. 21 Time inversion symmetry means that there is a Ut such that T (−t) = T (t + Ut) for all t.

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the time-period T itself, the shape of T (t) does not change, i.e. Tˆ (h) := T (Th) ;

(2.56)

is a T-independent function of its dimensionless argument h with period 1. Addressing small T, i.e. fast temperature oscillations, it is physically plausible that the system cannot follow any more these oscillations and thus behaves for T → 0 like in the presence of a constant averaged temperature T 1 1 W T := dt T (t) = dh Tˆ (h) : (2.57) T 0 0 Within this so-called sudden approximation we thus recover an e5ective Smoluchowski–Feynman ratchet dynamics (2.6). In other words, we expect that x ˙ → 0 for T → 0. This behavior is conFrmed by the analytical perturbation calculation in Appendix C, which yields moreover the leading-order small-T result [118] L 2 x ˙ =T B d x V (x)[V (x)]2 + O(T3 ) ; (2.58) 1

0

h ˆ − Tˆ (h)= ˆ TW )]2 4L 0 dh[ 0 d h(1 B := L : L

3 0 d x eV (x)=kB TW 0 d x e−V (x)=kB TW

(2.59)

Note that B is a strictly positive functional of T (t) and V (x) and is independent of T. The most remarkable feature of (2.58) is that there is no contribution proportional to T independently of whether T (t) is symmetric under time inversion or not. More according to our expectation is the fact that the current vanishes for very weak thermal noise, as a closer inspection of (2.58) implies: Similarly as for the weak noise analysis in Section 2.5.1, for TW → 0 the particles can never leave the local minima of the potential V (x). In the opposite limit, i.e. for TW → ∞, the potential should play no role any more and one expects again that x ˙ → 0, cf. Section 3.7. A more careful perturbative analysis of the high-temperature limit conFrms this expectation. On the other hand, Eq. (2.58) predicts a Fnite limit for TW → ∞, implying that the limits TW → ∞ and T → 0 cannot be interchanged in this perturbative result. In other words, the correction of order O(T3 ) in (2.58) approaches zero for any Fnite TW as T → 0, but is no longer negligible if we keep T Fxed (however small) and let TW → ∞. The above predictions are compared with accurate numerical solutions in Fig. 2.7 for a representative case, showing very good agreement. 2.11. Current inversions The most basic qualitative prediction, namely that generically x ˙ = 0, is a consequence of Curie’s principle. In this section we show that under more general conditions than in Section 2.6.3, even the sign of the current x ˙ may be already very diCcult to understand on simple intuitive grounds, not to speak of its quantitative value. This leads us to another basic phenomenon in Brownian motor systems, namely the inversion of the current direction upon variation of a system parameter. Early observations of this e5ect have been reported in [35,37,39,42,159,160]; here we illustrate it once more for our standard example of the temperature ratchet.

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<x>

0.03

.

0.02

0.01

0 0.1

1

10

Fig. 2.7. Average particle current x ˙ versus period T for the temperature ratchet dynamics (2.3), (2.34), (2.47), (2.51) in dimensionless units (see Section A.4 in Appendix A). Parameter values are F = 0, = L = kB = 1, V0 = 1=2 , TW = 0:1, A = 0:7. Solid: Time- and ensemble-averaged current (2.53) by numerically evolving the Fokker–Planck equation (2.52) until transients have died out. Dotted: Analytical small-T asymptotics from (2.58).

Since the quantity B from (2.59) is positive, it is the sign of the integral in (2.58) which determines the direction of the current. For the speciFc ratchet potential (see Eq. (2.3) and Fig. 2.2) used in Fig. 2.7 this sign is positive, but one can easily Fnd other potentials V (x) for which this sign is negative. By continuously deforming one potential into the other one can infer that there must exist an intermediate V (x) with the property that the particle current x ˙ is zero at some Fnite T-value. In the generic case, the x-curve ˙ passes with a Fnite slope through this zero-point, implying [118] the existence of a so-called “current inversion” of x ˙ as a function of T. An example of a potential V (x) exhibiting such a current inversion is plotted in Fig. 2.8 and the resulting current in Fig. 2.9. As compared to the example from Fig. 2.2, the modiFcation of the ratchet potential in Fig. 2.8 looks rather harmless. Especially, the explanation of a positive current x¿0 ˙ for large T according to Fig. 2.5 still applies. However, for small-to-moderate T this modiFcation of the potential has dramatic consequences for the current in Fig. 2.9 as compared to Fig. 2.7. Once a current inversion upon variation of one parameter of the model (T in our case) has been established, an inversion upon variation of any other parameter (for instance the friction coeCcient

) can be inferred along the following line of reasoning [161]: Consider a current inversion upon variation of T, say at T0 , as given, while is kept Fxed, say at 0 . Let us next consider T as Fxed to T0 and instead vary about 0 . In the generic case the current x ˙ as a function of will then go through its zero-point at 0 with a Fnite slope, meaning that we have obtained the proposed current inversion upon variation of , see Fig. 2.10. In other words, Brownian particles with di5erent sizes will have di5erent friction coeCcients and will thus move in opposite directions when exposed to the same thermal environment and the same ratchet potential. Had we not neglected the inertia e5ects m x(t) M in (2.1), such a particle separation mechanism also with respect to the mass m could be inferred along the above line of reasoning, and similarly for any other dynamically relevant particle properties.

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2

0.015

0.01

<x>

V ( x ) / V0

1

0

.

0.005

-1 0

-2 -1

-0.5

0

0.5

1

0.1

x/ L

1

10

Fig. 2.8. The ratchet potential V (x) = V0 [sin(2 x=L) + 0:2 sin(4 (x=L − 0:45)) + 0:1 sin(6 (x=L − 0:45))]. Fig. 2.9. Same as Fig. 2.7 but for the ratchet potential from Fig. 2.8.

0.001

<x>

0.0005

.

0

-0.0005

1

1. 5

2

2.5

η

Fig. 2.10. Same as in Fig. 2.9 but with a Fxed period T = 0:17 (i.e. close to the inversion point in Fig. 2.9) and instead with a varying friction coeCcient .

Promising applications of such current inversion e5ects for particle separation methods, based on the ratchet e5ect, are obvious. Another interesting aspect of current inversions arises from the observation that structurally very similar molecular motors may travel in opposite directions on the same intracellular Flament (see Section 7). If we accept the temperature ratchet as a crude qualitative model in this context (cf. Section 2.6.3), it is amusing to note that also this feature can be qualitatively reproduced: If two types of molecular motors are known to di5er in their ATP consumption rate 1=T, or in their friction coeCcient , or in any other parameter appearing in our

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temperature ratchet model, then it is possible 22 to Fgure out a ratchet potential V (x) such that they move indeed in opposite directions. For a more general discussion of current inversion e5ects we refer to Section 3.6 below. Additional material on the temperature ratchet model is contained in Section 6.3.

3. General framework In Sections 3–9 we will review theoretical extensions and their experimental realizations of the concepts which were introduced by means of particularly simple examples in Section 2. In the present section, we provide a Frst overview and general framework for the more detailed discussion in the subsequent sections: The main classes of ratchet models and their physical origin are introduced. Symmetry considerations regarding the occurrence or not of a Fnite particle current (ratchet e5ect) are a second important issue, complemented by a general method of tailoring current inversions. Finally, a general treatment is provided for the asymptotic regimes of small and large noise-strength and of weak non-equilibrium perturbations. SpeciFc examples and applications of these general concepts are mostly postponed to later sections. 3.1. Working model In hindsight, the essential ingredient of the ratchet e5ect from Section 2.6.2 was a modiFcation of the Smoluchowski–Feynman ratchet model (2.6) so as to permanently keep the system away from thermal equilibrium. We have exempliFed this procedure by a periodic variation of the temperature (2.48) but there clearly exists a great variety of other options. In view of this example, the following guiding principles should be observed also in more general cases: (i) We require spatial periodicity and either invariance or periodicity under translations in time. (ii) All forces and gradients have to vanish after averaging over space (“coarse graining” over many spatial periods), over time (in the case of temporal periodicity), and over statistical ensembles (in the case of random =uctuations). (iii) The system has to be driven permanently out of thermal equilibrium and there should be no symmetries which prohibit a ratchet e5ect a priori. According to Curie’s principle we can therefore expect the generic appearance of a Fnite particle current x. ˙ (iv) In view of the title of our present study, we will mostly (not exclusively) focus on models with a Fnite amount of thermal noise. 23 According to these preliminary considerations, we adopt as our basic working model the overdamped one-dimensional stochastic dynamics

22 23

x(t) ˙ = −V (x(t); f(t)) + y(t) + F + (t) ;

(3.1)

(t)(s) = 2 kB T(t − s) ;

(3.2)

See also Section 3.6 for a detailed proof. Note that (iv) is not a consequence of (iii), as demonstrated by any dissipative driven system at zero temperature.

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where is the viscous friction coeCcient and V (x; f) := V (x; f)=9x. With respect to its spatial argument x, the potential is periodic for all possible arguments f(t), i.e. V (x + L; f(t)) = V (x; f(t))

(3.3)

for all t and x. Along the same line of reasoning as in Section A.4 of Appendix A, inertia effects are neglected and thermal =uctuations are modeled by uncorrelated (white) Gaussian noise (t) of zero average and intensity 2 kB T (see also Section 3.4.1 below). Finally, F is a constant “load” force. Since such a bias violates the above requirement (ii), it should not be considered as part of the system but rather as an externally imposed perturbation in order to study its response behavior. We furthermore assume that f(t) and y(t) are either periodic or stochastic functions of time t. In the case that one or both of them are a stochastic process, we make the simplifying assumption that this process is stationary, and in particular statistically independent of the thermal noise (t) and of the state of system x(t). With the symbol · we henceforth indicate an ensemble average over realizations of the stochastic dynamics (3.1), i.e. a statistical average with respect to the thermal noise (t) and in addition with respect to f(t) and=or y(t) if either of them is a stochastic process. The quantity of central interest is the average particle current (cf. (2.15)) x ˙ := x(t) ˙ :

(3.4)

In most cases 24 we will furthermore focus on the behavior of the particle current in the long-time limit t → ∞ (cf. Section 2.4). If both f(t) and y(t) are random processes in time, then the existence of a stationary long-time limit and its uniqueness are taken for granted. If f(t) and=or y(t) is a periodic function of t, then the existence of a unique periodic long-time behavior is assumed and a time average is tacitly incorporated into x ˙ (cf. Eq. (2.53)). Both, for random and periodic processes, this long time limit of the current can usually be identiFed, due to ergodicity reasons, with the time-averaged velocity of a single realization x(t) of the stochastic dynamics (3.1), i.e. with probability 1 we have that x ˙ = lim

t→∞

x(t) ; t

(3.5)

independent of the initial condition 25 x(0).

24

There are only very few investigations on transient features of ratchet systems [162–167]. −1 t Proof. The time-averaged current from (3.4) can be rewritten as x =lim ˙ x(t ˙ ) dt =limt → ∞ x(t)−x(0) =t. t→∞ t 0 The random process x(t)√− x(0) exhibits on top of the systematic drift x(t) − x(0) =t a certain random dispersion (or di5usion) of the order 2De5 t for large t, cf. Eq. (3.7). Due to the division by t it follows that this dispersion is negligible, i.e. x ˙ = limt → ∞ [x(t) − x(0)]=t = limt → ∞ x(t)=t with probability 1 for any realization x(t). 25

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A further quantity of interest is the e1ective di1usion coe
De5 := lim

(3.6)

For −V (x(t); f(t))+y(t) ≡ 0, the e5ective di5usion coeCcient (3.6) agrees with the bare coeCcient (2.11), independent of F. In general, its determination is a diCcult time-dependent problems 26 and we will restrict ourselves to a few special cases. On a suCciently coarse grained scale in space, the motion of the particle x(t) takes the form of single “hopping” events which are independent of each other and equally distributed. According to the central limit theorem [100], a statistical ensemble of particles x(t) with initial condition x(0) = x0 thus approaches a Gaussian distribution [117,168–174] 1 [xt ˙ − x 0 ]2 √ (3.7) P(x; t) exp − 4De5 t 4 De5 t for large times t. As far as the objective of particle separation is concerned, we see that not only a large di5erence or even opposite sign of the velocities x ˙ is important (cf. Section 3.6), but also the e5ective di5usion coeCcients and the time t (or, equivalently, the length xt ˙ of the experimental device) play a crucial role [34,170,172,174,175] see also Section 6.6. A purely di5usive (x ˙ = 0) particle separation scheme will be discussed in Section 5.11. Once in a while, certain extensions of the above framework will appear, e.g. an additional Fnite inertia term m x(t) M on the left-hand side of (3.1) or two instead of one spatial dimensions, see e.g. in Sections 5.8 and 5.9, respectively. Furthermore, models with a time- or space-dependent temperature in (3.2) will be discussed in Sections 6.1– 6.3, and similarly in Section 6.4) models with a timeor space-dependent friction. Deviations of the spatial periodicity (3.3) may arise in the form of some amount of quenched spatial disorder (Section 6.8) or as a superposition of several periodic contributions with incommensurate periods (Section 4.5.1). The case of a spatially discretized state variable is reviewed in Section 6.7. A class of models with a non-trivial dependence of the process f(t) upon the state x(t) of the system appears in Section 6.2 and in Section 7. Generalizations of a more drastic nature are addressed in Sections 8 and 9. If y(t) is a periodic function of time, say y(t + T) = y(t) ; then we can assume without loss of generality, that T dt y(t) = 0 ; 0

(3.8)

(3.9)

26 While for the current it is suCcient to consider an auxiliary dynamics with periodic boundary conditions, which approaches a stationary (if f(t) and y(t) are random processes) or periodic long time limit (cf. Section 2.4), no such simpliFcation is possible with respect to the e5ective di5usion coeCcient. In particular, the e5ective di5usion coeCcient is in general no longer related to the mobility via a generalized Einstein relation (2.11), i.e. De5 = kB T 9x =9F ˙ only holds when f(t) ≡ 0, y(t) ≡ 0, and F = 0.

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thanks to the free constant F in (3.1). Similarly, if y(t) is a stationary stochastic process then we can assume that y(t) = 0 :

(3.10)

Without loss of generality, it is also suCcient to concentrate on f(t) which are unbiased in the same sense as in (3.9), (3.10). As far as unbiased stationary random processes are concerned, two examples are of particular importance due to their archetypal simplicity. To be speciFc, we will use the symbol f(t), while completely analogous considerations can of course be immediately transcribed to y(t) as well. The Frst example is a so-called symmetric dichotomous noise or telegraphic noise [63,176 –178], i.e. a stochastic process which switches back and forth between two possible “states” +# and −# with a constant probability $ per time unit. In the stationary state the distribution of the noise %(f) := (f − f(t))

(3.11)

is thus given by %(f) = 12 [(f − #) + (f + #)] ;

(3.12)

independent of the time t in (3.11). One furthermore Fnds that the correlation is given by f(t) f(s) = #2 e−|t −s|=& ; where & := 12 $ is the correlation time and 27 ∞ #2 := f2 (t) = dff2 %(f) −∞

(3.13)

(3.14)

is the variance (independent of t). Being abundant in natural systems as well as in technological applications, a stationary Gaussian distributed noise f(t) is clearly a second type of random =uctuations that warrants to be analyzed in more detail. In the simplest case, these stationary Gaussian =uctuations are furthermore unbiased, and Markovian. 28 According to Doob’s theorem [100], f(t) is thus a so-called Ornstein–Uhlenbeck process [99,101], characterized by a stationary probability distribution %(f) = (2 #2 )−1=2 e−f

2

=2#2

(3.15)

and the same correlation as in (3.13). So, the variance #2 and the correlation time & are the model parameters for both, dichotomous noise and Ornstein–Uhlenbeck noise.

27 28

Note that # in (3.14) is consistent with (3.12) and (3.15). The future of f(t) only depends on its present state, not on its past [101].

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3.2. Symmetry 3.2.1. De8nitions The potential V (x; f(t)) is called spatially symmetric or simply symmetric if there exists a Ux such that V (−x; f(t)) = V (x + Ux; f(t))

(3.16)

for all x and t. In other words, we will use the notions “symmetry” and “asymmetry” as synonyms for “spatial isotropy” and “anisotropy”, respectively. A further important symmetry regards the unbiased tilting process y(t): If y(t) is periodic in time and there exists a Ut such that −y(t) = y(t + Ut) for all t then we call y(t) “inversion symmetric” or simply symmetric. By performing the symmetry transformation twice, it follows that y(t) = y(t + 2Ut) and under the assumption that T is the fundamental time-period, i.e. the smallest &¿0 such that y(t + &) = y(t), the symmetry condition takes the form [39] − y(t) = y(t + T=2) :

(3.17)

If y(t) is a stationary stochastic process, then we call it symmetric if all statistical properties of the process −y(t) are the same as those of y(t), symbolically indicated as − y(t) = y(t) :

(3.18)

Examples are the symmetric dichotomous noise and the Ornstein–Uhlenbeck process as introduced at the end of the preceding subsection, or the symmetric Poissonian shot noise from Section 5.5. Note that the assumption of an unbiased y(t), see (3.9), (3.10), does not yet imply that y(t) is symmetric. Regarding nomenclature, an asymmetric potential is also called a ratchet potential. On the other hand, the dynamics (3.1) will be termed Brownian motor, ratchet dynamics, or simply ratchet not only if the potential V (−x; f(t)) is asymmetric but also if the driving y(t) is asymmetric, while the potential may then be symmetric. 3.2.2. Conclusions From the deFnition (3.16) it follows that a L-periodic potential V (x; f(t)) is symmetric if and only if it is of the general form V (x; f(t)) =

∞

an (f(t)) cos(2 nx=L) :

(3.19)

n=1

Here and in the following, trivial freedoms in the choice of the x- and V -origins are neglected. In the speciFc case (3.16) this means that we have silently set Ux = 0 and a0 (f(t)) = 0 in (3.19). Similarly, one sees that the symmetry condition (3.17) for a periodic, deterministic driving y(t) is equivalent to a Fourier representation of the general form 2 nt + )n : (3.20) bn cos y(t) = T n=1;3;5;:::

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In the case that y(t) is a stochastic process, the symmetry condition (3.18) is equivalent to the requirement that all its odd moments vanish [179,180], i.e. y(t1 ) y(t2 ) · · · y(t2n+1 ) = 0

(3.21)

for all integers n ¿ 0 and all times 29 t1 ; t2 ; : : : ; t2n+1 . Basically, the reason for this is that the stochastic process y(t) is completely speciFed by the set of all its multiple-time joint probability distributions (Kolmogorov-theorem) and those are in turn completely Fxed by all its moments [100]. On this basis, the equivalence of (3.18) and (3.21) follows. If the potential V (x; f(t)) respects the symmetry condition (3.19) and the driving y(t) either (3.20) or (3.21) then we can conclude that the long-time-averaged particle current (3.5) vanishes in the absence of a static tilt F in (3.1), i.e. x(t) x ˙ = lim = 0: (3.22) t→∞ t For a proof, we recall that the current in (3.5) is independent of the initial condition x(0) and we may thus choose x(0) = 0. If both, V (x; f(t)) and y(t) are symmetric according to (3.16) – (3.18), or equivalently (3.19) – (3.21), then it follows that a realization x(t) of the random process (3.1), (3.2) with F = 0 in (3.1) occurs with the same probability as its mirror image −x(t). Hence, we can infer from (3.5) that x ˙ = −x, ˙ implying (3.22). In other words, the main conclusion of this subsection is that if both, the potential V (x; f(t)) and the driving y(t) are symmetric according to (3.16) – (3.21) then the average particle current (3.5) is zero. If the potential and the driving y(t) do not both satisfy their respective symmetry criteria, then, according to Curie’s principle, a Fnite average current is expected in the generic case. The exceptional (non-generic) cases with zero current (3.22) in spite of a broken symmetry are either in some sense “accidental” [12] (analogous to the current inversion in Fig. 2.9) or can be traced back to certain “hidden” symmetry reasons of a more fundamental and systematic nature. Examples of the latter type will be the subject of Sections 3.5 and 6.4.1, see also the concluding remarks in Section 10. The generalization of these symmetry considerations to the case of a quasiperiodic driving y(t) is due to [181], while an extension to two-dimensional systems (cf. Section 5.9) and models with an internal degree of freedom (cf. Section 6.5) is contained in [182,183] and [184], respectively. 3.3. Main ratchet types In this section we introduce the classiFcation scheme underlying the organization of Sections 4 – 6. Some general physical considerations complementing this abstract classiFcation are summarized in Section 3.4. As already discussed in Section 2.6, of foremost interest is usually the current x ˙ in the long-time limit in the absence of a static tilt F in (3.1). If both, the potential V (x; f(t)) and the tilting force y(t) are symmetric, then a vanishing current will be the result (see preceding subsection). The following classiFcation of the di5erent types of ratchet models is on the one hand, based on the systematic breaking of this symmetry, on the other hand, it follows to some extent the historically grown, non-systematic nomenclature. 29

Here and in what follows we tacitly assume that all multiple-time moments of the process y(t) exists.

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There are two fundamental classes of ratchet models arising from (3.1). The Frst one are models with y(t) ≡ 0, which we denote as pulsating ratchets. The second are models with f(t) ≡ 0, called tilting ratchets [49]. Within the realm of pulsating ratchets (y(t) ≡ 0), the Frst main subclass is obtained when f(t) in (3.1) is additive, i.e. V (x; f(t)) = V (x) [1 + f(t)] :

(3.23)

Such models carry the name 9uctuating potential ratchets. The summand 1 is a matter of convention, re=ecting a kind of “unperturbed” contribution to the total potential. The class of =uctuating potential ratchets contains as special case the on–o1 ratchets when f(t) can take only two possible values, one of them being −1 (potential “o5”). Without loss of generality, the other value can then be assumed to be +1. One readily sees that the potential V (x; f(t)) on the left-hand side of (3.23) satisFes the symmetry condition (3.16) if and only if V (x) on the right-hand side of (3.23) is symmetric as well. Furthermore, it is obvious that a symmetric V (x) in (3.23) always results in a vanishing current x, ˙ whatever the properties of f(t) are. We will therefore focus on the simplest non-trivial scenario, namely asymmetric potentials V (x) in combination with symmetric f(t). As the word “=uctuating potential” already suggests, we will mainly focus on random f(t), though periodic f(t) are in principle meant to be equally covered by this name. A second subclass of pulsating ratchets, called traveling potential ratchets, have potentials of the form V (x; f(t)) = V (x − f(t)) :

(3.24)

The most natural choice, already suggested by the name “traveling potential”, are f(t) with a systematic long time drift u := limt → ∞ f(t)=t. As a consequence, f(t) can only be a veritable periodic function or stationary stochastic process after subtraction of this systematic drift. We will call such a model a genuine traveling potential ratchet scheme. This slight extension of our general framework will be justiFed by our demonstration that such a model is exactly equivalent either to a tilting ratchet or to a so-called improper traveling potential ratchet, for which already the “original” f(t) is a periodic function or a stationary stochastic process. Within a traveling potential ratchet scheme, the potential V (x; f(t)) on the left-hand side of (3.24) never satisFes the symmetry criterion (3.16), independently of whether the potential V (x) on the right-hand side is symmetric or not. Both, the genuine and improper schemes are therefore interesting to study since a symmetric potential V (x) is su
(3.25)

in (3.1). When V (x) is a ratchet potential, then we will restrict ourselves mostly to symmetric y(t). If y(t) is a stochastic process, we speak of a 9uctuating force ratchet. The case of a tilting ratchet with a periodic driving y(t) is of particular experimental relevance and carries the obvious name rocking ratchet [42].

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Coming to symmetric potentials V (x) in (3.25), a broken symmetry of y(t) turns out to be necessary and generically also suCcient for a Fnite current x. ˙ We will use the name asymmetrically tilting ratchet if y(t) is not symmetric, independently of whether it is a periodic function or a stochastic process, and independently of whether V (x) is symmetric or not. A further important class of ratchets is given by models of form the (3.1), (3.3) with both f(t) ≡ 0 and y(t) ≡ 0 but instead with a space- or time-dependent temperature T in (3.2). They carry the names Seebeck ratchets and temperature ratchets, respectively. In the case of a space-dependent temperature, T (x) is assumed to have the same periodicity L as the potential V (x). In the case of a time-dependent temperature T (t), again a periodic or stochastic, stationary behavior is assumed. We anticipate that models of this type are obviously not pulsating ratchets in the original sense, but—as will be demonstrated in Sections 6:1 and 6:3—they can be mapped onto genuine pulsating ratchets. Also discussed in this context (Section 6.2) will be the so-called Feynman ratchets, i.e. the extension of the isothermal Smoluchowski–Feynman ratchet and pawl from Fig. 2.1 to the non-equilibrium case involving simultaneously two thermal baths at di5erent temperatures. Starting with a faithful two-dimensional model, which is in fact equivalent to a generalized =uctuating potential scheme, additional simpliFcations give rise to a one-dimensional, Seebeck ratchet-like approximative description. Finally, the case of a varying friction coeCcient in (2.11) (temporal and=or spatial) is denoted as friction ratchet. In Section 6.4.1, we show that such a modiFcation of the Smoluchowski–Feynman ratchet model (2.5), (2.6) does not break the detailed balance symmetry and thus does not admit a ratchet e5ect, 30 in contrast to a modiFed, so-called memory-friction modelling as discussed in Section 6.4.3. We remark that the main idea of the above classiFcation scheme is the identiFcation of di5erent basic minimal models. Clearly, there are many possible combinations and generalizations, e.g. a simultaneously pulsating and tilting ratchet or the simultaneous breaking of more than one symmetry. Especially, there exist numerous pulsating ratchet schemes involving potentials V (x; f(t)) which go beyond the special cases of =uctuating potential and traveling potential ratchets. Such generalizations will not be systematically analyzed since no fundamentally new phenomena are expected. They are, however, realized in some interesting experimental systems and will be discussed in such speci8c contexts. 3.4. Physical basis The physical situations in which a model of form (3.1) – (3.3) may arise are extremely diverse. Therefore, a systematic discussion makes little sense and we restrict ourselves in this section to a few general remarks before turning to the various concrete systems in the subsequent chapters. The stochastic process x(t) in (3.1) has as state space the entire real axis and for simplicity is often called a “Brownian particle”. While in some cases, x(t) indeed represents the position of a true physical particle, in others it may also refer to some quite di5erent type of collective degree of freedom or relevant (slow) state variable. Examples which we will encounter later on are the chemical reaction coordinate of an enzyme, the geometrical conFguration or some other internal degree of freedom of a molecule, the position of the circular ratchet in Fig. 2.1 with respect to 30

Especially, such a modiFcation requires a correct handling of the non-trivial overdamped limit m → 0 in (2.1), see Section 6.4.1.

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the pawl, the Josephson phase in a SQUID (superconducting quantum interference device), and the collective angular variable in phenomenological models for pinned charge density waves. In many cases the state variable x(t) is thus originally of a phase-like nature with a circle as state space. The expansion to the real axis is immediate and has the additional advantage of counting the number of revolutions. Accordingly, the periodicity (3:3)—a central property of our model—may have its root either in a true spatial periodicity of the physical system or in the phase-like nature of the original state variable. 3.4.1. Thermal environment Another central feature in our working model (3.1), (3.2) is the presence of a thermal environment. In this section we continue and extend our discussion from Section 2.1.2 (see also Section A.1 in Appendix A) regarding the physical origin of the particularly simple form of the system–bath interaction in (3.1), (3.2), namely an additive white Gaussian noise and an additive viscous dissipation proportional to the instantaneous system velocity. Adopting a phenomenological approach, in many cases [66,99,101] such an ansatz has proven to provide a rather faithful modeling, justiFed by its agreement with experimental measurements and the intuitive physical picture that has emerged on the basis of those observations. A di5erent approach starts with a microscopic modeling of the system of actual interest and its thermal environment. In the following we brie=y sketch the main steps of such an approach. For a somewhat more detailed illustration of these general concepts for speciFc physical examples we also refer to Sections 7:2, 7:3, and 8:1. On the one hand, such a microscopic foundation provides a physical picture of why the phenomenological modeling (3.1), (3.2) is successful in such a wide variety of di5erent systems. On the other hand, a feeling for the conditions under which such a modeling is valid is acquired as well as an idea of how to modify the model when they break down. Our starting point is a Hamiltonian of the general form H=

N pj2 p2 + Vb (x; x1 ; : : : ; xN ) ; + Vs (x) + 2m 2m j j=1

(3.26)

where x and p are the coordinate and momentum of the actual system of interest, while xj and pj are those of the numerous (N 1) microscopic degrees of freedom of the environment. The last term in (3.26) is a general interaction potential, including the coupling between system and environment. To keep things simple, we restrict ourselves to a single relevant (i.e. “slow”) state variable x(t), e.g., the cartesian coordinate of a particle in the absence of magnetic Felds or the Josephson phase in a SQUID. We remark that in other cases, e.g. the chemical reaction coordinate of an enzyme, the geometrical conFguration, or some other internal degree of freedom of a molecule, the respective “slow” relevant state variable x(t) is usually a generalized coordinate (a non-trivial function of the cartesian coordinates of the nuclei, cf. Section 7.2), and similarly for the “fast” bath degrees of freedom xj (t). As a consequence, the kinetic energy terms are of a more complicated form than in (3.26) and with respect to the potential terms there exists no longer a meaningful distinction between the “actual system of interest” and the “environment plus the system–bath coupling”. In those cases, our general line of reasoning remains still valid, but the detailed calculations become more involved [92,93,150,185].

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Elimination of the bath degrees of freedom: Having set the stage (3.26), our next goal is to get rid of the environmental degrees of freedom xj (t). To this end, we start by formally solving the respective equations of motions for any prescribed function x(t) and initial conditions )0 := (x1 (0); p1 (0); : : : ; xN (0); pN (0)) at time t0 = 0. In other words, we can write down (formal) solutions xj (t; [x(t )]; )0 ) which are at the same time functions of t and )0 and functionals of the (explicitly still unknown) system dynamics x(t ) for t ∈ [0; t]. Introducing these solutions into the equation of motion for the system x(t) is equivalent to a Newtonian dynamics of the general structure 31 m x(t) M = f(x(t); t; [x(t )]; )0 ) :

(3.27)

In most cases, an explicit analytical expression for f(x(t); t; [x(t )]; )0 ) is not available 32 since this would require analytical solutions xj (t; [x(t )]; )0 ) of a high dimensional chaotic dynamics and would in fact comprise the derivation of the basic principles of equilibrium statistical mechanics as special case. Rather, one proceeds the other way round, exploiting the fact that the environment is a thermal equilibrium heat bath and thus statistical mechanical principles can be invoked. Namely, one assumes that the systems initial conditions x(0) and p(0) are arbitrary but Fxed, while the initial state of the bath )0 is randomly sampled from a canonical probability distribution 33 P()0 ) ˙ exp{−H (x(0); p(0); )0 )=kB T }. It is via this randomness of the environmental initial conditions )0 that the system dynamics (3.27) acquires itself a stochastic nature. Denoting the average over those initial conditions by ˜ f(x(t); t; [x(t )]) := f(x(t); t; [x(t )]; )0 ) ;

(3.28)

we can decompose the right-hand side of (3.27) into a sum of three terms, m x(t) M = −V (x(t)) − h(x(t); t; [x(t ˙ )]) + (x(t); t; [x(t )]; )0 ) ;

(3.29)

where the Frst term is determined by the instantaneous state of the system, the second by its past history, and the third term is of microscopic origin, giving rise to the stochastic nature of the dynamics. Their explicit deFnitions are ˜ V (x(t)) := − f(x(t); t; [x(t ) ≡ x(t)]) ;

(3.30)

˜ ˜ h(x(t); t; [x(t ˙ )]) := − f(x(t); t; [x(t )]) + f(x(t); t; [x(t ) ≡ x(t)]) ;

(3.31)

˜ (x(t); t; [x(t )]; )0 ) := f(x(t); t; [x(t )]; )0 ) − f(x(t); t; [x(t )]) :

(3.32)

Here, [x(t ) ≡ x(t)] means that the function x(t ) keeps the same value x(t) for all times t ∈ [0; t] and is understood as a formal functional argument rather than an actual solution of the real system 31

The explicit but formal expression of f(x(t); t; [x(t )]; )0 ) in terms of the potentials in (3.26) and the formal solutions xj (t; [x(t )]; )0 ) is straightforward but of no further use, see below. Especially, f(x(t); t; [x(t )]; )0 ) in (3.27) has nothing to do with f(t) from (3.1). 32 The only solvable exception—the so-called harmonic oscillator bath—arises when Vb (x; x1 ; : : : ; xN ) in (3.26) is a quadratic function of its arguments and thus the bath-dynamics is not chaotic, see Section 8.1. 33 The physical origin of this canonical description is a “superbath” to which the bath of actual interest is weakly coupled.

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dynamics (3.29). Further, the modiFed functional argument [x(t ˙ )] on the left-hand side of (3.31) is justiFed by the fact that any function x(t ) with t ∈ [0; t] can be reconstructed from the knowledge of x(t) and x(t ˙ ). Finally, we remark that the source of randomness )0 enters via the “noise” (3.32), which has a vanishing mean value by construction. Observing that for x(t ) ≡ x(t) the bath keeps its initial canonical probability distribution and expressing the force on the right-hand side of (3.27) in terms of the potentials in (3.26) one can infer from (3.28) and (3.30) that V (x) = Vs (x) − kB T ln

N

d xj exp{−Vb (x; x1 ; : : : ; xN )=kB T } :

(3.33)

j=1

In general, the bare system potential is thus renormalized (dressed) by the eliminated degrees of freedom of the environment and plays a role similarly to a free energy rather than a (bare) energy [92,93,150,186]. However, if the potential Vb (x; x1 ; : : : ; xN ) is translation invariant (i.e. equal to Vb (x+ ; x1 +; : : : ; xN +) for all ) then the renormalization in (3.33) boils down to an irrelevant additive constant. Linearized friction and thermal #uctuations: While all so far formal manipulations are still exact, we Fnally make two approximations with respect to the “friction” term (3.31). First, we functionally expand h(x(t); t; [x(t ˙ )]) with respect to x(t ˙ ). Considering that h(x(t); t; [x(t ˙ ) ≡ 0]) = 0 (cf. (3.31)) and that t ∈ [0; t], the leading-order approximation is t h(x(t); t; [x(t ˙ ) ≡ 0]) x(s) ˙ : (3.34) h(x(t); t; [x(t ˙ )]) ds x(s) ˙ 0 Second, we exploit the assumed property that the relevant state variable x(t) changes “slowly” in comparison with the environment, hence x(s) ˙ x(t) ˙ for all s-values which notably contribute in (3.34) (Markov approximation). By closer inspection one sees that within the same approximation the remaining integral does no longer explicitly depend on t. As a result, we approximately Fnd a friction term of the following general form: h(x(t); t; [x(t ˙ )]) (x(t))x(t) ˙ :

(3.35)

As far as the omitted corrections on the right-hand side of (3.35) are not incidentally identically zero, by neglecting them we are tampering with the original equilibrium environment with the consequence of a (possibly very small but generically non-vanishing) breaking of thermal equilibrium and thus a violation of the second law of thermodynamics, see also Sections 2.9 and 8.1. This shortcoming can only be remedied by a corresponding adjustment of the =uctuations in (3.32) in the following way: Along a similar line of reasoning as in [97] (see also Section A.1 in Appendix A) one can show that the speciFc structure (3.29), (3.35) of the dynamics together with the requirement that the environment is at thermal equilibrium (respects the second law of thermodynamics) uniquely determine all statistical properties of those properly adjusted =uctuations appearing in (3.29). Namely, they are necessarily an unbiased Gaussian white noise whose correlations satisfy a =uctuation– dissipation relation of the form (x(t); t; [x(t )]; )0 )(x(s); s; [x(s )]; )0 ) = 2 (x(t))kB T(t − s) :

(3.36)

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As already noticed below (2.5), the function (x) may thus be viewed as the coupling strength to the thermal environment. If the potential Vb (x; x1 ; : : : ; xN ) is known to be translation invariant then not only the renormalization of the potential in (3.33) reduces to an irrelevant additive constant but also the spatial dependence of the friction coeCcient (x) disappears. In the overdamped limit m → 0 we thus exactly recover our “unperturbed” working model 34 (3.1), (3.2). This omission of the inertia term in (3.1) is usually a quite satisfactory approximation for the typically very small systems under consideration, cf. Section A.4 in Appendix A. A noteworthy exception is the case of a SQUID system, 35 for which both the overdamped limit (3.1) as well as the case with Fnite inertia describe realistic experimental situations of interest, see Section 5.10. The translation invariance of Vb (x; x1 ; : : : ; xN ) and thus the x-independence of the system–bath coupling arises naturally if the periodic potential in (3.1) and the thermal environment have di5erent physical origins. Since this is the case in most concrete examples which we will consider or at least it can be assumed without missing basic new e5ects, we will mostly focus on an x-independent friction coeCcient henceforth. Prominent examples with x-dependent friction coeCcients (x) are discussed in Sections 6.4.2 and 7.3. One basic assumption in our so far discussion has been the existence of a clear-cut separation between the characteristic time scales governing the “slow” system variable and those of the environment, with the consequence of a memoryless friction mechanism and uncorrelated thermal =uctuations. However, there exist physical systems for which this assumption is not fulFlled. One reason may be that one has overlooked additional relevant “slow” state variables and thus one simply has to go over to a higher dimensional vector x(t) in the above calculations. However, in some cases the necessary dimensionality of x(t) may become very high, while those additional dimensions are actually of no further interest, so that keeping a memory-friction and correlated noise may be more convenient. Restricting ourselves to the simpler case with a translation invariant potential Vb (x; x1 ; : : : ; xN ), approximation (3.34) takes the general form t h(x(t); t; [x(t ˙ )]) ds (t ˆ − s)x(s) ˙ ; (3.37) −∞

where we have assumed that x(t) ˙ = 0 for all 36 t 6 0 in order to uniquely deFne the evolution of the integro-di5erential equation (3.29), and hence the lower integration limit could been extended to −∞. Similarly as in (3.36), the assumption of thermal equilibrium then implies [97] that the properly adjusted =uctuations appearing in (3.29) are necessarily an unbiased Gaussian noise whose correlation satisFes a =uctuation–dissipation relation of the form (x(t); t; [x(t )]; )0 )(x(s); s; [x(s )]; )0 ) = (|t ˆ − s|)kB T :

(3.38)

Examples of this type will be discussed in Sections 6.4.3 and 8.1. 34 For the sake of notational simplicity only we have not included f(t); y(t), and F into the deFnition of Vs (x) form (3.26). 35 The reason is that the “e5ective inertia” in a SQUID has a “macroscopic” origin, namely the capacitance of the considered circuit, cf. Section 5.10. 36 This can be physically realized by means of a time-dependent potential Vs (x; t) in (3.26) which keeps x(t) at a Fxed position for t 6 0 and switches to the actual potential of interest for t¿0.

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It should be emphasized that the dynamics (3.29) reproduces the correct equilibrium distribution P(x; p) ˙ exp{−[p2 =2m + V (x)]=kB T } in the long-time limit, independently of the choice of (x) or (t) ˆ in (3.35) – (3.38). Especially, this distribution is exactly identical to the steady state result for the original system (3.30) – (3.32) before making any approximations [93]. Moreover, the second law of thermodynamics is strictly satisFed in all cases. It is only away from equilibrium that the speciFc choice of (x) or (t) ˆ becomes important 37 and that the approximations made in (3.35) – (3.37) may have a noticeable e5ect. In general, the above program of identifying “slow” and “fast” variables, establishing the microscopic model (3.26), and determining 38 (x) or (t) ˆ according to (3.34) cannot be practically carried out [93]. The same applies for a well-controlled justiFcation of the approximation (3.34), although this linearization turns out to provide remakably good approximations in a large variety of di5erent systems. One reason may be the fact that in most cases only terms of odd order in the system velocity will contribute to the omitted corrections on the right-hand side of (3.34) due to symmetry reasons. In view of those practical diCculties we are thus in some sense back at a phenomenological modeling which draws its legitimation from the comparison with experimental Fndings. However, as already mentioned, the microscopic modeling provides a general framework (functional form) for a large class of approximate models and a feeling for their wide range of applicability as well as for possible reasons in case they fail. 3.4.2. Non-equilibrium perturbations There are two main types of possible “perturbations” of the “unperturbed” equilibrium system (3.1) with f(t) ≡ 0; y(t) ≡ 0, and F = 0. The Frst acts essentially like the force F in (3.1), i.e. the system x gains (or looses) energy if it is displaced by one spatial period L. For instance, this may be a homogeneous force acting on a true Brownian particle or an angular momentum-type perturbation if x was originally of a phase-like nature. In any case, such a perturbation interacts directly with the state variable x. The unbiased, time-dependent part of such a perturbation gives rise to the “tilting force” y(t) and the systematic part to the “static force” F in (3.1). The second possible type of perturbations interacts directly with the system variable x but does not lead to an energy change if x is displaced by one period L. A simple example is an electrical dipole with a single rotational degree of freedom in a homogeneous electrical Feld. Another option is a perturbation which does not directly interact with the state variable x, but rather a5ects the physical mechanism responsible for the periodic potential in (3.1). Either some “internal degree of freedom” of the system x is excited, which modiFes the interaction with the periodic potential [13], or the periodic potential itself may be a5ected by the perturbation [187]. For instance, an electrical Feld may change the internal charge distribution (electrical polarization) of a neutral Brownian particle or of the periodic substrate with which it interacts. This type of perturbation gives rise to a “pulsating potential” V (x; f(t)) in (3.1). Depending on the details of the system, either one of the three basic types (=uctuating, improper traveling, or genuine traveling 37 An obvious example is the mobility in the absence of the system potential Vs (x), independently of whether the system is close to or far from equilibrium. Other observables which signiFcantly depend on the choice of (x) or (t) ˆ are escape rates (even so-called equilibrium rates) [66], as well as the particle current and the e5ective di5usion from (3.4) and (3.6). 38 The explicit determination of V (x) according to (3.33) may still be feasible.

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potential) arises in its pure form, or a combination thereof, possibly even with a tilting ratchet admixture, is encountered. One possible origin of those di5erent types of “perturbations” may be an experimentally applied external 8eld. While periodic signals then clearly represent the standard case, random perturbations have been realized as well [188]. Another possibility is a system-intrinsic source of “perturbations”, usually of stochastic nature. The origin of such an intrinsic noise source may be either a non-equilibrium heat bath or a second thermal heat bath 39 at a di5erent temperature than the (t)-bath. As far as the tilting ratchet scheme is concerned, the coexistence of such an extra heat bath and the thermal (t)-bath, which both interact directly with the state variable x but practically not with each other, may be experimentally tailored, but is not very common in natural systems. Exceptions are electrical circuits, where non-equilibrium =uctuations, e.g. dichotomous noise [193] or shot noise (see Vol. 1 of [115]) may naturally arise, and experimental analog electronic circuits for dichotomous [194,195] or Gaussian [196] colored noise. More common are sources of noise which manifest themselves via an internal degree of freedom and thus lead to a pulsating ratchet scheme. Examples are catalytic chemical reactions with reactant and product concentrations far from their equilibrium ratio, or excitations induced by electromagnetic irradiation. Another example is a modiFed Feynman ratchet as discussed in Section 6.2. In such cases, the coexistence of two practically independent sources of the noises (t) and f(t) in (3.1) is indeed realistic. At Frst glance, the property (3.3) that the potential is changing its shape in perfect synchrony over arbitrary distances x might appear somewhat strange. However, this is in fact very natural if either x is of a phase-like character or if the pulsating potential mechanism is caused by an internal degree of freedom of the system x. Also experimentally imposed external perturbations usually do not cause an asynchronous pulsating potential scheme. Asynchronously pulsating potentials [197–202] can only be expected if x is a space-like variable and if the potential is subjected to independent “local” non-equilibrium noise sources, or in a speciFcally tailored experimental setup. In the case of stochastic “perturbations” f(t) or y(t) in (3.1), we have assumed stationarity and especially x-independence of their statistical properties. Similarly as for the thermal noise (t), this re=ects the assumption that their origin is a “huge” heat bath which is practically not in=uenced by the behavior of the “small” system x(t). A more drastic assumption in (3.1) is the implicit omission of a back-coupling mechanism (“active decoupling”) [11,12,15] to the f(t)- or y(t)-heat bath, analogous to the dissipation mechanism in the case of the equilibrium (t)-bath. This means that the coupling to this former bath is very weak and that this bath is very far away (“highly excited”) from equilibrium with respect to the (t)-bath at temperature T . Only then, the e5ect of the =uctuations f(t) or y(t) are still appreciable while the corresponding back-coupling e5ects are negligible. 40 In Section 7 we will encounter a speciFc model where such a back-coupling mechanism is fully taken into account (see Section 7.3.1). Furthermore, it will be demonstrated explicitly how 39

Microscopic models for two (or more) coexisting thermal heat baths at di5erent temperatures have been discussed in [189 –192]. In the case of a tilting ratchet scheme it turns out that a ratchet e5ect (cf. Section 2.6.2) is only possible for a correlated (non-white) thermal noise y(t) and a concomitant memory friction term, see Section 6.4.3. A generalization of these microscopic models to the case that one bath is out of equilibrium is also possible. 40 For example, the origin of f(t) or y(t) may be a second thermal equilibrium bath at a temperature much higher than T . Though such a model may not be very realistic it is of great conceptual appeal as one of the simplest models for a system far from equilibrium [189]: Two thermal equilibrium baths are connected through a single degree of freedom x(t) and can be exploited to do work. A concrete example is the Feynman ratchet in Section 6.2.

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this back-coupling may again become negligible as the corresponding source of noise is driven far away from equilibrium (see Section 7.4.2). Another example with a non-trivial back-coupling appears in Section 6.2. Note that if f(t) or y(t) are such that a periodic perturbation of the system arises, then those back-coupling e5ects are also omitted in our model (3.1) but in this case such an omission is very common. The actual justiFcation for doing so, however, follows in fact along the same line of reasoning as for random perturbations. In all those various cases, it is clear that the system can never reach a thermal equilibrium state even in the long-time limit: either this is prohibited by a permanent periodic perturbation or a second heat bath out of equilibrium or at equilibrium but with a temperature di5erent from T . In either case, the second law of thermodynamics cannot be applied, i.e. the symmetry of detailed balance is violated. In the absence of any other prohibitive symmetries, which we have systematically broken by our classiFcation scheme (cf. Section 3.3), we thus expect the generic occurrence of the ratchet e5ect x ˙ = 0 according to Curie’s principle. The corresponding intuitive microscopic picture is a permanent energy =ow from the source of the perturbations f(t) or y(t)—be it a periodic external driving or a second heat bath—into the thermal bath at temperature T via the single common degree of freedom x(t). 3.5. Supersymmetry In this section we continue our symmetry considerations from Section 3.2, where we have seen that breaking thermal equilibrium, or equivalently, breaking the symmetry of detailed balance in whatever way, in a periodic, asymmetric system, is generically suCcient for the ratchet e5ect to manifest itself: In general, the occurrence of a Fnite current in such systems is the rule rather than the exception, in accord with Curie’s principle. We thus more and more return to Smoluchowski and Feynman’s point of view that away from thermal equilibrium, the absence rather than the presence of directed transport in spite of a broken symmetry is the truly astonishing situation. In this section, an entire class of such intriguing exceptional cases is identiFed which do not exhibit a ratchet e5ect in spite of broken thermal equilibrium and broken symmetry. Especially, such systems (cf. Eqs. (3.1), (3.2) with F = 0) exhibit zero current x ˙ for any choice of the friction , the temperature T , the amplitude and characteristic time scale of the drivings f(t) and y(t), etc., much like the symmetric systems from Section 3.2. In contrast to usual current inversions (cf. Sections 2.11 and 3.6), no Fne-tuning of those parameters is thus required in order that x ˙ = 0. 3.5.1. De8nitions We begin with the following deFnitions: We call a potential V (x; f(t)) with a periodic function f(t) supersymmetric if there exist Ux; Ut; UV such that −V (x; f(t))=V (x+Ux; f(−t +Ut))+UV for all x and t. If f(t) is a stochastic process then we call the potential V (x; f(t)) supersymmetric if for any x all statistical properties of −V (x; f(t)) and V (x + Ux; f(−t)) + UV are the same (no Ut is needed since f(t) is stationary). Especially, a static potential is supersymmetric if −V (x)= V (x + Ux) + UV for all x. Note that while we can and will choose the t- and V -origins such that Ut = 0 and UV = 0, the same is not possible for Ux. In fact, by applying the above deFned supersymmetry transformation twice, we can conclude that V (x + 2Ux; f(t)) = V (x; f(t)) for all x and t. Under the assumption that L is the fundamental period of V (x; f(t)), i.e. the smallest z¿0 with

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V (x + z; f(t)) = V (x; f(t)), we can henceforth focus on Ux = L=2. In summary, the supersymmetry criterion can thus be symbolically indicated (cf. (3.18)) for both, periodic and stochastic f(t) as − V (x; f(t))=V ˆ (x + L=2; f(−t)) :

(3.39)

Turning to the driving y(t), we will call it supersymmetric if for a periodic y(t) we have that −y(t) = y(−t + Ut) for all t and an appropriate Ut, which can be transformed to zero as usual. For a stochastic y(t) we speak of supersymmetry if −y(t) and y(−t) are statistically equivalent. In other words, supersymmetry means for both, periodic and stochastic y(t), a parity-time-invariance of the form − y(t)=y(−t) ˆ :

(3.40)

Regarding our above introduced notion of supersymmetry we remark that for undriven (f(t) ≡ 0; y(t) ≡ 0; F = 0) systems (3.1), a connection with supersymmetric quantum mechanics [203,204] has been Frst pointed out in [205] and has been further developed in [206,207], see also the [208] for a review. The basic idea is to transform the Fokker–Planck equation (cf. Section 2.2) associated with the undriven stochastic dynamics (3.1) into a SchrModinger-type equation [99 –101,209,210]. By replacing in this equation the potential by its supersymmetric partner potential (in the quantum mechanical sense) a new SchrModinger equation emerges which can be transformed back into a new Fokker–Planck equation. The potentials of the original and the new Fokker–Planck equations then coincide (up to irrelevant shifts Ux and UV ) if and only if the supersymmetry condition (3.39) is satisFed. In the presence of a periodic driving y(t) (but still f(t) ≡ 0; F = 0) in the stochastic dynamics (3.1), a similar line of reasoning has been developed in [211], yielding the supersymmetry condition (3.40). The case of various stochastic drivings y(t) has been addressed in [212,213]. Here, we will borrow the previously established notion of “supersymmetry” for the conditions (3.39), (3.40), but we will neither exploit nor further discuss their connection with quantum mechanical concepts. 3.5.2. Main conclusion We now come to the central point of this section: We consider the general stochastic dynamics (3.1), (3.2) with F = 0 together with the usual assumptions on f(t) and y(t) from Section 3.1. By introducing z(t) := x(−t)+L=2, we can infer that z(t) ˙ := − x(−t), ˙ i.e. the averaged currents satisfy 41 z ˙ = −x. ˙ In doing so, we have exploited that only deterministic and=or stationary stochastic processes appear in (3.1), (3.2), hence the evolution of the dynamics backward in time does not give rise to any problem. Especially, −(−t) is statistically equivalent to the forward Gaussian white noise (t). On the other hand, if both V (x; f(t)) and y(t) are supersymmetric according to (3.39), (3.40) then one can readily see that z(t) satisFes the same dynamics (3.1) as x(t). Due to the self-averaging property of the current in (3.5) it follows that z ˙ = x. ˙ In view of our previous Fnding z ˙ = −x ˙ we arrive at our main conclusion: if both V (x; f(t)) and y(t) are supersymmetric according to (3.39), (3.40) then the average particle current x ˙ is zero, see also [214 –217]. 41

Note that it is not possible to derive this conclusion z ˙ = −x ˙ from (3.5). The reason is that the initial and Fnal times exchange their roles when going over from x(t) to z(t) and thus the implicit assumption in (3.5) that the initial time is kept Fxed while t → ∞ is no longer fulFlled for z(t). The properly generalized version of (3.5) reads x ˙ = limt−t0 → ∞ [x(t) − x(t0 )]=[t − t0 ], from which one readily recovers z ˙ = −x . ˙

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2

V(x,f(t))

1

0

-1

-2 -1

-0.5

0

0.5

1

x/ L

Fig. 3.1. Example of a supersymmetric potential V (x; f(t)) (in arbitrary units) of the type (3.41) at an arbitrary but Fxed f(t)-value.

We emphasize again that the conclusion x ˙ = 0 only holds true if either both, the potential and the driving are symmetric or if both of them are supersymmetric. In any other case, x ˙ = 0 is expected generically. Especially, a symmetric but not supersymmetric potential in combination with a supersymmetric but not symmetric driving generically implies x ˙ = 0 (see Sections 3.5.3 and 5.12 for more details and examples). 3.5.3. Examples Next we turn to the discussion of examples. Our Frst observation is the following completely general implication of the supersymmetry condition (3.39): For any minimum of V (x; f(t)), say at x = xmin , there exists a corresponding maximum at x = xmin + L=2 and vice versa. 42 For the rest, the condition (3.39) is still satisFed by a very large class of potentials and their exhaustive characterization on an intuitive level seems rather diCcult. Here, we restrict ourselves to two suCcient (but not necessary) simple criteria, which are still very general, namely: 1. The potential V (x; f(t)) is of the general form 2 nx + n (f(t)) V (x; f(t)) = 0n (f(t)) cos and f(t) time-inversion invariant ; L n=1;3;5;::: (3.41)

where time-inversion invariance of f(t) means, in the same sense as in (3.40), that f(−t)=f(t). ˆ A typical example of this type (3.41) of supersymmetric potential V (x; f(t)) is depicted in Fig. 3.1. Note that in general not only the shape of V (x; f(t)) but also the location of the extrema may still 42

Since this property holds separately for any given f(t)-value, xmin and xmax may in general still depend on t.

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be di5erent for any f(t)-value. One readily sees that (3.41) indeed implies (3.39). For =uctuating potential ratchets, i.e. V (x; f(t)) = V (x)[1 + f(t)], and especially for static potentials V (x), also the inverse can be shown, i.e. (3.41) is an exhaustive characterization of supersymmetric potentials in these special cases, but not in general. 2. A second class of supersymmetric potentials V (x; f(t)) is obtained by means of the representation V (x; f(t)) = V+ (x; f(t)) + V− (x; f(t)) ; V± (x; f(t)) :=

V (x; f(t)) ± V (x; −f(t)) ; 2

(3.42) (3.43)

i.e. the potential is decomposed into symmetric and antisymmetric contributions with respect to f(t), V± (x; −f(t)) = ±V± (x; f(t)) :

(3.44)

Then, the following conditions are suCcient for the potential V (x; f(t)) to be supersymmetric: 2 nx + n (f(t)) V+ (x; f(t)) = 0n (f(t)) cos and V− (x + L=2; f(t)) L n=1;3;5;::: = V− (x; f(t))

and

f(t) supersymmetric :

(3.45)

One readily veriFes that these conditions (3.45) in combination with (3.42), (3.44) indeed imply (3.39), i.e. V (x; f(t)) is supersymmetric. A simple example is a supersymmetric f(t) and V (x; f(t)) = V1 (x) + V2 (x)f(t) ;

(3.46)

where V1 (x) is a static supersymmetric potential (cf. Eq. (3.41) and Fig. 3.1) and where V2 (x) is an arbitrary L=2-periodic function. In other words, in (3.42) the potential V+ (x; f(t)) is independent of f(t) and V− (x; f(t)) is linear in f(t). Next, we come to the supersymmetry conditions 43 for the driving y(t). If y(t) is periodic then condition (3.40) of supersymmetry is equivalent to a Fourier representation of the general form y(t) =

∞

$n sin(2 nt=T) :

(3.47)

n=1

A typical example of such a supersymmetric y(t) is depicted in Fig. 3.2. For a stochastic y(t) we can rewrite (3.40) as y(t1 ) y(t2 ) · · · y(tn ) = (−1)n y(−t1 )y(−t2 ) · · · y(−tn )

(3.48)

for all integers n ¿ 1 and all times t1 ; t2 ; : : : ; tn (see also the discussion below Eq. (3.21)). Note that out of the three possible symmetry properties of y(t), namely (ordinary) symmetry, supersymmetry, and time-inversion invariance, two always imply the third. All three invariance properties are indeed 43

They can of course be immediately transcribed into corresponding supersymmetry conditions for f(t) as well.

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y(t)

1

0

-1

-2 -0.5

0

0.5

1

1.5

t/

Fig. 3.2. Example of a supersymmetric T-periodic driving y(t) (in arbitrary units) of the type (3.47).

satisFed for many particularly simple examples y(t) which we will treat in more detail below, for instance symmetric dichotomous noise and Ornstein–Uhlenbeck noise (see Section 3.1), as well as symmetric Poissonian shot noise (see Section 5.5). Note also that arbitrary linear combinations of supersymmetric drivings are again supersymmetric. A few speciFc examples which, prima facie quite astonishingly, produce zero current due to supersymmetry reasons are worth mentioning: The Frst set of examples are tilting ratchets (f(t) ≡ 0) with a supersymmetric potential like in Fig. 3.1 and a periodic driving y(t) like in (3.47), see also Fig. 3.2, or with a symmetric dichotomous noise y(t), an Ornstein–Uhlenbeck noise y(t), or a symmetric Poissonian shot noise y(t). On the other hand, a symmetric, but not supersymmetric potential V (x) (e.g. (3.19) with a1 = 0 and a2 = 0) in combination with a supersymmetric but not symmetric driving y(t) (e.g. (3.47) with $1 = 0 and $2 = 0) does generically produce a Fnite current x, ˙ see Section 5.8. A summary of the symmetry considerations for tilting ratchets with periodic drivings (i.e. rocking ratchets and asymmetrically tilting ratchets) is depicted in Fig. 3.3. In order to bring out the essential features as clearly as possible, we have chosen in this Fgure stylized, non-smooth potentials V (x) and drivings y(t) and we have restricted ourselves to time-periodic y(t). In the case of a pulsating ratchet (y(t) ≡ 0), a symmetric dichotomous noise f(t), an Ornstein– Uhlenbeck noise f(t), a symmetric Poissonian shot noise f(t), or a periodic f(t) of the form (3.47) yields x ˙ = 0 if one of the following two conditions is met: (i) The potential V (x; f(t)) is for any given f(t)-value of the form (3.39), see also Fig. 3.1. We recall that not only the shape of V (x; f(t)) but also the location of the extrema may be di5erent for any f(t)-value, i.e. both =uctuating potential ratchets and (improper) traveling potential ratchets are covered. (ii) The potential V (x; f(t)) respects supersymmetry when f(t) = 0 and is augmented for f(t) ≡ 0 by a =uctuating potential term V2 (x)f(t) with an arbitrary L=2-periodic function V2 (x), see (3.46).

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V (x)

y (t)

1

1

1

1

0

0

-1

-1

-1

0

1

-1

2

0

-1

-1

0

-1

1

3

0

-1

-1

0

1

-1

4

1

0

1

4

1

0

0

-1

-1

-1

0

y (t)

V (x) 1

1

3

1

0

-1

0

y (t)

V (x) 1

1

2

1

0

-1

0

y (t)

V (x) 1

105

0

1

-1

Fig. 3.3. Summary of the symmetry considerations for tilting ratchets with potentials Vi (x) and periodic drivings yi (t) (in arbitrary units). i = 1: symmetric and supersymmetric. i = 2: symmetric but not supersymmetric. i = 3: supersymmetric but not symmetric. i = 4: neither symmetric nor supersymmetric (but still satisfying (3:9)). The particle current (3:5) vanishes for arbitrary combinations of potentials and drivings which are either both symmetric or both supersymmetric. For any other combination of potentials and drivings, a Fnite current arises generically.

3.5.4. Discussion As long as V (x; f(t)) and y(t) are supersymmetric, the property x ˙ = 0 is robust with respect to any change of the friction , temperature T , amplitude and characteristic time scale of the drivings f(t) and y(t) etc. Much in contrast to ordinary current inversions, we thus Fnd x ˙ = 0 without 8ne-tuning any of these model parameters. The same is of course true for symmetric instead of supersymmetric V (x; f(t)) and y(t). Note that this conclusion is no contradiction to Curie’s principle

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since a generic variation within the entire class of admitted ratchet models also involves a change of V (x; f(t)) and y(t) such that these symmetries are broken, see also the concluding discussion in Section 10. In the above respect, but also upon comparison of (3.19) – (3.21) with (3.41), (3.47), (3.48), the formal structure and the consequences of symmetry and supersymmetry are remarkably similar. There is, however, also one fundamental di5erence which appears if an additional inertia term m x(t) M is included on the left-hand side of the general ratchet dynamics (3.1): While symmetry implies x ˙ = 0 even in the presence of inertia e1ects, the same conclusion no longer applies in the case of supersymmetry. For instance, a rocking ratchet with a cosine potential V (x) and a driving y(t) like in Fig. 3.2 implies x ˙ = 0 in the overdamped limit [31,218,219] but generically x ˙ = 0 if inertia is included [220]. In the opposite limit of a deterministic Hamiltonian rocking ratchet dynamics (Fnite inertia, vanishing dissipation and thermal noise) a condition [221] reminiscent of supersymmetry will be discussed in Section 5.8. In the intermediate regime of Fnite inertia and dissipation, no comparable symmetry concept is known. Since the current changes always continuously upon variation of any model parameter, it follows that for any suCciently small deviations from a perfectly supersymmetric situation, e.g. in the presence of a very small ineria term, the current x ˙ will still be arbitrarily small [215]. In the following we focus again on the overdamped limit. 3.5.5. Generalizations We close with a brief look at the ratchet classes with both f(t) ≡ 0 and y(t) ≡ 0 in (3.1) but instead with a varying temperature T in (3.2): In the case of Seebeck ratchets, characterized by a space-dependent, L-periodic temperature T (x), we speak of a (spatially) symmetric system if both, V (x) and T (x) satisfy the symmetry condition (3.16) with the same Ux, which may be transformed to zero as usual, i.e. V (−x) = V (x)

and

T (−x) = T (x) [symmetry] :

(3.49)

Similarly, supersymmetry is deFned by the following condition for the potential together with a modiFed such condition for the temperature: − V (x) = V (x + L=2)

and

T (−x) = T (x + L=2) [supersymmetry] :

(3.50)

Along the same line of reasoning as in Section 3.2, i.e. by considering the mirror image −x(t) of x(t), one readily Fnds that the average current x ˙ indeed vanishes if the symmetry conditions (3.49) are satisFed. On the other hand, by considering z(t) := x(−t) + L=2, one veriFes that x ˙ =0 if supersymmetry (3.50) is respected. Finally, in the case of temperature ratchets, characterized by a time-dependent temperature T (t), a zero current x ˙ = 0 is recovered provided that either V (−x) = V (x) [symmetry] ;

(3.51)

independently of the properties of T (t), or that − V (x) = V (x + L=2)

and

T (t) time-inversion invariant [supersymmetry] :

(3.52)

Comparison with (3.16) and (3.41) conFrms once more the similarity between pulsating ratchets and temperature ratchets (see also Section 6.3).

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Further generalization to higher dimensional systems are also possible but not further pursued here, see also Section 9.2.3. 3.6. Tailoring current inversions The argument which we have invoked at the end of Section 2.11 can be considerably generalized as follows: We consider any ratchet model of the general form (3.1) – (3.3), usually (not necessarily 44 ) with F = 0 in (3.1), and possibly also with an x- and=or t-dependent temperature T in (3.2). Next, we focus on an arbitrary parameter of the model and we prescribe an arbitrary reference value 0 . Under the only assumption that two potentials Vi (x; f(t)), i=0; 1, with opposite currents x ˙ at =0 exist, we can then construct a third potential, say V30 (x; f(t)), with the property that the current x ˙ as a function of the parameter exhibits a current inversion at the prescribed reference value 0 . The proof of this proposition is almost trivial. Namely, we deFne a set of potentials V3 (x; f(t)) := 3V1 (x; f(t)) + (1 − 3)V0 (x; f(t)) ;

(3.53)

parametrically dependent on 3 ∈ [0; 1]. In other words, the potentials V3 (x; f(t)) continuously interpolate between the above deFned two potentials Vi (x; f(t)), i = 0; 1, with opposite current directions at = 0 . Under the tacit assumption that the current x ˙ changes continuously upon variation of 3, it follows that it vanishes at a certain intermediate potential V30 (x; f(t)). We remark that this assumption is very weak: For instance, one can show that a non-vanishing thermal noise (t) in (3.1) is suCcient, but by no means necessary. Since the sign of x ˙ is robust against small changes of Vi (x; f(t)), it can furthermore be taken for granted that V30 (x; f(t)) is a generic potential in the sense that the dynamics (3.1) is neither symmetric nor supersymmetric, nor exhibits any other “accidental” symmetry. In other words, we are dealing with the generic case that, upon variation of the parameter , the current x ˙ exhibits an isolated zero, i.e. a genuine current inversion, at = 0 . If the condition that two potentials Vi (x; f(t)) with opposite current directions at = 0 exist is not fulFlled, then also a current inversion at 0 is obviously not possible, i.e. this condition is both necessary and suCcient. For instance, if the driving y(t) is symmetric and the temperature T independent of x then we know that V (−x; f(t)) yields a current opposite to that associated with V (x; f(t)). Hence, we can choose as V0 (x; f(t)) any potential with x ˙ = 0 and as V1 (x; f(t)) a slightly deformed modiFcation of V0 (−x; f(t)) to conclude that a current inversion exists always. We may also consider some characteristic property of the driving y(t) as variable and instead leave all the other ingredients (especially the potential) of the ratchet dynamics (3.1) Fxed. If the existence of two special drivings yi (t), i = 0; 1, with opposite currents at = 0 is known, then we can prove along the same line of reasoning as above that there exists at least one 3 ∈ (0; 1), say 30 , such that y3 (t) := 3y1 (t) + (1 − 3)y0 (t) ;

44

(3.54)

Note that current inversions upon variation of any model parameter can obviously be enforced by applying an appropriately chosen external force F [54,195,222,223].

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produces a current inversion at the arbitrarily prescribed reference parameter value 0 . For instance, if V (x; f(t)) is symmetric (and the temperature T constant) then we know that an asymmetric y(t) generically produces a current x ˙ = 0 and −y(t) a current in the opposite direction. Hence, an appropriate asymmetric driving y(t) can be tailored which separates particles on opposite sides of an arbitrarily prescribed parameter value 0 . Since y(t) is typically generated by an externally applied Feld, such a separation scheme may be of considerable experimental interest. At this point it is worth recalling that once a current inversion upon variation of one model parameter has been established, the existence of an inversion upon variation of any other parameter follows along the same line of reasoning as in Section 2.11. Current inversions upon changing certain parameters of the system have been studied for the Frst time in the context of photovoltaic e5ects in non-centrosymmetric materials [159,160]. Early observations in simple theoretical models as we study them here are due to [35,37,39,42]. Since then the search and control of current inversions has been attracting much attention with respect to the possibility of new particle separation technologies based on the ratchet e5ect. Moreover, multiple current inversions have been exempliFed, e.g. in [170,223–232]. In the latter case, particles with parameter values within a characteristic “window” may be separated from all the others. The Frst systematic investigation of such multiple inversions from [232] suggest that it may always be possible to tailor an arbitrary number of current inversions at prescribed parameter values. However, a corresponding generalization of our rigorous proof has not yet been established. Our method of tailoring current inversions implies that, in general, the direction of the particle current, and even more so its quantitative magnitude, depends in a very complicated way on many details of the ratchet potential V (x; f(t)) and=or on the driving y(t). In this respect, the leading order small-T behavior of the temperature ratchet in (2.58) is still a rather simple example. Therefore, any heuristic “explanation” or simple “rule” regarding current directions should make us suspicious unless it is accompanied by a convincing (and usually rather severe) restriction on the admitted potentials V (x; f(t)) and=or the driving y(t). Otherwise, one can typically construct even quite innocent looking counter-examples of such a “rule”. The above described procedure is a very simple and universal tool for the construction of current inversions per se. However, little control over the more detailed dependence of the current as a function of the considered parameter is possible in this way. For instance, the maximal magnitudes of the currents may be very di5erent in the positive and negative directions. Likewise, we can hardly avoid ending up with a quite complicated looking potential V30 (x) and=or driving y30 (t). For both purposes, “symmetrically” shaped current inversions as well as “simple” potentials and=or drivings which do the job, more detailed analytical predictions are invaluable. For instance, the results depicted in Fig. 2.9 have not been obtained directly by the above construction scheme. Rather, the approximation (2.58) has been exploited in order to obtain such an “innocent” looking ratchet potential with a current inversion. 3.7. Linear response and high-temperature limit For vanishing f(t), y(t), and F we recover a Smoluchowski–Feynman ratchet in (3.1), yielding zero current (3.5) in the long-time limit (steady state), see Sections 2:1–2:4. In the case of a tilting ratchet scheme, an interesting question regards the linear response behavior in the presence of a weak but Fnite driving y(t) (while f(t) and F are still zero), i.e. the behavior of the averaged long-time

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current (3.5) in linear order 45 y(t). In the case of a symmetric driving y(t) (cf. (3.17) and (3.18)) no such linear contribution can arise, since the averaged long-time current is clearly invariant under inversion of the driving amplitude, y(t) → −y(t). In the general (asymmetric) case, we can expand y(t) into a Fourier series, which does not exhibit a constant term since y(t) is unbiased, see (3.9) or (3.10). In linear order y(t), the net current follows simply by summing up all the contributions of the single Fourier modes. Since each Fourier mode corresponds to a symmetric driving, the resulting net current is zero. A similar line of reasoning applies for pulsating ratchets with a weak driving f(t) (and y(t) = F = 0). An exception is a genuine traveling potential scheme with a systematic long-time drift u := limt → ∞ f(t)=t. In this case, the above Fourier expansion of f(t) cannot be applied any more and indeed a Fnite linear order f(t) contribution to the current is observed generically, see Section 4.4.1. In other words, for tilting ratchets, 9uctuating force ratchets, and improper traveling potential ratchets no directed current occurs within the linear response regime (linear order y(t) and f(t), respectively). The same conclusions obviously extends to systems with simultaneously small but Fnite y(t) and f(t) (but still F = 0). Due to their equivalence with =uctuating potential ratchets, the conclusion also carries over to temperature ratchets (cf. Section 6.3) with a small perturbation of the temperature T (t) about the (Fnite) average value TW . In the above line of reasoning we have tacitly assumed analyticity of the current with respect to the amplitude of the perturbations y(t) and f(t) and that a Fourier expansion 46 of these perturbations is possible. (Especially, for a stochastic process, the word “weak perturbation” refers to its intensity, but not necessarily to its instantaneous value at any given time t.) Though this may be diCcult to rigorously justify in general, a more careful analysis of each speciFc case (known to the present author) shows that the conclusion of vanishing linear response remains indeed correct. Another limit which admits a completely general conclusion is that of asymptotically large temperatures T in (3.1), (3.2): Again one Fnds that the current x ˙ always approaches zero in the limit T → ∞. While this result is physically rather suggestive (the e5ect of the potential V (x; f(t)) is completely overruled by the noise (t)) the technical details of the mathematical proof go beyond the scope of this review. In those numerous cases for which the current also vanishes for T → 0, a bell-shaped x-versus-T ˙ curve is thus recovered. We Fnally remark that in the limit of asymptotically strong drivings y(t) and=or f(t), no generally valid predictions are possible. 3.8. Activated barrier crossing limit For many of the above-deFned classes of ratchets (3.1) it may turn out that in the absence of the thermal Gaussian noise (T = 0), the particle x(t) is conFned to a restricted part of one spatial period for all times. In the presence of a small amount of thermal noise, the particle will be able to cross the previously forbidden regions by thermal activation. Yet, such events will be rare and after each thermally activated transition from one spatial period into an adjacent one, the particle will again remain there for a long time. Since the duration of the actual transition events is negligible in comparison with the time the particle spends in a quasi steady state (metastable state) between 45

Formally, this amounts to replacing y(t) by 4y(t) and then performing a series expansion in 4 while keeping y(t) Fxed. 46 Or any other series expansion in terms of symmetric basis functions.

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the transitions, it follows that the probability for a transition per time unit can be described in very good approximation by a constant rate. 47 Denoting the rate for a transition to the right by k+ and to the left by k− , the average particle current readily follows 48 as x ˙ = L [k+ − k− ] :

(3.55)

For a special case, this result has been derived explicitly already in Eq. (2.44). Exploiting the rate description for the transitions once more, also the e5ective di5usion coeCcient (3.6) can be readily evaluated [100] with the result De5 =

L2 [k+ + k− ] : 2

(3.56)

The very same conclusions (3.55), (3.56) hold true if transitions are not excluded but still very rare at T = 0. This may be the case for instance in a tilting ratchet (f(t) ≡ 0) when y(t) is a Gaussian random process with a small intensity dty(t)y(0). On the other hand, genuine traveling potential ratchets will turn out to support an appreciable particle current typically even for T = 0 and are henceforth excluded. The evaluation of the current (3.55) and the di5usion coeCcient (3.56) has thus been reduced to the determination of certain rates k across rarely visited regions between some type of e5ective local potential wells or periodic attractors (if f(t) or y(t) is periodic in t). In the case that both f(t) and y(t) are stochastic processes (possibly one of them identically zero), this problem of thermally activated surmounting of a potential barrier with randomly =uctuating shape has attracted considerable attention since the discovery of the so-called resonant activation e5ect [238], see [68] for a review. On condition that a rate description of the barrier crossing problem is possible, i.e. the transitions are rare events especially in comparison with the time scale of the barrier =uctuations [68], the =uctuating barrier crossing problem is thus equivalent to determining the current and di5usion in a ratchet model. A large body of analytical results on the former problem [239 –254] are thus readily applicable for our present purposes. Particularly closely related to the resonant activation e5ect are the theoretical works [158,255] on externally driven molecular pumps (cf. Section 4:6:1) and their experimental counterparts in [256,257]. If f(t) and=or y(t) are periodic in time with a large period T then a close connection to the phenomenon of stochastic resonance [62] can be established. If the time period T is not very large, then this problem of thermally activated escape over an oscillating potential barrier represents a formidable technical challenge [258]. The few so far available results on weak [259,260], slow (but beyond the adiabatic approximation) [261], fast [262], and general oscillations [263,264] can again be readily adapted for our present purposes via (3.55), (3.56). 47

If f(t) or y(t) is periodic in time, then we will have a quasi periodic behavior between transition events and the transition probability is only given by a constant after “coarse graining” over one time-period. 48 We tacitly focus here on the simplest and most common case with just one metastable state per period L. For more general cases, the current and the e5ective di5usion coeCcient can still be expressed in closed analytical form in terms of all the involved transition rates, but the formulas become more complicated, see [233–235] and further references therein. For special cases, such formulas have been repeatedly re-derived, e.g. in [236,237].

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One basic conclusion of all the above-mentioned analytical studies of the thermally activated escape over =uctuating or oscillating barriers is that those rates become exponentially small with decreasing thermal noise strength, much like in the simple explicit example (2.45), (2.46). On top of that, also the ratio k+ =k− generically becomes either exponentially small or large. In other words, practically only transitions in one direction occur in the course of time and x(t) basically simpliFes to a unidirectional Poissonian hopping process [100]. This type of unidirectionality is a distinct weak-noise feature. As soon as the thermal noise-strength increases, the stochastic trajectory x(t) always displays appreciable displacement in both directions. Though a substantial part of the present author’s contributions to the Feld under review is concerned with the evaluation of such thermally activated rates over =uctuating or oscillating barriers, we desist from going into any further details of these technically rather involved theories. 4. Pulsating ratchets In this section we focus on the pulsating ratchet scheme, i.e. we consider a stochastic dynamics of the form

x(t) ˙ = −V (x(t); f(t)) + (t) ;

(4.1)

complemented by the =uctuation–dissipation relation (3.2) for the thermal noise (t) and the periodicity condition (3.3) for the potential. Further, f(t) is assumed to be an unbiased time-periodic function or stationary stochastic process. As compared to the general working model (3.1) we have set the load force F equal to zero on the right-hand side of (4.1) since this case is usually of foremost interest. 4.1. Fast and slow pulsating limits We consider an arbitrary (unbiased, i.e. f(t) = f(0) = 0) stochastic process f(t) and assume that its stationary distribution %(f) (see (3.11)) and thus especially its variance f2 (t) = f2 (0) (see (3.14)) is always the same, while its correlation time 49 ∞ dtf(t)f(s) ; (4.2) & := −∞ 2f2 (t) characterizing the decay of the correlation f(t) f(s) = f(t − s)f(0), can be varied over the entire positive real axis. More precisely, we assume that the process f(t) can be written in the form ˆ f(t) = f(t=&) ;

(4.3)

ˆ where f(h) is a suitably deFned, Fxed reference process with dimensionless time-argument h, cf. (2.56). The statistical properties of the process f(t) then depend solely on the parameter &, while %(f) is &-independent. One readily sees that the examples of dichotomous and Ornstein–Uhlenbeck noise from (3.12) – (3.15) are of this type. This so-called constant variance scaling assumption 49

By means of a calculation similar to that in Section A.2 of Appendix A one can show that the intensity and hence the correlation time (4.2) is always non-negative.

dtf(t)f(s)

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[265] is “natural” in the same sense as it is natural to keep in a time-periodic perturbation f(t) the amplitude Fxed while the frequency is varied, cf. Section 2.10. For very small &, the noise f(t) changes very quickly, while its strength dtf(t)f(s) tends to zero (see also footnote 49). Thus, for & → 0 (fast pulsating limit) there will be no e5ect of f(t) in (4.1), i.e. we recover a Smoluchowski–Feynman ratchet with x ˙ = 0. Similarly, for & → ∞ the noise becomes very slow, f(t) f = const: (adiabatic approximation, see also Section 2.10). Since for any Fxed value of f we have again a Smoluchowski–Feynman ratchet in (4.1), the result x ˙ =0 subsists after averaging over all those Fxed f-values according to %(f). The very same conclusion x ˙ = 0 holds for periodic functions f(t) in the limits of asymptotically long and short periods T and can be furthermore extended also to generic traveling potential ratchets (cf. Section 4:4:1). In other words, we arrive at the completely general result that for any type of pulsating ratchet (4.1), the current x ˙ disappears both in the fast and slow pulsating limit. We exemplify the leading-order correction to this asymptotic result x ˙ = 0 for the case that f(t) is an arbitrary stationary stochastic process with very small correlation time &. Generalizing an argument form [68,246] we can, within a leading-order approximation, substitute in (4.1) the random process V (x; f(t)) for any Fxed x by an uncorrelated Gaussian process with the same mean value ∞ V0 (x) := V (x; f(t)) = df%(f)V (x; f) (4.4) and the same intensity

−∞

dt C(x; t), where the correlation C(x; t) is deFned as

C(x; t) := V (x; f(t))V (x; f(0)) − [V0 (x)]2 :

(4.5)

An alternative representation of C(x; t) analogous to the second equality in (4.4) is given by ∞ C(x; t) = df1 df2 %(f1 ; f2 ; t) V (x; f1 ) V (x; f2 ) ; (4.6) −∞

where %(f1 ; f2 ; t) is the joint two-time distribution of the stationary process f(t), i.e. %(f1 ; f2 ; t) := (f(t) − f1 ) (f(0) − f2 ) : Referring to [68,246] for the calculational details, one obtains in this way the result ∞ L d x V0 (x) −∞ dt C(x; t) L 0 x ˙ = 2 :

kB T L d x eV0 (x)=kB T L d x e−V0 (x)=kB T 0 0

(4.7)

(4.8)

Note that the e5ective potential V0 (x) is again L-periodic. Under the conditions that T ¿0 and that (4.5) decays exponentially in time, the result (4.8) gives the leading order contribution for small correlation times & of f(t). Using (4.3) we can infer that ∞ ∞ ˆ h) ; dt C(x; t) = & dh C(x; (4.9) −∞

−∞

ˆ ˆ t) is deFned like in (4.5) but with f(h) where C(x; instead of f(t) and thus the integral on the right-hand side of (4.9) is &-independent. In other words, the asymptotic current (4.8) grows linearly with the correlation time &.

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In the special case that V (x; f(t)) is of the =uctuating potential ratchet form V (x)[1 + f(t)], the small-& result from [266] is readily recovered from (4.8), namely L 2L&f2 (t) 0 d x [V (x)]3 ; (4.10) x ˙ = L L

2 kB T 0 d x eV (x)=kB T 0 d x e−V (x)=kB T see also [267]. A similar leading order correction of the adiabatic approximation x ˙ = 0 is possible for large & [266] but leads to a complicated formal expression which still depends on much more details of the noise than in (4.8). For periodic driving f(t), expansions along the lines of Appendix C can be undertaken, see Section 6.3. Technically similar calculations can also be found in [268,269] for fast and in [261] for slow periodic driving. We will not pursue this task here any further, 50 ; 51 since the main conclusion follows from the already complicated enough result (4.8), namely that a Fnite current x ˙ is generically expected for driving signals f(t) with a Fnite characteristic time scale and that its sign and magnitude depend very sensitively on the details of the potential V (x) and the driving f(t). 4.2. On–o1 ratchets The on–o5 ratchet scheme has been introduced in a speciFc theoretical context in 1987 by Bug and Berne [32] and has been independently re-invented as a general theoretical concept in 1992 by Ajdari and Prost [34]. In its latter form it is of archetypal simplicity and the predicted occurrence of the ratchet e5ect has been veriFed by several experimental implementations. The model is a special case of the overdamped one-dimensional Brownian motion dynamics (4.1), namely

x(t) ˙ = −V (x(t))[1 + f(t)] + (t) ;

(4.11)

where V (x) is spatially periodic, asymmetric “ratchet” potential. The function f(t) is restricted to the two values ±1, so the potential in (4.11) is either “on” or “o5”. In the simplest case the potential V (x) has one maximum and minimum per spatial period L (for examples see Figs. 2.2 and 4.1), the potential di5erence between maxima and minima is much larger than the thermal energy kB T , and f(t) is a time-periodic function with long sojourn-times in the +1-state (potential “on”). Under these premises the analysis of the model proceeds in complete analogy to Fig. 2.6. Qualitatively, a net pumping of particles into the positive direction will occur if the minima of V (x) are closer to their neighboring maxima to the right than to the left (“forward on–o5 ratchet”), otherwise into the negative direction. 52 Quantitatively, the average current x ˙

50 In the special case of slow on–o5 and slow, traveling potential ratchets, the qualitative behavior will become intuitively clear later on. 51 If V (x; f(t)) is of the =uctuating potential ratchet form V (x)[1 + f(t)] it is possible to show that x ˙ vanishes faster than proportional to &−1 in the slow driving limit & → ∞ for both, periodic and stochastic f(t). In the latter case, this conclusion is also contained implicitly in the calculations of [266]. 52 A computer animation (Java applet) which graphically visualizes the e5ect is available on the internet under [270].

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V(x)

1

0

-1

-2 -1.5

-1

-0.5

0

0.5

1

1.5

x/ L

Fig. 4.1. Schematic illustration of a piecewise linear “saw-tooth” ratchet potential V (x) (in arbitrary units), consisting of two continuously matched linear pieces per period L, one with negative and one with positive slope, but otherwise asymmetric.

can be readily expressed in closed analytical form [34] apart from an error function, which has to be evaluated numerically. Similarly, one can readily evaluate [34] the e5ective di5usion coeCcient (3.6). Depending on their friction coeCcient , di5erent species of particles which are initially mixed (say x(0) = 0 for all of them) will exhibit after a time t di5erent displacements xt ˙ and dispersions √ 2De5 t and thus can be separated for suCciently large t, see also Eq. (3.7). For more general potentials V (x) and drivings f(t), things become more complicated. As seen in Section 4.1, the current x ˙ approaches zero both for very fast and slow switching of f(t) between ±1. As long as the potential V (x) is suCciently similar to the simple examples from Figs. 2.2 and 4.1, a single maximum of x ˙ develops at some intermediate switching time. For more complicated, suitably chosen potentials V (x), the existence of current inversions follows from Section 3.6 and has been exempliFed in [271], see also [216,272]. For a few additional quantitative analytical results we also refer to the subsequent Section 4.3 on =uctuating potential ratchets, which includes the on–o5 scenario as a special case. 4.2.1. Experimental realizations The theoretically predicted pumping e5ect x ˙ = 0 has been demonstrated experimentally by Rousselet et al. [38] by means of colloidal polysterene latex spheres, suspended in solution and exposed to a dielectric “ratchet” potential, created by a series of interdigitated “Christmas-tree” electrodes which were deposited on a glass slide by photolithography and which were turned on and o5 periodically. With one adjustable parameter, accounting for the uncertainty about the shape of the one-dimensional “e5ective” potential V (x), the agreement of the measured currents x ˙ with the theory from [34] turned out to be quite good for all considered particle diameters between 0.2 and 1 m.

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A very similar experimental setup was used by Faucheux and Libchaber [273] (see also [778]) but they went one step further in that they studied solutions containing two di5erent species of particles at a time, and demonstrated experimentally that they can be separated. Again, the quantitative Fndings depend on the particle size not only via the corresponding viscous friction coeCcient but also via slightly di5erent “e5ective” potentials V (x). A further experimental veriFcation of the on–o5 ratchet scheme by Faucheux et al. [274] works with single 1:5 m diameter polysterene spheres, conFned to an e5ective one-dimensional “ratchet” potential by laser–optical trapping methods (optical tweezers). The characteristics of such an optical ratchet potential can be determined with satisfactory accuracy, leading to a quite good quantitative agreement of the observed ratchet e5ect with the simple theory from [34] without any adjustable parameters. A Frst step towards a practically usable pumping and separation device was achieved by GorreTalini et al. [275] (see also [276]). Due to their well deFned geometry, also in this experiment latex spheres diluted in water with diameters mostly between 0.5 and 2:5 m were used, but in principle nothing speaks against replacing them e.g. by micrometer sized biological objects like DNA, viruses, or chromosomes. The main ingredient for creating the (electrostatic) “ratchet” potential is an optical di5raction grating, commercially available in a variety of periods and asymmetries. This setup overcomes many of the practical drawbacks of its above-mentioned predecessors, agrees well with the simple theory from [34] without any Ftting, and the predicted separating power may well lead to the development of a serious alternative to existing standard particle separation methods. As regards transport and separation of DNA, all the so far discussed setups are expected to cease working satisfactorily for DNA fragments below one kilobase. By means of an interdigitated electrode array 53 Bader and coworkers [277,278] successfully micromachined an on–o5 ratchet on a silicon-chip, capable to transport DNA molecules of 25 –100 bases in aqueous solution. With some improvements, such a “lab-on-a-chip” device could indeed provide a feasible alternative to the usual electrophoresis methods for nucleic acid separation, one of the central preprocessing tasks, e.g. in the Human Genome Project [277–279]. As pointed out in [276], not only the separation of large DNA molecules by present standard methods seems to have become one of the major bottlenecks in sequencing programs [280], but also chromosomes, viruses, or cells exhibiting major biological di5erences may only di5er very little from the physicochemical viewpoint, thus making their separation highly challenging [281,282]. 4.3. Fluctuating potential ratchets The =uctuating potential ratchet model has been introduced practically simultaneously 54 in 1994 by Astumian and Bier [15] and independently by Prost et al. [13]. The model is given by the same type of overdamped dynamics as the on–o5 ratchet in (4.11) except that the amplitude modulations f(t) are no longer restricted to the two values ±1. In other words, the time-dependent potential V (x)[1 + f(t)], to which the Brownian particle x(t) is exposed, has always the same shape but its 53

About 100 metallic strips perpendicular to the transport axis with alternating smaller and larger spacings are either alternatingly charged (“on”) or all uncharged (“o5 ”). 54 The closely related work by Peskin et al. [17] came somewhat later and mainly focuses on the special case of an on–o5 ratchet.

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amplitude changes in the course of time between two or more values. Thus the on–o5 scheme is included as a special case throughout the considerations of this section. One central and completely general feature of the =uctuating potential ratchet scheme readily follows from (4.11). Namely, within each spatial period there are x-values where V (x) is zero. These points cannot be crossed by x(t) in the overdamped dynamics (4.11) without the help of the =uctuations (t). In other words, in an overdamped 9uctuating potential ratchet, thermal noise is indispensable for a 8nite current x. ˙ A second, completely general conclusion has been pointed out already in Section 3.3, namely that only asymmetric potentials V (x) admit a ratchet e1ect. For the latter reason, we will mainly concentrate on the simplest non-trivial case of an asymmetric potential V (x) in combination with a symmetric driving f(t). As usual, one option in (4.11) is a periodic driving f(t), see e.g. [271,283,554]. For asymptotic results for fast periodic driving f(t) we refer to Section 6.3. In the following we focus, in accordance with most of the existing literature, on the two particularly simple examples of a stationary random processes f(t) as introduced in Section 3.1, see also the asymptotic result (4.10) for general fast stochastically =uctuating potentials. A well-known experimental phenomenon which can be interpreted by means of a =uctuating potential ratchet scheme is the photoalignement (absorbtion-induced optical reorientation) of nematic liquid crystals [284 –286,779], exemplifying the general considerations in [13]. Another realization by means of a Josephson junction device similarly to that proposed in [287] (cf. Section 5.7.3) is presently planned by the same authors. The large variety of potential applications of the =uctuating potential ratchet mechanism for small-scale pumping devices is the subject of [283]. 4.3.1. Dichotomous potential 9uctuations The case that the amplitude modulations f(t) in (4.11) are given by a symmetric dichotomous noise (see (3.12), (3.13)) has been considered in [13,15,17]. A numerical simulation of the time discretized stochastic dynamics (4.11) similarly as in equation (2.7) is straightforward. Equivalent to this stochastic dynamics is the following master equation for the joint probability densities P± (x; t) that at time t the particle resides at the position x and the dichotomous process f(t) is in the states ±#, respectively: 9 9 (1 + #)V (x) k B T 92 P+ (x; t) = P+ (x; t) + P+ (x; t) ; 9t 9x

9x2 − $P+ (x; t) + $P− (x; t) ; 9 (1 − #)V (x) k B T 92 9 P− (x; t) + P− (x; t) ; P− (x; t) = 9t 9x

9x2 − $P− (x; t) + $P+ (x; t) :

(4.12)

(4.13)

The derivation of the Frst two terms on the right-hand side of (4.12) and (4.13) is completely analogous to the derivation of the Fokker–Planck equation (2.14): The Frst terms (drift terms) account for the Liouville-type evolution of the probability densities P± (x; t) under the action of the respective force Felds −V (x)[1 ± #]. The second terms (di5usion terms) describe the di5usive broadening of the probability densities due to the thermal white noise (t) of strength 2 kB T in (4.11). The last two terms in (4.12) and (4.13) are loss and gain terms due to the transitions of f(t) between the two “states” ±# with transition probability (=ip rate) $.

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For the marginal density P(x; t) := P+ (x; t) + P− (x; t) one recovers from (4.12), (4.13) a master equation of the general form (2.17), whence the particle current (2.19) follows. Like in Section 2.4, one sees that it suCces to focus on spatial periodic “reduced” distributions Pˆ ± (x + L; t) = Pˆ ± (x; t) in order to calculate the average current x ˙ and that in the long-time limit t → ∞ a unique steady st ˆ state P ± (x) is approached. However, explicit analytical expressions for the probability densities and the particle current in the steady state can only be obtained in special cases. A prominent such case is that of a “saw-tooth” ratchet potential as depicted in Fig. 4.1. Within each interval of constant slope V (x), the steady state solutions of (4.12), (4.13) can be readily determined which then have to be matched together. For the straightforward but rather tedious technical details of such a calculation we refer to [15,288]. The resulting expression for the current x ˙ is awkward and not very illuminating. Qualitatively, the results are like for the on–o5 scheme (see also Fig. 2.6): If the local minima of the saw-tooth potential are closer to their neighboring maxima to the right than to the left, then the current is positive for all Fnite temperatures T and =ipping rates $ of the dichotomous noise, and (cf. Section 3.1) the current approaches zero if T or $ tends either to zero or to inFnity [13,15]. As long as the potential is suCciently similar to a saw-tooth potential such that one can identify essentially one steep and one =at slope per period L, qualitatively unchanged results are obtained, and similarly for more general potential =uctuations f(t). In this sense, the “natural” current direction in a pulsating ratchet is given by the sign of the steep potential slope. For more general potentials V (x), the same qualitative asymptotic features are expected for large and small T and $. However, as shown in Section 3.6, the current direction may change as a function of any model parameter. So, general hand weaving predictions about the sign are impossible, not to mention quantitative estimates for x, ˙ as already the small-& result (4.10) demonstrates. A case of particular conceptual interest is the singular perturbation expansion about the unperturbed situation with vanishing thermal noise, T = 0, and vanishing particle current x ˙ = 0. On condition that the local potential extrema subsist for all values of f(t), i.e. |#|¡1, the transitions between neighboring minima of the potential will be very rare for small T . As a consequence, the current x ˙ can be described along the activated barrier crossing approach from Section 3.8. Specifically, for our present situation of thermally assisted transitions across barriers with dichotomous amplitude =uctuations, the results of the singular perturbation theory from [254] are immediately applicable. While for the case of a saw-tooth ratchet potential V (x), which is subjected to symmetric dichotomous amplitude =uctuations f(t) in (4.11), current inversions can be ruled out [15], for more complicated =uctuations f(t) this is no longer the case: The occurrence of a current inversion in a saw-tooth potential V (x) has been demonstrated for asymmetric (but still unbiased) dichotomous noise [216,271,272], for sums of dichotomous processes [266], and for a three-state noise [289]. Originally, the inversion e5ect has been reported in those works [216,266,271,272,289] for certain parameters characterizing the noise f(t), but according to Section 3.6 the e5ect immediately carries over to any parameter of the model. 4.3.2. Gaussian potential 9uctuations Next we turn to the second archetypal driving f(t), namely correlated Gaussian noise of the Ornstein–Uhlenbeck type (3.13), (3.15). If both, the thermal noise and the Ornstein–Uhlenbeck

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noise-driven barrier =uctuations are suCciently weak, transitions between neighboring potential wells of V (x) are rare, and a considerable collection of analytical results [68,243,244,246 –248] (see also further references in [68])—obtained in the context of the “resonant activation” e5ect—can be immediately applied via (3.55). Fast #uctuations (multiplicative white noise): In Section 4.1 we have discussed for a general =ashing ratchet scheme the asymptotics for a stochastic f(t) with small correlation time & under the assumption that the distribution (3.11) of the noise is kept Fxed upon variation of & (constant variance scaling). In the special case that the general =ashing potential V (x; f(t)) in (3.1) depends linearly on its argument f(t), as it is the case in our present =uctuating potential ratchet model (4.11), a second interesting scaling for small & is possible: Namely, one keeps the intensity dtf(t)f(0) of the noise constant upon variation of &. The distinguishing feature of this so-called constant intensity scaling [265] is the emergence of a sensible white noise limit & → 0 in the sense that the e5ect of the noise f(t) approaches a non-trivial limiting behavior (for constant variance scaling the noise has no e5ect for & → 0). Since the limit depends on the detailed properties of the noise f(t), we focus on the archetypal example of Ornstein–Uhlenbeck noise (3.13), (3.15) with a constant (&-independent) intensity Q := #2 =& ;

(4.14)

i.e. the variance #2 acquires an implicit dependence on &. In other words, the correlation (3.13) takes the form f(t)f(s) = 2Q& (t − s) ;

(4.15)

where 1 −|t |=& e (4.16) 2& approaches a Dirac delta function for & → 0. While this choice of f(t) is clearly of little practical relevance, it gives rise to a model that shares many interesting features with more realistic setups, but, in contrast to them, can be solved in closed analytical form [290]. Furthermore, this exactly solvable model will serve a the basis for our discussion of collective phenomena in Section 9.2. A mathematically similar setup, however, with as starting point a rather di5erent physical system, will also be encountered in Section 6.1. The fact that the coupling strength V (x(t)) of the noise f(t) in (4.11) depends on the state x(t) of the system (so-called multiplicative noise) makes the white noise limit & → 0 for constant intensity scaling subtle (so-called Ito-versus-Stratonovich problem [63,99 –101]). The basic reason is [291] that the & → 0 limit does not commute with the mˆ → 0 limit in (2.1) and the Ut → 0 limit in (2.7). We will always assume in the following that the mˆ → 0 and Ut → 0 limits are performed Frst and that & becomes small only afterwards. For the sake of completeness, we mention that this sequence of limits amounts [291] to treating the multiplicative noise in (4.11) in the so called sense of Stratonovich [63,99 –101], though we will not make use of this fact in the following but rather implicitly derive it. No such problem arises as far as the (additive) thermal noise (t) in (4.11) is concerned. Hence, we can replace for the moment the Dirac delta in (3.2) by the pre-Dirac function & (t) from (4.16) and postpone the limit & → 0 according to our purposes. It follows that in (4.11) the sum of the & (t) :=

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two independent, unbiased, & -correlated Gaussian noises −V (x(t))f(t) and (t) are statistically equivalent 55 to a single unbiased Gaussian multiplicative noise & (t) of the form V (x(t)) + g(x(t))& (t) ;

2 1=2 V (x) kB T +Q g(x) :=

x(t) ˙ =−

(4.17)

(4.18)

with correlation & (t)& (s) = 2& (t − s) : By means of the auxiliary variable x d xW ; y(x) := W 0 g(x)

(4.19)

(4.20)

it follows from (4.17) that y(t) := y(x(t)) satisFes the stochastic dynamics y(t) ˙ =−

d )(x(y(t))) + & (t) ; dy

where x(y) is the inverse of (4.20) (which obviously exists) and where x W V (x) d xW 2 )(x) := :

g (x) W 0

(4.21)

(4.22)

Next we perform the white noise limit & → 0 in (4.21). The corresponding Fokker–Planck equation for P(y; t) follows by comparison with (2.6) and (2.14), reading d 9 9 92 (4.23) P(y; t) = )(x(y)) P(y; t) + 2 P(y; t) : 9t 9y dy 9y The probability densities P(x; t) and P(y; t) are connected by the obvious relation [101] ∞ dy (x − x(y))P(y; t) = P(y(x); t)=g(x) : P(x; t) = −∞

(4.24)

Upon introducing (4.24) into (4.23) we Fnally obtain a Fokker–Planck equation (2.17) for P(x; t) with probability current 9 V (x) + g(x) g(x) P(x; t) : (4.25) J (x; t) = −

9x

Proof. The two noises 1 (t) := − V (x(t))f(t) + (t) and 2 (t) := 2g(x(t))& (t) with g(x) from (4.18) and & (t) from (4.19) are both Gaussian, have zero mean, and the same correlation, thus all their statistical properties are the same. 55

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Along the same calculations as in Section 2.4 one now can derive for the reduced density (2.22) in the steady state the result [112–115] (see also (2.36)) e−)(x) x+L e)(y) st ˆ ; (4.26) P (x) = N dy g(x) x g(y) where the normalization N is Fxed through (2.25), and for the particle current (2.26) one Fnds x ˙ = LN[1 − e)(L)−)(0) ] :

(4.27)

Discussion: Our Frst observation is [290,292] that the sign of the current is completely determined by the (reversed) sign of )(L) − )(0) = )(L) (note that )(0) = 0). Especially, the current vanishes if and only if )(L) = )(0) = 0. As expected, this is the case in the absence of the potential =uctuations (Q = 0, cf. Section 2.4) or if V (x) is symmetric. In any other case, we can infer that the current will be typically non-zero, notwithstanding the fact that only white noises are acting. These basic qualitative conclusions become immediately obvious upon inspection of the transformed dynamics (4.21): The e5ective potential )(x(y)) from (4.22) is periodic in y if and only if )(L)=0. In this case an e5ective Smoluchowski–Feynman ratchet (2.6) arises with the result y=0. ˙ If )(L) = 0, we are dealing in (4.21) with a tilted Smoluchowski–Feynman ratchet (cf. Section 2.5), yielding a current y ˙ with a sign opposite to that of )(L). Considering that the ensemble average x ˙ is equivalent to the time average of a single realization (3.5) and similarly for y(t), it follows from (4.20) that L x ˙ dx y ˙ = ; (4.28) L 0 g(x) especially the sign of x ˙ must be equal to that of y. ˙ Remarkably, this exact relation (4.28) between the currents in the original (4.17) and the transformed (4.21) dynamics remains valid for arbitrary (not necessarily Gaussian white) noises & (t) [293]. As an example we consider a piecewise linear potential with three continuously matched linear pieces per period L with the following slopes: V (x) = −1 for −2¡x¡0, V (x) = 3 for 0¡x¡1, and V (x) = 2 − 3 for 1¡x¡2, where 3 ∈ (0; 2) is a parameter that can be chosen arbitrarily. Hence, the potential V (x) has a minimum at x = 0 and barriers of equal height 2 at x = ±2. Outside this “fundamental cell” of length L = 4 the potential is periodically continued. 56 One then Fnds from (4.22) that )(L) =

2Q(1 − 32 )[Q3(2 − 3) − 3T ] : (T + Q)(T + Q32 )(T + Q(2 − 3)2 )

(4.29)

As expected, )(L) and thus the current (4.27) vanishes for 3 = 1, while for any 3 = 1 the quantity )(L) changes sign at Q = 3T=3(2 − 3). It follows that the current x, ˙ considered as a function of either T or Q undergoes a current reversal. Similar reversals as a function of any model parameter follow according to Section 3.6.

56

In doing so we are working in dimensionless units (cf. Section A.4 in Appendix A) with UV =2, L=4, and =kB =1.

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A somewhat similar observation of a current inversion arising upon considering a slight modiFcation of a symmetric saw-tooth potential V (x) has been reported for dichotomous (and other) potential =uctuations in [229], see also [216,271,272,294]. A modiFed ratchet model dynamics, without any thermal noise and no particular relation between g(x) and V (x) comparable to (4.18), but instead with a Gaussian multiplicative colored noise of Fnite correlation time & in (4.17) has been studied numerically and by means of various approximations in [292,293]. The physical viewpoint in these works is thus closely related to that of the Seebeck ratchet scheme from Section 6.1 rather than the =uctuating potential ratchet model of the present section. 4.4. Traveling potential ratchets In this section we consider a special case of the stochastic dynamics (4.1) of the form

x(t) ˙ = −V (x(t) − f(t)) + (t) :

(4.30)

As usual, we are concentrating on the overdamped limit and the thermal =uctuations are modeled by unbiased Gaussian white noise (t) of strength 2 kB T . Further, V (x) is a periodic, but not necessarily asymmetric potential with period L. Thus, the e5ective potential experienced by the particle in (4.30) is traveling along the x-axis according to the function f(t), which may be either of deterministic or stochastic nature. Upon introducing the auxiliary variable y(t) := x(t) − f(t) ;

(4.31)

in (4.30), we obtain ˙ + (t) ;

y(t) ˙ = −V (y(t)) − f(t)

(4.32)

whence the average velocity x ˙ of the original problem follows as ˙ + y x ˙ = f ˙ :

(4.33)

4.4.1. Genuine traveling potentials As a Frst example of a so-called genuine traveling potential ratchet we consider a deterministic function f(t) of the form f(t) = ut :

(4.34)

In other words, the potential V (x − f(t)) in (4.30) is given by a periodic array of traps (local minima of the potential), traveling at a constant velocity u along the x-axis. Hence, (4.30) models basically the working principle of a screw or a screw-like pumping device—both invented by Archimedes [295]—in the presence of random perturbations. Qualitatively, we expect that the Brownian particle x(t) will be dragged into the direction of the traveling potential traps. Next, we note that with (4.34), the auxiliary y-dynamics (4.32) describes the well-known overdamped motion in

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a “tilted washboard” potential 57 [99,112–115]. Quantitatively, upon comparison of (4.31) – (4.34) with (2.34) – (2.37) the average velocity in the steady state takes the form LkB T [e uL=kB T − 1] : x ˙ = u − L x+L

0 d x x dy e[V (y)−V (x)+(y−x) u]=kB T

(4.35)

Though this formula looks somewhat complicated, one sees that typically x ˙ = 0 with the expected exceptions that either u = 0 or V (x) ≡ 0. Especially, a broken spatial symmetry of the potential V (x) is not necessary for a 8nite current x. ˙ Furthermore, thermal noise is not necessary either: For T → 0 one obtains directly from (4.31), (4.32) that  L u− if u + V (x)= = 0 for all x ; L (x)= ) d x=(u + V x ˙ = (4.36) 0  u otherwise : This deterministic behavior already captures the essential features of the more involved Fnite-T expression (4.35): Namely, x ˙ has always the same sign as u but is never larger in modulus, in agreement with what one would have naively expected. 58 Furthermore, there are two basic “modes” of motion in (4.36). In the Frst case in (4.36), i.e. for large speeds u, the Brownian particle is only “loosely coupled” to the traveling potential (cf. Section 2.7), it behaves like a swimmer a=oat on the surface of the ocean and may thus be called a Brownian swimmer [296]. In the second case in (4.36), i.e. for small speeds u, we have a Brownian surfer [296], “riding” in a tightly coupled way on the traveling potential 59 in (4.30), (4.34). The current x ˙ tends to zero for both very small and very large speeds u, and has a maximum at the largest u which still supports the surFng mode. More general types of genuine traveling potential ratchets are obtained by supplementing the uniformly traveling contribution in (4.34) by an additional unbiased periodic function of t or by a stationary random process. In the transformed dynamics (4.32) this yields a tilting ratchet mechanism with an additional constant external force − u as treated in Section 5 in more details. Applications: Out of the innumerable theoretical and experimental applications of the above described simple pumping scheme (4.34) we can only mention here a small selection. The most obvious 57

Note that a positive velocity u corresponds to a washboard “tilted to the left”. Particles x(t), moving opposite to the traveling potential in (4.30), or, equivalently, particles y(t) sliding down the ˙ = −u) would indeed be quite unextilted washboard faster than in the absence of a potential V (y) (faster than −f(t) pected. Since we did not Fnd a rigorous proof in the literature, we give one herewith: If a function f(x) is concave on an interval I , i.e. f(3x + (1 − 3)y) 6 3f(x) + (1 − 3)f(y) for all 3 ∈ [0; 1] and x; y ∈ I , then it follows by induction that f(N −1 Ni=1 Xi ) 6 N −1 Ni=1 f(Xi ) for all X1; :::; N ∈ I . Choosing u¿V (x)= for all x (especially u¿0), f(x) = 1=x, 1 1 I = [0; ∞], and Xi = u + V (i=N ) it follows for N → ∞ that u−1 = [ 0 d x (u + V (x))]−1 6 0 d x (u + V (x))−1 . Working in dimensionless units with L = = 1 (cf. Section A.4 in Appendix A) we can infer that x ˙ ¿ 0 in (4.36) if u ¿ 0, and similarly x ˙ 6 0 if u 6 0. Choosing f(x) = exp{−x}, I = R, Xi = V (i=N ) − V (i=N + z) it follows that 1 1 1 x+1 1 = exp{ 0 d x [V (x + z) − V (x)]} 6 0 d x exp{V (x + z) − V (x)} for any z. Hence, 0 d x x dy exp{u(y − x) + V (y) − 1 1 1 ˙ ¿ 0 in (4.35) V (x)} = 0 d z exp{uz} 0 d x exp{V (x + z) − V (x)} ¿ 0 d z exp{uz} = [exp{u} − 1]=u. It follows that x if u ¿ 0, and similarly x ˙ 6 0 if u 6 0. 59 In the corresponding noiseless tilted washboard dynamics (4.32), the particle y(t) permanently travels downhill in the Frst case, while it quickly comes to a halt in the second. 58

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one is a particle, either suspended in a =uid or =oating on its surface, in the presence of a traveling wave [296 –301]. The resulting drift (4.36) has been predicted in the deterministic case (T =0) already by Stokes [302]. In the presence of thermal =uctuations (T ¿0), the closed analytical solution (4.35) has been pointed out in [298,300]. The e5ect of 8nite inertia has been discussed in [296], based on the known approximative solutions for the corresponding tilted washboard dynamics [99]. A model with an asymmetric potential V (x) and a dichotomous instead of a Gaussian white noise (t) in (4.30) has been studied in [301] with the possibility of current inversions as its most remarkable feature, i.e. particles can now even move in the direction opposite to that of the velocity u. This result can be readily understood as a consequence of the ratchet e5ect in the equivalent =uctuating force ratchet (4.32), (4.34), cf. Section 5.5.2. Further generalizations for superpositions of traveling waves in arbitrary dimensions are due to [297,303], which will be discussed in somewhat more detail in Section 4.5.1. The mesoscopic analog of Archimedes’ water pump is the adiabatic quantum electron pump by Thouless [304]. This theoretical concept has been realized in a quantum dot experiment in [305], triggering in turn further theoretical studies [306 –308]. Similar single electron pumps, however, operating in the classical regime [305], have been realized in [309 –312]. For additional closely related single-electron pumping experiments, see also [313,314]. A theoretical analysis of di5usion (unpredictability) in clocked reversible computers in terms of a Brownian motion in a traveling potential is given in [315]. Experimental studies of Brownian particles (2 m diameter polysterene spheres in water), moving on a circle in the presence of a traveling optical trap, have been reported in [316] and are in good agreement with the simple theoretical model (4.30), (4.34). Single-electron transport by high-frequency surface acoustic waves in a semiconductor heterostructure device has been demonstrated for example in [317]. A more sophisticated variant with excitons (electron–hole pairs) instead of electrons, which is thus able to transport “light”, has been realized in [318]. Though an asymmetry in the periodic potential V (x) is not necessary to produce a current in (4.30), (4.34), one may of course consider traveling ratchet-shaped potentials nevertheless. This situation has been addressed for instance in [319,320], leading to interesting e5ects for traveling wave trains of Fnite spatial extension (i.e. V (x) → 0 for x → ± ∞) which are re=ected at a wall and then pass by the same particle again in the opposite direction [296]. One basic e5ect of pumping particles by a traveling potential is a concentration gradient. The inversions, namely making a potential travel by exploiting a particle =ux (e.g. due to a concentration gradient) is also possible [321], as exempliFed by the chiral dynamics of a “molecular wind-mill” [322]. Note that useful (mechanical) work can be gained in either way. A number of further applications in plasma physics and quantum optics have been compiled in [296]. In fact, a great variety of engines are operating in a cyclic manner with a broken symmetry between looping forward and backward and may thus be classiFed as ratchet systems, typically of the traveling potential type. The examples of screws, water pumps, propellers, or equally spaced traveling cars (representing the traveling potential minima for the passengers) demonstrate a certain danger of invoking an overwhelming practical relevance while the underlying basic principle may become trivial from the conceptual viewpoint of contemporary theoretical physics. Furthermore, these largely mechanical examples together with our above Fndings that neither a broken symmetry nor thermal noise are necessary, current inversions are impossible, and the tight x(t)-to-f(t)-coupling

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(see Section 2.7) in the most important case of small speeds u show that many characteristic aspects of the Brownian motor concept are actually absent in genuine traveling potential ratchets. While the general framework in Section 3.1 has been purposefully set such as to include traveling potential ratchets, they are at the boarder line in so far as they often involve a quite obvious a priori preferential direction of transport. The boarder between the realm of Brownian motors and that of “pumping between reservoirs” is deFnitely crossed when in addition the spatial periodicity and noise e5ects play no longer a central role. 4.4.2. Improper traveling potentials Next, we turn to the simples type of a so-called improper traveling potential ratchet, arising through a modiFcation of the driving function f(t) from (4.34) of the form t ∞ f(t) = ut − dt ni L(t − &i ) ; (4.37) 0

i=−∞

where, the coe
˙ = lim f

(4.38)

so that x ˙ = y ˙ in (4.33), i.e. on the long term the discontinuous jumps in (4.37) have to counterbalance the continuous change ut. This condition is satisFed if and only if TunW = L ;

(4.39)

where we have introduced k

1 ni ; k → ∞ 2k + 1

nW := lim

(4.40)

i=−k k

1 (&i+1 − &i ) : T := lim k → ∞ 2k + 1

(4.41)

i=−k

Especially, the so deFned limits (4.40), (4.441) are assumed to exist and to be independent of the considered realization in the case that the summands are randomly sampled. Regarding examples, the simplest choice of the coeCcients ni in (4.37) is ni = 1 for all i. On the other hand, the simplest choice of the times &i arises if they are regularly spaced. Then T in (4.41) is obviously equal to this spacing, i.e. &i = i T + const :

(4.42)

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Another simple option are random &i with a Poisson statistics, i.e. the probability to have m time points in a time interval of duration t ¿ 0 is (t=T)m −t=T Pm (t) = : (4.43) e m! Then, in accordance with (4.41), T is again the mean value of &i+1 − &i . Returning to general processes (4.39) with integers ni respecting (4.38) or equivalently (4.39), we now come to the central point of this section, consisting in the following very simple observation: Since the discontinuous jumps of the driving f(t) in (4.37) are always integer multiples of the period L and the potential V (x) is L-periodic, the jumps of f(t) in (4.37) do not have any e1ect whatsoever on the stochastic dynamics (4.30)! In other words, the genuine traveling potential ratchet (4.30), (4.34) is equivalent to the improper traveling potential ratchet (4.30),(4.37), (4.38). Especially, the results (4.35), (4.36) for the current x ˙ can be taken over unchanged. Due to (4.33), (4.38) the ˙ same results are moreover valid for y. ˙ With (4.37) the term f(t) in the y-dynamics (4.32) takes the form ∞ ˙ =u− f(t) ni L(t − &i ) : (4.44) i=−∞

In other words, we have found that the dynamics (4.32), (4.44) with (4.38) or equivalently (4.39), admits a closed analytic solution. In the special case of a Poisson statistics (4.43) the random process (4.44) is called a Poissonian shot noise [178,323–326]. Its mean value is zero owing to (4.38), and its correlation is found to be n2 L2 ˙ f(s) ˙ f(t) = (t − s) ; T

(4.45)

k

1 2 ni ; k → ∞ 2k + 1

n2 := lim

(4.46)

i=−k

i.e. the shot noise is uncorrelated in time (white noise). The conclusion that a stochastic dynamics (4.32) in the presence of a white shot noise [327–330] or an unbiased periodic driving of the form (4.44) (with ni being integers) and simultaneously a white Gaussian noise (t) is equivalent to the Brownian motion in a traveling potential or in a tilted washboard potential and thus exactly solvable is to our knowledge new. Generalizations, equivalences, applications: Generalizations of the above arguments are obvious and we only mention here a few of them. First, an arbitrary periodic f(t) in (4.30) is equivalent to ˙ a dynamics (4.32) with a homogeneous periodic driving force f(t). Due to the periodicity of f(t), this force is unbiased in the sense of (4.38) and due to (4.33) the currents of the original (4.30) and the transformed dynamics (4.32) are thus strictly equal. Such a dynamics (4.32) will be considered under the labels “rocking ratchets” and “asymmetrically tilting ratchets” in Section 5. Both these classes of ratchets are thus equivalent to a Brownian motion in a back-and-forth traveling periodic ˙ potential. As we will see, a Fnite current x ˙ = y ˙ = 0 generically occurs if V (x) and=or f(t) is asymmetric (see Section 3.2) and unless both of them are supersymmetric according to Section 3.5. An inversion of the current upon variation of an arbitrary parameter of the model can be designed along the same line of reasoning as in Section 3.6.

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The above exempliFed procedure of transforming a genuine into an improper traveling potential ratchet is obviously very general. Due to (4.38) this transformed model can then be mapped once more onto an unbiased tilting ratchet scheme (4.32). In short, genuine and improper traveling potential models are equivalent to each other and moreover equivalent to a tilting ratchet. Finally, we turn to a modiFcation of the (genuine) uniformly traveling potential (4.30), (4.34), namely the case that f(t) and thus the periodic potential advance in discrete steps: t ∞ f(t) = dt i (t − &i ) : (4.47) 0

i=−∞

As seen before, steps i = ni L do not have any e5ect on the dynamics (4.30). Thus, the simplest non-trivial case arises when two subsequent steps add up to one period L : 2i = iL;

2i+1 = iL + 3; 3 ∈ (0; L) :

(4.48)

We furthermore assume that the jumping times &i are regularly spaced &2i = i T;

&2i+1 = iT + &; & ∈ (0; T) :

(4.49)

For 3 = L=2 and & = T=2 the signal f(t) in (4.47) is thus a discretized version of (4.34) advancing at equidistant steps in time and in space with the same average speed u = L=T. For other values of 3 and &, every second step is modiFed. More steps per period, random instead of deterministic times &i and many other generalizations are possible but do not lead to essential new e5ects. Recalling that jumps of f(t) by multiples of L do not a5ect the dynamics (4.30), we can infer that (4.47) – (4.49) is equivalent to 0 if t ∈ [0; &) ; f(t) = 3 if t ∈ [&; T) ; f(t + T) = f(t) ;

(4.50)

i.e. f(t) periodically jumps between the two values 0 and 3. Such a genuine traveling potential advancing in discrete steps is thus equivalent to a periodic switching between two shifted potentials. The periodicity of f(t) in (4.50) implies (4.38), thus the current in (4.33) agrees with that of the ˙ transformed dynamics (4.32), featuring an unbiased additive force f(t) with -peaks of weight 3 at t = iT and weight −3 at t = iT + &. As before, this equivalent dynamics establishes the connection of a stepwise traveling potential with the rocking and asymmetrically tilting ratchet schemes from Section 5. Without going into the details of the proofs we remark that: (i) A symmetric potential V (x) in combination with & = T=2 in (4.49) implies x ˙ = 0 for any 3 in (4.48). (ii) A symmetric potential V (x) at temperature T =0 implies x=0 ˙ for any & and 3. (iii) For T ¿0, 3 = L=2, & = T=2 a current x ˙ = 0 is generically expected [214]. (iv) If V (x) is asymmetric [40,331] or f(t) supports more than two e5ectively di5erent discrete states (cf. (4.50)), i.e. when transitions between more than two shifted potentials are possible [332,333], then a ratchet e5ect x ˙ = 0 is expected generically. Apart from those peculiarities, one expects basically the same qualitative behavior as for the uniformly traveling potential case (4.34). Quantitative results, conFrming this expectation, are

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exempliFed in [40,333]. Like in the continuously traveling potential case, there are again two basic modes of motion (cf. the discussion below (4.36)): One which is “loosely” coupled to the traveling potential (Brownian swimmer), and one in which the particle “rides” on the traveling wave (Brownian surfer). This clear cut distinction is washed out by the thermal noise. The detailed dependence of x ˙ on model parameters like 3, &, or shows furthermore certain traces [40] of the discrete jumps in (4.47) which are not present in the continuously traveling counterpart (4.34). An experimental realization of directed motion by switching between two shifted ratchet shaped potentials has been reported in [334]. The moving particle is a mercury droplet of about 1 mm in diameter and the two shifted ratchet potentials are created by suitably positioned electrodes. Both, for periodic and stochastic switching between the two ratchet potentials, the measured displacements agree very well with the simple T = 0 theory from [40]. The same ratchet scheme has been implemented experimentally for m-sized latex spheres in [276]. The setup is similar to the one by Rousselet et al. [38] and by Faucheux and Libchaber [273] described in Section 4.2.1 and thus the same uncertainties when comparing measurements with theory arise. The main di5erence in [276] is that now two superimposed “Christmas-tree electrodes” are used, shifted against each other so as to generate the two shifted ratchet potentials by switching the applied voltage. A further di5erence with the on–o5 experiments [38,273] is that in such a traveling potential-type setup [276] thermal =uctuations are negligible in very good approximation (cf. Section 4.4.1). For two di5erent species of highly diluted particles (latex spheres with 0.2 and 0:5 m diameters) the theoretically predicted e5ect that for a suitable choice of parameters, only one of them appreciably moves, was qualitatively veriFed in the experiment [276]. 4.5. Hybrids and further generalizations 4.5.1. Superpositions of traveling potentials In this section we consider combinations and other extensions of the =uctuating potential and traveling potential ratchet schemes from (4.11) and (4.30). As a Frst example, we consider a pulsating potential ratchet dynamics (4.1) involving superpositions of several traveling potentials [297,303] V (x; f(t)) =

Vi (x − ui t) ;

(4.51)

i

Vi (x + Li ) = Vi (x) :

(4.52)

At variance with all other cases considered in this chapter, the potential (4.51) is thus not periodic in the spatial variable x unless the periods Li are all commensurable with each other. The starting point for an approximate treatment is an expansion of the single-potential result (4.35) up to the Frst non-trivial order in V (x)=kB T , which turns out to be the second order [297,298]. The next salient point is that for several potentials one simply can, within the same approximation, add up the contributions from all the single potentials provided that their traveling speeds ui and periods Li are, in modulus, di5erent from each other. In other words, up to second order, no mixed contributions in the amplitudes of the traveling potentials appear [297,303]. Basically, the reason is that the mismatch of the di5erent temporal and spatial periods only leads to oscillating mixed terms which average out to zero in the long run. Proceeding along this line of reasoning, the Fnal result

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for the net current x ˙ is Li Li 0i d x Vi2 (x) d x x+Li dy Vi (x)Vi (y) 0i (y−x)=Li ; ui − e x ˙ = 2 0i − 1 2 L (k T ) e L L (k T ) i B i i B 0 0 x i 0i := ui Li =kB T ;

(4.53)

where it is assumed that both terms in the square brackets are small in comparison to unity, i.e. Vi (x)=kB T needs to be small, but also 0i should be not too large in modulus. SpeciFcally, for sinusoidal traveling potentials of the general form 2 V (x; f(t)) = (4.54) Ai sin (x − ui t) + )i ; Li i one obtains

−1 1 L i ui 2 2 x ˙ = ui A i 1 + : 2(kB T )2 i 2 kB T

(4.55)

Thus, already with two superimposed potentials with opposite speeds u1 and u2 and |u1 | = |u2 |, |L1 | = |L2 | one can tailor the two amplitudes A1 and A2 such that the current (4.55) will change its direction either upon variation of the temperature T or, at a Fxed but Fnite T , upon variation of the friction coeCcient . While for transport based on a single traveling potential, thermal =uctuations are not important (cf. Section 4.4.1), they are thus indispensable for this type of particle separation scheme [297]. There is no reason to expect that the above e5ect is restricted to potentials of small amplitudes, but beyond this regime quantitative analytical progress becomes cumbersome. Qualitatively, the following very simple prediction is worth mentioning: We consider in (4.1) a potential that is given as a linear combination of two potentials, moving uniformly in opposite directions, i.e. f(t) = ut and V3 (x; f(t)) := 3V1 (x + ut) + (1 − 3)V0 (x − ut) ;

(4.56)

where 3 is a control parameter and the spatial periods of V0 (x) and V1 (x) may or may not agree. Similarly as in Section 3.6 one sees that for a “generic” choice of V0 (x) and V1 (x) (no “accidental symmetries” of V3 (x; f(t)) for any 3 ∈ (0; 1)) a 3-value must exist at which the current x ˙ exhibits an inversion upon variation of an arbitrarily chosen parameter of the model (4.1). Note that in contrast to the prediction from (4.53) the present conclusion holds even if the thermal noise (t) in (4.1) is zero, see Eq. (4.36). Another variation with one static and one traveling potential, i.e. V (x; f(t)) := V0 (x) + V1 (x − ut) ;

(4.57)

has been analyzed in [335] in the zero temperature limit (t) ≡ 0 in (4.1). If at least one of the two potentials V0 (x), V1 (x) is asymmetric and their relative amplitudes are properly chosen then the traveling potential is able to drag the particle x(t) in (4.1) into one direction. However, if the traveling direction is reversed (u → −u), the particle cannot be dragged in that direction anymore due to the asymmetry of the potential. The possibility of such a behavior becomes particularly obvious in the case of small speeds u such that the particle tends to follow one of the instantaneous local minima of the total potential V (x; f(t)) in (4.57): For a very small amplitudes of V1 (x), the particle

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clearly cannot be dragged into either direction, while for very large amplitudes it can be dragged into both directions. Thus there must be an intermediate amplitude of transition from localized to commoving behavior. Due to the spatial asymmetry, this transition amplitude is typically not the same for positive and negative speeds and commoving behavior in only one direction is recovered. 4.5.2. Generalized pulsating ratchets and experimental realizations In the remainder of this section we focus again on periodic potentials (3.3) in the genuine pulsating potential ratchet setting (4.1). Still, the various possibilities of how to choose V (x; f(t)) obviously rule out an exhaustive discussion. We will restrict ourselves to a few representative examples which cover most of the existing experimental and theoretical literature and which already exhibit all main features that one may possibly expect in more general cases. The simplest example is a hybrid of a uniformly traveling and simultaneously =uctuating potential ratchet of the form ˜ V (x; f(t)) = V (x − ut)[1 + f(t)] ;

(4.58)

˜ := f(f(t)=u) ˜ where f(t) := ut and the auxiliary function f(t) is assumed to be a periodic function of its argument. By means of the same transformation of variables as in (4.31) – (4.33) one can map this model onto a purely =uctuating potential ratchet with a superimposed tilt. Thus a Fnite current x ˙ is generic and the possibility of current inversions is also immediately obvious. A similar hybrid of a traveling and simultaneously =uctuating potential ratchet arises if f(t) is not given by ut in (4.58) but instead increases in discrete steps like in (4.47). In the simplest case, a model which switches either regularly or randomly between two di5erent potentials Vm (x) (m = 1; 2) arises (cf. Eq. (4.50)), which both have the same shape but are shifted against each other and moreover di5er in their amplitudes. Observing that the on–o5 ratchet is a special case and exhibits current inversions for suitably tailored potentials [40] (cf. Section 4.2), the same property follows for the present more general situation. Going just one step further, one may consider in (4.1) the case of a periodic or random switching between two potentials Vm (x) (m = 1; 2) which have the same spatial period L but are otherwise completely independent of each other. The generic occurrence of a non-vanishing current x ˙ and the existence of current inversions for suitably chosen potentials is obvious. An experimental study of such a system has been performed by Mennerat-Robilliard et al. [336]. Laser-cooled rubidium atoms in the presence of two suitably chosen counterpropagating electromagnetic waves can switch between two e5ective optical potentials Vm (x) with the above described properties. The switching is caused by absorption–spontaneous emission cycles of the rubidium atoms and results in an average velocity x ˙ of the atoms of about 0:1 m=s. While the simple stochastic dynamics (4.1) with two alternating potentials Vm (x) is suCcient for a qualitative explanation of the observed results, a quantitative comparison would require a semiclassical or even full quantum mechanical treatment (see also Section 8.4). Another generalization are the so-called asynchronously pulsating ratchets [197] (see also Sections 3.4.2 and 6.7) and especially the so-called sluice–ratchet scheme [198,199,201]. In this latter case, the amplitudes of every second potential barrier are periodically oscillating in perfect synchrony, similarly as for a =uctuating potential ratchet. The rest of the potential barriers are also synchronously oscillating in the same way, but with a time-delay of T=4 (where T is the time-period). Thus, the Brownian particle x(t) moves forward somewhat similar to a ship in an array of sluices and may

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achieve 100% eCciency in the adiabatic limit T → ∞ (see Section 6.9). An experimental realization by semiconductor superlattice heterostructures is due to [200,202]. We close with three promising experimental implementations of a pulsating ratchet scheme on a molecular scale which have so far been partially realized. The Frst one is based on the single triptycene[4]helicene molecules which we already encountered in Section 2.1. By means of certain chemical processes which basically play the role of the non-thermal potential =uctuations in the pulsating ratchet scheme, Kelly et al. [337–339] achieved a unidirectional intramolecular rotary ◦ motion. The system is so far only a “partial” Brownian motor in that only rotations by 120 have been realized. Monodirectional rotation in a helical alcene molecule with a deFnite chirality (broken symmetry) has been investigated by Koumura et al. [338,340]. Basically, ultraviolet radiation induces transitions between two ratchet-shaped potentials which are identical except that they are shifted by half a period. In other words, a photochemical two-state pulsating (or traveling) potential ratchet scheme as anticipated theoretically in [13,28] is realized. Chemically, the light-induced switching between the two potentials corresponds to a cis–trans isomerization, and each such transition is followed by a thermally activated relaxation process. While experimentally, each of the partial steps of a full cycle has so far been only demonstrated separately, there seems no reason why the system should not be able to also rotate continuously. Finally, Gimzewski et al. [341,342] have visualized single propeller-shaped molecular rotors (hexa-tert-butyl decacyclene) deposited on a Cu-surface by means of scanning tunneling microscopy (STM). Under appropriate conditions, the molecule is observed to perform a thermally driven rotary Brownian motion within an environment which gives rise to a highly asymmetric, ratchet-shaped e5ective potential of interaction with the rotor. In principle, we are thus dealing with another molecular realization of a Smoluchowski–Feynman ratchet and pawl gadget (cf. Section 2.1), but in the present case the time resolution of the STM was too low to conFrm the absence of a preferential direction of rotation. As the authors in [341,342] propose, with the help of a second non-thermal source of noise, for example a tunnel current, it should be possible to realize a ratchet e5ect in terms of a preferential direction of rotation. Considering that such a tunnel current would not directly interact with the angular state variable of the system but rather with some internal degree of freedom (of the environment), a pulsating ratchet scheme is expected according to our general analysis from Section 3.4.2. 4.6. Biological applications: molecular pumps and motors Consider an isothermal chemical reaction in the presence of a catalyst protein, i.e. an enzyme. In the simplest case, the reaction can be described by a single reaction coordinate, cycling through a number of chemical states. A suitable working model is then an overdamped Brownian particle (reaction coordinate) in the presence of thermal =uctuations in a periodic potential. The local minima within one period represent the chemical states and looping once through the chemical cycle in one or the other direction is monitored by a forward or backward displacement of the reaction coordinate by one spatial period. Completing a cycle in one direction means that one entity of reactant molecules have been catalyzed into product molecules, while a cycle in the other direction corresponds to the reverse chemical reaction.

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Since our so simpliFed model is nothing else than a Smoluchowski–Feynman ratchet (2.6), the absence of a net current signals that we are dealing with a chemical process at equilibrium, i.e. the concentrations of reactants and products are at their equilibrium values and are not changing on the average under the action of the enzyme. If the concentrations of the reactants and products are away from their equilibrium (detailed balance) ratio, then the catalyst molecule will loop through the chemical reaction cycle preferably in one way, namely such that the reaction proceeds towards equilibrium. In the corresponding Smoluchowski–Feynman ratchet model, the periodic potential has to be supplemented by a constant tilt, 60 resulting in a stochastic dynamics of the form (2.34). Note that while the environment of the catalyst is out of equilibrium as far as the concentrations of reactants and products are concerned, the properties of the random environmental noise and of the dissipation mechanism in (2.34) are still the same as for the equilibrium system (2.6). Usually, one or several transitions between chemical states may also be (in a probabilistic or deterministic sense) accompanied by a change of the geometrical shape (mechanical conFguration) of the catalyst molecule (“mechanochemical coupling”). Transitions between such conFgurations may then be exploited for doing mechanical work. Due to the preferential direction in which these transitions are repeated as time goes on, one can systematically accumulate useful mechanical energy out of the chemical energy by keeping up the non-equilibrium concentrations of reactants and products. This conversion of chemical into mechanical energy reminds one of the working of a macroscopic combustion engine, except that everything is taking place on a molecular scale and thus thermal =uctuations must be added to the picture. Similarly as for the chemical reaction coordinate, in the simplest case the changes of the geometrical conFguration can be described by a single mechanical coordinate, originally living on a circle but easily convertible to a periodic description on the real axis. In the absence of chemical reactions, another Smoluchowski–Feynman ratchet dynamics (2.6) for the mechanical state variable arises. One suggestive way to include the e5ect of the chemical reaction is the traveling potential scheme (4.30), where x(t) and f(t) are the mechanical and chemical state variables, respectively. Thus, the traveling potential proceeds in a preferential direction in accord with the chemical reaction and thereby is dragging the mechanical coordinate along the same preferential direction. 61 Another possibility is that, instead of producing a traveling potential, the chemical process gives rise to a =uctuating potential to which the mechanical coordinate is exposed, or an even more general type of pulsating periodic potential (4.1). This general scheme seems to be indeed exploited by nature for numerous intracellular transport processes [23,24]. An example are “molecular pumps” (enzymes) in biological membranes, which transfer ions or small molecules from one side of the membrane to the other by catalyzing ATP (adenosine triphosphate) into ADP (adenosine diphosphate) and Pi (inorganic phosphate) [187,343]. Another example, also fueled by ATP, are “molecular motors” which are able to travel along 60

For a proper description of an out of equilibrium catalytic cycle on a more sophisticated level, see Section 7.3.2. This description is in terms of a discrete chemical state variable m. On the same level, a consistent description in terms of a continuous state variable x does not seem to exist (see also [186]) unless it is basically equivalent to the discrete description (activated barrier crossing limit, see Section 3.8). 61 Properly speaking, there is also a back-reaction of the mechanical on the chemical state variable. A more detailed discussion of the present modeling framework is given in Section 7.

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intracellular polymer Flaments. A detailed discussion of the latter example will be presented in Section 7, especially in the Fnal Section 7.7. Finally, we remark that in principle nothing speaks against the possibility that the general scheme could be realized not only for enzymes (proteins) but also for much simpler kinds of catalysts. 4.6.1. Externally driven molecular pumps Molecular pumps are enzymes that use energy from ATP hydrolysis to create and maintain concentration gradients of ions or other small molecules like sugars (glucose) or amino acids across membranes. As discussed before, such a chemical process requires that the concentrations of reactants and products are kept away from their equilibrium ratio. In living cells, this task is accomplished by intracellular “energy factories”, maintaining the concentration of ATP about six decades above its thermal equilibrium value. Experimentally, there exists another interesting option [16,187], namely to suppress ATP hydrolysis either by low temperature or by bringing the ATP concentration close to its equilibrium value and instead apply an external time-dependent electric Feld. Without the Feld and in the absence of ATP hydrolysis, we thus recover the Smoluchowski–Feynman model (2.6) for the mechanical state variable x(t) of the molecular pump. 62 Since ATP hydrolysis is suppressed, the chemical state variable, previously denoted by f(t), can be omitted in the following discussion and the same symbol f(t) is now used for the external Feld. We Frst consider the case that the Feld f(t) only couples to the mechanical coordinate x(t) of the enzyme but not to the pumped molecule (e.g. ions are excluded if f(t) is an electrical Feld). As a consequence, the e5ective potential V (x; f(t)) experienced by the mechanical state variable x(t) changes its shape as a function of f(t) but will maintain always the same spatial periodicity. The corresponding model dynamics is thus of the general form (4.1). Though the detailed shape of the =uctuating periodic potential V (x; f(t)) is usually not known, the occurrence of a ratchet e5ect is generically expected for a very broad class of periodic or randomly =uctuating external driving signals f(t). In other words, the molecular pump starts to loop in one or the other preferential direction and so pumps molecules across the membrane. An external driving can thus substitute for the chemical energy from ATP hydrolysis to power the molecular pump, i.e. f(t) in (4.1) may represent either the chemical reaction coordinate or the external driving Feld, the main consequence x ˙ = 0 is the same. The more general case that the external Feld not only induces a pumping of molecules across the membrane but also leads to a production of ATP out of ADP and Pi is discussed in [21,22]. Finally, ATP-driven pumping may also induce electrical Felds in the vicinity of the enzyme [21,22]. If the substance transported by the molecular pump itself couples to the external Feld f(t), e.g. a ion when f(t) is an electrical Feld, then the total potential experienced by the mechanical state variable x(t) acquires a tilt in addition to the periodic contribution. If the mechanical coordinate x(t) does not couple to the Feld, then the periodic contribution to the total potential is always the same and we recover the tilting ratchet scheme from Section 5. If the Feld a5ects both the transported substance and the enzyme, a combined pulsating and tilting ratchet mechanism will result. 62

We recall that the mechanical coordinate represents the geometrical shape of the enzyme. Since ions or molecules are mechanically transferred through the membrane, a displacement of the mechanical coordinate monitors at the same time the pumping of ions or molecules.

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For periodic Felds f(t), this e5ect has been discovered experimentally in [19,20,188] and explained theoretically in [21,22,344,345] by means of a model with a discretized mechanical coordinate, hopping between four states at certain rates which change under the in=uence of the Feld f(t). Employing the same type of models, the possibility that also a randomly =uctuating Feld f(t) of zero average may put molecular pumps to work was Frst predicted in [158,255] and subsequently veriFed experimentally in [256,257]. Though these and later discussions [346 –348] are conducted mainly in the language of discrete state kinetic models, the underlying physical picture is equivalent to the spatially continuous ratchet paradigm [16,187,294,349,350]. Indeed, a plot reminiscent of a =uctuating potential ratchet (restricted to a fraction of the full spatial period) appears already in [255] and a full-=edged traveling ratchet scheme is depicted in [351]. Note also the close connection to the resonant activation e5ect from Section 3.8. 5. Tilting ratchets 5.1. Model At the focus of this chapter is the one-dimensional overdamped stochastic dynamics

x(t) ˙ = −V (x(t)) + y(t) + (t) :

(5.1)

Here, as discussed in detail in Section 3.1, V (x) is a L-periodic potential, (t) is a white Gaussian noise of strength 2 kB T , and y(t) is either an unbiased T-periodic function or an unbiased stationary random process (especially independent of (t) and x(t)). With respect to the load force F from (3.1), we immediately focus on the case of main interest F = 0. According to Curie’s principle (Section 2.7), noise induced transport is expected when the system is permanently kept away from thermal equilibrium and does not exhibit a spatial inversion symmetry. Within the model (5.1), these requirements can be met in two basic ways: The Frst option is an asymmetric “ratchet-potential” V (x) in combination with a perturbation y(t) which is symmetric under inversion y(t) → −y(t) (see Section 3.2 for a precise deFnition), amounting to a “=uctuating force ratchet” if y(t) is a random process, and to a “rocking ratchet” if y(t) is periodic in t. The second option is a spatially symmetric V (x) in combination with a broken symmetry of y(t), called an “asymmetrically tilting ratchet”. A few models which go beyond the basic form (5.1) are also included in the present section: Before all, this concerns the discussion of photovolatic e5ects in Section 5.6. Moreover, the in=uence of Fnite inertia is discussed in Section 5.8, while two-dimensional generalizations are the subject of Section 5.9. 5.2. Adiabatic approximation The simplest situation in (5.1) arises if the changes of y(t) in the course of time are extremely slow [11]. Then, at any given instant t, the particle current has practically the same value as the steady state current (2.37) for the tilted Smoluchowski–Feynman ratchet (2.34) with a static tilt F = y(t). Like in Section 2.10, this so-called adiabatic approximation thus corresponds to an accompanying steady state description in which the time t plays the role of a parameter.

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For a periodic driving y(t + T) = y(t), the time-averaged current (3.5) in the adiabatic approximation thus follows as [11,42,52] T 1 1 x ˙ = dt v(y(t)) = dh v(y(h)) ˆ ; (5.2) T 0 0 v(y) :=

L 0

LkB T [1 − e−Ly=kB T ] ; x+L d x x d z e[V (z)−V (x)−(z−x)y]=kB T

y(h) ˆ := y(hT) :

(5.3) (5.4)

Similarly as in Section 2.10, it is assumed that apart from the variation of the time-period T itself, the shape of y(t) does not change, i.e. y(h) ˆ is a T-independent function of h. As a consequence, the right-hand side of (5.2) is independent of T, in close analogy to Eq. (2.57). In the zero-temperature limit T → 0, one Fnds similarly as in (4.36) that 

L   if y = V (x) for all x ;  L d x=(y − V (x)) 0 v(y; T = 0) = (5.5)   0 otherwise : For Fnite but very small temperatures T this result is only slightly modiFed if y = V (x) for all x. In the opposite case, there are (y-dependent) solutions x = xmax and x = xmin of y = V (x) with the property that xmax maximizes V (x) − xy within the interval [xmin ; xmin + L] and xmin minimizes V (x) − xy within [xmax − L; xmax ], cf. Section 2.5.1. From (2.40) – (2.46) we can read o5 that v(y) = L[k+ − k− ] =

L|V (xmax ) V (xmin )|1=2 −UV (y)=kB T e [1 − e−yL=kB T ] ; 2

UV (y) := V (xmax ) − V (xmin ) − (xmax − xmin )y

(5.6) (5.7)

for suCciently small temperatures such that kB T UV (y); UV (y) − yL. If y(t) in (5.1) is an unbiased stochastic process with an extremely large correlation time (cf. (4.2)) ∞ dt y(t)y(s) ; (5.8) & := −∞ 2y2 (t) then one obtains along the same line of reasoning as in (5.2) the adiabatic approximation [352] ∞ x ˙ = dy %(y)v(y) : (5.9) −∞

Here, %(y) is the distribution of the noise (cf. (3.11)) %(y) := (y − y(t))

(5.10)

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and it is required that %(y) does not change upon variation of the correlation time &. We have encountered this so-called constant variance scaling assumption already in Section 4.1 and it is obviously the counterpart of the T-independence of y(h) ˆ from (5.4) in the case of a periodic driving y(t). For general analytic conclusions, the adiabatic expressions (5.2) or (5.9) are still too complicated, one has to plot concrete examples with the help of (5.3) numerically. Only in particularly simple special cases one may also be able to directly predict the direction of the current. Such an example arises if y(t) can take only two possible values ±y0 with very rare deterministic or random =ips, and V (x) exhibits a very simple ratchet proFle, consisting essentially of one steep and one =at slope (see e.g. Fig. 2.2 or 4.1). Upon increasing y0 , the condition y = 0 for all x in (5.5) will be Frst satisFed either for y =y0 or y =−y0 with a resulting T =0 current in (5.2) or (5.9) with a sign equal to that of the =at slope. The intuitive picture is simple: out of the two tilted asymmetric potentials V (x) ∓ y0 x, one does not exhibit any extrema and thus supports a permanent downhill motion, while the other still exhibits extrema which act as motion-blocking barriers. One may go one step further and again decrease y0 until both V (x) ∓ y0 x exhibit potential barriers and thus prohibit deterministic motion. One readily sees that the barrier induced by the steeper slope of V (x) is higher than that induced by the =atter slope. With (5.6) it follows again that for weak thermal noise the current goes into the direction of the =at slope of V (x). Similarly, for an asymmetrically tilting ratchet with only two possible values y± for y(t) and a symmetric potential V (x), the sign of the bigger slope y± in modulus dictates the sign of the current x. ˙ Numerical evaluations [11,15,42,116,182,183,193,224,228,236,265,352–357] of the adiabatic expressions (5.2) or (5.9) for more complicated drivings y(t) but still relatively “simple” potentials like in Figs. 2.2 and 4.1 lead to analogous conclusions. Another noteworthy feature arises if only small y-values are known to play a signiFcant role in the expression for the adiabatic current (5.2) or (5.9). For T = 0 it immediately follows from (5.5) that x ˙ = 0. For T ¿0 and suCciently small y, one may linearize (5.3) to obtain L −1 L 2 −V (x)=kB T V (x)=kB T v(y) = yL

dx e dx e : (5.11) 0

0

Since y(t) is unbiased, see (3.9) or (3.10), one recovers again x ˙ = 0 from (5.2) or (5.9), in agreement with the general prediction from Section 3.7. 5.3. Fast tilting In the case of a stochastic process y(t) with a very small correlation time (5.8) one may proceed under the assumption of constant variance scaling along the same line of reasoning like for the fast pulsating ratchet scheme in Section 4.1. Thus, we can replace in leading-order & the random precess y(t) by a white Gaussian noise with the same intensity dt y(t)y(0) = 2&y2 (0). Like in Section 4.3.2, the resulting two independent Gaussian white noises in (5.1) can be lumped into a single Gaussian white noise. We thus recover an e5ective Smoluchowski–Feynman ratchet, implying that in leading order & no current x ˙ is obtained. 63 Since it is not possible to extend the above 63

Strictly speaking, our argument is only valid for T ¿0. The conclusion, however, also remains true for T = 0, see at the end of Section 5.5.

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simple type of argument to higher orders in &, such higher order results have to be derived separately for each speciFc type of noise y(t). Similarly, for periodic perturbations y(t) one Fnds zero current in leading order of the period T and one has to proceed to higher orders. The main conclusion of those various expansions, reviewed in more detail in the next section, is that the current x ˙ for fast tilting ratchets vanishes in leading order and depends on the detailed properties of y(t) in higher orders, i.e. no simple universal results as for the pulsating ratchets in Section 4.1 are accessible. 5.4. Comparison with pulsating ratchets From Section 5.2 we can infer as a Frst major di5erence in comparison with the pulsating ratchet scheme that for tilting ratchets, a 8nite current x ˙ is generically observed in the limit of adiabatically slow tilting. Since in experiments it is often diCcult to go beyond the adiabatic regime, this feature is an invaluable advantage of the tilting ratchet paradigm. An interesting exceptional class of asymmetrically tilting ratchets will be discussed in Section 5.12. Our second conclusion is that the “natural” current direction in 9uctuating force and rocking ratchets is given by the sign of the 9at potential slope. Comparison with Section 4.3.1 shows that this “natural” direction is just opposite to the “natural” direction in a 9uctuating potential ratchet. A similar “natural” direction can be identiFed for asymmetrically tilting ratchets. However, precise criteria of “simplicity” such that this natural current direction is actually realized are not available. 64 Opposite current directions can deFnitely been observed in more complicated potentials V (x) and also for “simple” potentials outside the adiabatic regime. Examples will be given later and can also been constructed along the lines of Section 3.6. A third major di5erence in comparison with the =uctuating potential ratchet model is that thermal noise is not indispensable for the occurrence of the ratchets e1ect provided suCciently large tilting forces y(t) appear in (5.2) or (5.9) such that a Fnite velocity in (5.5) is possible. This feature is of particular conceptual appeal in the case of a stationary stochastic process y(t) with unrestricted support of %(y), e.g. a Gaussian distributed noise. In the absence of the thermal noise (t) in (5.1) we obtain a ratchet e5ect for a system in a non-equilibrium environment of archetypal simplicity, 65 see also Section 3. Such models have been extensively studied under the label colored noise problem, see [67] for a review. In Section 5.3 we have found that (within a constant variance scaling scheme) the current vanishes in leading order of the characteristic time scale in the fast tilting limit. Along a completely analogous line of reasoning one sees that for a Gaussian noise driven =uctuating potential ratchet within a constant intensity scaling scheme the current still vanishes in the white noise limit, while it remains Fnite for a =uctuating potential ratchet, see Section 4.3.2 (for a traveling potential ratchet this limit is not well deFned). In other words, both in the fast and slow driving limits, pulsating and tilting ratchets behave fundamentally di1erent. 64 Such precise criteria would probably be very complicated (in the worst case a huge lookup table) and thus of little practical use and moreover di5erent for any type of ratchet. On the other hand, there will be also many “complicated” examples which nevertheless lead to a “natural” current direction. 65 We may always consider y(t) + (t) in (5.1) as a single noise, stemming from one and the same non-equilibrium heat bath, but for T = 0 this viewpoint is not very “natural”.

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5.5. Fluctuating force ratchets In this section we consider the tilting ratchet scheme (5.1) with a spatially asymmetric, L-periodic potential V (x) and a =uctuating force y(t) which is given by a stationary stochastic process, symmetric under inversion y(t) → −y(t) (in the statistical sense, see Section 3.2), hence in particular unbiased (3.10). Physically, this gives rise to a model of paradigmatic simplicity for a system under the in=uence of a non-thermal heat bath. Similarly as for the so-called “colored noise problem” [67], the setup is mainly of conceptual interest, while its direct applicability to real systems is limited, see also Sections 3.4.2 and 5.5.2. As argued in the preceding section, if y(t) is another Gaussian white noise then we are dealing with an e5ective Smoluchowski–Feynman ratchet. Hence, to obtain directed transport one either has to invoke a correlated (non-white), Gaussian or non-Gaussian noise (“colored noise”), giving rise to a so-called correlation ratchet 66 [265,353], or a white, non-Gaussian noise. As far a Gaussian colored noise is concerned, its properties are completely Fxed by the Frst and second moments y(t) = y(0) and y(t)y(s) = y(t − s)y(0) [100,101]. Focusing on unbiased stationary examples, the distribution is thus always given by (3.15), while the correlation y(t)y(s) can be chosen largely arbitrarily. 67 The simplest example is Ornstein–Uhlenbeck noise with an exponentially decaying correlation (3.11). A standard example of a non-Gaussian colored noise is the symmetric dichotomous noise from (3.12) – (3.13). A further example of interest is its generalization with three instead of two states [358], i.e. the noise y(t) can take three possible values, y(t) ∈ {−B; 0; B}. The transition rates from ±B to 0 are deFned as 1=& and the backward rates from 0 to ±B as 3=&, k±B → 0 = 1=& ;

k0 → ±B = 3=& :

(5.12)

This so-called three state noise is thus characterized by the three parameters B; &; 3¿0. Note that the so deFned & is proportional but in general not identical to the correlation time deFned in (5.8). The rather lengthy expression for the proportionality factor follows from a straightforward calculation but is of no further interest for us. In the context of the above three state noise, & will always be understood as given by (5.12) rather than (5.8). The special case of a dichotomous noise is recovered in the limit 3 → ∞. Finally, so-called symmetric Poissonian shot noise is deFned as [178,323–326] ∞ y(t) = yi (t − &i ) ; (5.13) i=−∞

where the “spiking times” &i are independently sampled (thus Poissonian) random numbers with average interspike distance k

1 (&i+1 − &i ) : k → ∞ 2k + 1

T := lim

(5.14)

i=−k

66

The same name has been introduced in [17] for a =uctuating potential ratchet in our present nomenclature. One obvious restriction is that the intensity dt y(t)y(s) and hence the correlation time in (5.8) must not be negative nor inFnite. 67

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Furthermore, the spiking amplitudes yi in (5.13) are random numbers, independent of each other and of the &i , distributed according to some symmetric distribution P(yi ). SpeciFcally, we will consider the example P(yi ) =

1 −|yi |=A e ; 2A

(5.15)

yielding a correlation of the form y(t)y(s) = 2 A2 T−1 (t − s) ;

(5.16)

i.e. this type of shot noise is uncorrelated (white noise) with two model parameters T and A. Yet, in close analogy to correlated noise (cf. Sections 4.1 and 5.2), the T-dependence of y(t) is of the form y(t=T) ˆ with a suitably deFned, T-independent Poissonian white shot noise y(t). ˆ Note that a similar (but asymmetric) type of Poissonian shot noise has already been encountered in (4.43) – (4.45) and will later appear again in the asymmetrically tilting ratchet scheme in Section 5.12.2. Throughout the present review, Poissonian shot noise (symmetric or not) will be employed as an interesting abstract example process of archetypal simplicity. For concrete applications in various contexts of electronic devices and solid state physics see for instance [137]. For models of chemical reactions and other transport processes in gases we refer to [359]. We furthermore remark that the above Poissonian symmetric shot noise can be recovered [326] as a limiting case of the three state noise (5.12) if & → 0; 3 → 0; B → ∞; T := &=23;

A := &B

Fxed :

(5.17)

5.5.1. Fast 9uctuating forces We Frst address the case of a correlation ratchet (colored noise y(t)) in the regime of small correlation times & in (5.8). Examples are a dichotomous noise or an Ornstein–Uhlenbeck noise, cf. (3.12) – (3.15). As mentioned before (see Section 5.3), a simple leading-order & argument as for the pulsating ratchet scheme in Section 4.1 yields the trivial result x ˙ = 0, i.e. the correlation ratchet is in some sense reluctant to obey Curie’s principle in the fast noise regime. Higher order & contributions require a separate perturbation calculation for each type of noise y(t), similar in spirit as the example in Appendix C. The result of such perturbation calculations for various types of noises y(t), among others symmetric dichotomous noise, three-state noise, and Ornstein–Uhlenbeck noise, can be written in the general form [35,49,352,358,360] L L &3 Ly2 (0) Y1 [y2 (0)=(kB T )2 ] 0 d x [V (x)]3 + Y2 0 d x V (x)[V (x)]2 x ˙ =− ; (5.18) L L

kB T d x eV (x)=kB T d x e−V (x)=kB T 0

0

where Y1; 2 are dimensionless and &-independent characteristic numbers of the speciFc noise y(t) under consideration. For instance, for a dichotomous process one has [49,352,360] Y1(DN) = 1;

Y2(DN) = 1 :

(5.19)

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For Ornstein–Uhlenbeck noise one can infer from [35,37,265,292,293,352] that Y1(OU) = 0;

Y2(OU) = 1 :

(5.20)

For the three-state noise from (5.12) one has [358] Y1(3) = [3) − )2 − 1]=)3 ;

Y2(3) = 1=) ;

(5.21)

where the so-called “=atness” is deFned as ) := y4 (0)=y2 (0)2 :

(5.22)

For the speciFc case of the three-state noise in (5.12) one obtains the result ) = 1 + 1=23, which has to be substituted in (5.21). The following assumptions are crucial for the validity of (5.18): (i) constant variance scaling of the colored noise y(t); (ii) Fnite thermal noise strength T ¿0; (iii) smooth potential V (x). It is not proven but may be expected as an educated guess that the general form (5.18) of the small-& asymptotics remains valid even beyond the various examples of colored noises y(t) so far covered in [35,49,265,352,358,360]. Turning to the case of the symmetric Poissonian white shot noise (5.13), (5.15) one readily recovers the asymptotic behavior [126] for small characteristic times in (5.14) from the behavior of the three-state noise (5.18), (5.21) in the limit (5.17). Remarkably, the result is then again given by the same formula as in (5.18) if one makes the “natural” replacement &3 y2 (0)2 → T3 A4

(5.23)

and with Y1(shot) = −1;

Y2(shot) = 0 :

(5.24)

Our Frst observation in (5.18) is that x ˙ vanished not only in leading order &, as already mentioned above, but also in second order, i.e. the fast 9uctuating force ratchet is very reluctant to produce a current. Second, the functional dependence on the potential V (x) in (5.18) becomes identical to the =uctuating potential asymptotics in (4.10) when Y2 → 0 (e.g. for shot noise) and identical to the asymptotics for the temperature ratchet in (2.58) when Y1 → 0 (e.g. for Ornstein–Uhlenbeck noise). This comparison gives also a quantitative =avor about the necessary caution to be observed when comparing “natural” directions in =uctuation force and =uctuating potential ratchets. Regarding the quantity Y1 in (5.18), it has been conjectured in [35,49] that, for a rather general class of colored noises y(t), it is given by a simple function of the =atness (5.22), e.g. Y1 = 2 − ) for dichotomous and Ornstein–Uhlenbeck noise, and by (5.21) for the three-state noise (5.12). So far neither a proof nor a counterexample seems to be known. The coeCcient Y2 depends in general on additional details of the noise y(t). E.g. for Gaussian noise (3.15) but with a correlation which is not given by the pure exponential decay (3.13), the =atness in (5.22) is obviously always the same, while the expression for Y2 is in general di5erent from the one in (5.20), as can be concluded from [37,227] (see also footnote 69 below).

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The direction of the current in (5.18) is determined by the characteristics Y1; 2 of the noise y(t) and the integrals over [V (x)]3 and V (x) [V (x)]2 . The latter fact makes once more explicit the warning from Section 3.6 that beyond the most primitive potential shapes, there exists no simple rules and natural directions any more, the sign of x ˙ depends on all the details of V (x) [265]. Another surprising observation [35] is that a current inversion solely upon changing the statistical properties of the noise y(t) is possible. An example [358] is the three-state noise (5.12) which in the shot noise limit (5.17) gives, according to (5.24), rise to a current direction in (5.18) opposite to that for the dichotomous noise limit 3 → ∞ (see (5.19)), at least for y2 (0)kB T . In fact, when y2 (0)kB T such an inversion upon changing the noise statistics will occur for any potential V (x) due to the factorization of the noise- and potential-properties in the numerator of (5.18) and is thus of a very di5erent nature than the inversion-tailoring procedure from Section 3.6. For the case of Ornstein–Uhlenbeck noise y(t), the existence of rather simple looking potentials V (x) has been Frst pointed out in [265] which give rise in the adiabatic limit & → ∞ to a current x ˙ in the corresponding “natural” direction (see Section 5.4), but in the opposite direction in the small-& limit according to (5.18), (5.20). As a consequence, a current inversion upon variation of the correlation time & has been predicted theoretically and veriFed by precise numerical results in [265]. An analogous theoretical prediction and numerical veriFcation in the case of dichotomous noise y(t) is due to [250]. Considering that for simple (saw-tooth-like, but smooth) potentials V (x), the “natural” current direction will deFnitely be recovered in the adiabatic limit & → ∞ (cf. Section 5.2), a current inversion as a function of the correlation time & follows also for a three-state noise with suitably chosen parameters in (5.12) [224], see also [352,358,361,362]. A similar conclusion was reached in [226] for a modiFed three-state noise y(t) with broken symmetry by cycling through the three states in a preferential sequence (see also [333]). We remark that the three-state noise y(t) from [226] is supersymmetric according to (3.40), hence V (x) must not be supersymmetric (but may still be symmetric) in order that x ˙ = 0. We recall that mere the existence of current inversions as exempliFed above are just special cases of our general current inversion tailoring procedure from Section 3.6. For a more detailed quantitative control of the e5ect, analytical approximations as exploited above are however invaluable. We conclude our discussion of the fast potential =uctuation asymptotics with some remarks regarding the validity conditions (i) – (iii) mentioned below (5.22). First, if the potential V (x) is not smooth, then the second integral in (5.18) is ill-behaving. The adequate small-& analysis becomes much more involved and yields an “anomalous” &5=2 leading-order behavior [49,360,363]. Paradoxically, a piecewise linear saw-tooth potential (Fig. 4.1), originally introduced as a stylized approximation of more realistic, smooth potentials in order to simplify the mathematics, actually makes the calculations more diCcult for & → 0. Second, we remark that while we are exclusively using here a constant variance scaling for the noise y(t), in the literature on the small-& asymptotics a constant intensity scaling is often (but not always) employed. Third, in the case T = 0, which we excluded so far, one Fnds within the realm of constant variance scaling that for small & the current x ˙ approaches zero faster than any power of & (for constant intensity scaling see [361]). 5.5.2. Speci8c types of 9uctuating forces Beyond the fast and adiabatically slow =uctuating force limits, there has been a great variety of analytical and numerical studies. We restrict ourselves to a brief overview of the main analytical results and numerically observed e5ects together with the few so far suggested or actually realized

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experimental systems. For a more detailed discussion of the interesting special features in each particular case we refer to the cited works. Dichotomous noise: For a dichotomous process y(t) (see (3.12), (3.13)) closed, though not very transparent analytical solutions are possible for T = 0 and arbitrary V (x) [35,193,352,364] and for arbitrary T and piecewise linear V (x) [15,288,352,365] along the same lines as for the =uctuating potential scheme described in Section 4.3.1. For T =0 one sees from (5.5), (5.9), (3.12) that in the adiabatic limit the current vanishes for small amplitudes # of y(t). Upon increasing #, the current x ˙ as a function of #, sets in continuously but with a jump in its derivative when one of the two e5ective potentials V (x) ∓ #x in (5.1) ceases to exhibit barriers against overdamped deterministic motion. A similar discontinuous derivative appears when the extrema of the other e5ective potential disappear. Upon adding in (5.1) a load force F (and keeping # Fxed), two analogous jumps in the “di5erential resistance” 9x=9F ˙ arise, while x ˙ itself is always continuous. The same features are recovered [193,195] not only in the adiabatic limit but for any Fnite correlation time &, basically because the noise y(t) may remain with small but Fnite probabilities in the same state +# or −# for arbitrary long times. If x ˙ = 0 for T =0, then a straightforward perturbation expansion for small but Fnite T is possible with the expected result of small corrections to the unperturbed result x. ˙ More challenging is the case that x ˙ = 0 for T = 0, calling for a so-named singular perturbation theory for small T , see Section 3.8. This task has been solved in [250] by a rate calculation based on WKB-type methods which become asymptotically exact for small T for both, arbitrary correlation times & and arbitrary (smooth) potentials V (x). The connection between the rates obtained in this way and the current then follows as usual from (3.55), yielding a very good agreement with accurate numerical results [250]. An experimental ratchet system with additive dichotomous =uctuations has been proposed by way of combining in an electric circuit two components that will both be discussed separately in more detail below: On the one hand, an asymmetric DC-SQUID (superconducting quantum interference device) threatened by a magnetic =ux gives rise to an e5ective ratchet-shaped potential for the phase, see [354] and Section 5.10. On the other hand, a point contact with a defect, tunneling incoherently between two states, can act as a source of dichotomous noise, see [193] and Section 5.12.2. Studies based on an experimental analog electronic circuit have been performed for negligibly small thermal noise T → 0 both in the overdamped limit as well as in the presence of a Fnite inertia term m x(t) M on the right-hand side of (5.1) in [194,195]. Inertia-like e5ects have also been theoretically addressed, both for dichotomously =uctuating potential and =uctuating force ratchets, in [366]. Gaussian noise: The simplest type of Gaussian noise y(t) is Ornstein–Uhlenbeck noise, characterized by an exponentially decaying correlation (3.13), (3.15). A Frst, numerical study of the corresponding correlation ratchet dynamics (5.1) has been reported in [11], recognizing as main difference in comparison, e.g. with dichotomous noise, the fact that even in the absence of the thermal noise (T = 0), a ratchet e5ect x ˙ = 0 arises generically for any Fnite intensity 68 of y(t). The case T = 0 has been further studied analytically for small & in [35,265,292,293] and especially in [37], indicating that even within the restricted class of Gaussian colored noises y(t), the direction

68

The reason is the inFnite support of the distribution %(f) in (3.15) as compared to the bounded support e.g. for dichotomous noise in (3.12), thus the potential barrier cannot block completely any transport.

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of the current may change solely by modifying the statistical properties of this Gaussian noise. 69 This prediction has been numerically corroborated and extended to the Fnite-T regime in [227,367], revealing moreover multiple current inversions beyond the realm of small &. Additional details of the T = 0 case have been studied theoretically in [131,196,368–370] and by means of an experimental analog electronic circuit in [196]. Very accurate numerical results over extended parameter regimes as well as two di5erent analytical approximations for arbitrary (smooth) potentials V (x) and Ornstein–Uhlenbeck noise of arbitrary correlation time & in the activated barrier crossing regime (i.e. weak noises (t) and y(t)) are contained in [265]. These approximations exploit the connection (3.55) between the particle current x ˙ and the rate expressions from a path-integral [247] and a so-called generalized uniFed colored noise approximation [248,249], originally derived in the context of the so-called “resonant activation” e5ect. While the path integral method yields qualitatively the correct behavior over the whole & regime, including the occurrence and location of current inversions, the generalized uniFed colored noise approximation is limited to small & values, where it is superior to the path integral approach. Supplementary studies along the same lines with particular emphasis on the above-mentioned accurate numerical methods and results are contained in [367,371]. For tilting ratchets driven by Ornstein–Uhlenbeck noise y(t), several groups have studied in detail the e1ect of 8nite inertia, i.e. if on the right-hand side of (5.1) a term m x(t) M is included 70 [119,156,222,223,372,373]. Analytically, this problem represents a considerable technical challenge and the results of various approximative approaches are not always compatible. The upshot of those analytical as well as numerical explorations is the convincing demonstration that also the particle mass is a parameter, upon variation of which the current may change sign, i.e. a mass-sensitive particle separation is feasible. Similar conclusions have been reached in [195,366] for dichotomous instead of Ornstein–Uhlenbeck noise y(t). Shot noise: The symmetric Poissonian shot noise (5.13) is of interest for several reasons. First, it demonstrates that the appearance of a net current in the =uctuating potential scheme (5.1) it is not necessary that the noise y(t) is correlated in time [126]. Second, its “natural direction” is typically opposite to that of correlated noise y(t) in the adiabatic limit. E.g. in a saw-tooth potential V (x), the current direction turns out to have the same qualitative features as for the on–o5 saw-tooth potential [126] treated in Section 4.2 if one identiFes the characteristic time T from (5.14) with the correlation time in the on–o5 scheme. An intuitive explanation of this prima facie astonishing similarity follows from the discussion of the three-state noise in [35] in combination with its shot noise limiting behavior according to (5.17). Since for shot noise there is no correlation time, and the noise distribution %(y) is not well deFned, an adiabatic limit in the sense of (5.9) does not exist. The regime of a slow time scale T in (5.14) is therefore of a fundamentally di5erent nature. Again analogous to the on–o5 scheme, one Fnds [126] that the current approaches zero as T becomes very large (both for constant variance and constant intensity scaling).

69 For an unbiased, stationary Gaussian process, the statistical properties are completely determined by its correlation y(t)y(s) = y(t − s)y(0) . While Y1 = 2 − ) in (5.18) is always zero, Y2 may change its sign upon modiFcation of the correlation. The prediction from [37] is that the sign of Y2 is given by that of dt t 2 y(t)y(0) . 70 The case without a white Gaussian noise (t) but instead with an Ornstein–Uhlenbeck noise y(t), an additional periodic (rocking) force, and possibly also a memory friction (cf. Section 6.4.3) has been considered numerically in [119].

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5.6. Photovoltaic e1ects In this section we discuss experimental ratchet systems which cannot be realistically captured by the simple model (5.1) but are is physically closely related to it. In non-centrosymmetric materials, photocurrents are induced by short-wavelength irradiation (optical or X-ray illumination) in the absence of any externally applied Felds [374]. Experimental observations of this so-called photovoltaic e1ect 71 in ferroelectrics, piezoelectrics, and pyrroelectrics such as BaTiO3 or LiNbO3 can be traced back at least to the mid-1960s. The basis of its correct theoretical explanation was laid 1974 by Glass et al. [27], recognizing that it is not a surface or interface e5ect—in contrast e.g. to photovoltaic e5ects occurring in n–p junctions (see Sections 6.1 and 8.4)—but rather a bulk phenomenon with the asymmetry of the crystal lattice 72 playing a central role. Furthermore, they already touch upon the points that the absence of thermal equilibrium is another crucial precondition, that the e5ect should be a general property of a large class of materials, 73 and that the e5ect may be an attractive new method of energy conversion in large-area pyroelectric polymers or ceramics, acting e.g. as “solar cells”. These basic ideas have been subsequently developed into a full-=edged theory by Belinicher, Sturman, and others. Several hundred experimental and theoretical papers on the subject are reviewed in [28,29] and various general conclusions therein are remarkably similar to those of our present paper. For instance, the counterpart of the Smoluchowski–Feynman Gedankenexperiment in this context corresponds to the question why a steady state photovoltaic e5ect cannot exist under isotropic thermal blackbody irradiation. To answer such questions it is pointed out 74 that in the absence of gradients in concentration, temperature, or light intensity : : : the current direction is controlled : : : by the internal symmetry. It constitutes the generation of a directed current in a uniform medium on homogeneous illumination : : : in any medium (without exception) that lacks a center of symmetry : : : The absence of a center of symmetry : : : results in a current in virtually any nonequilibrium stationary state. There is no current in thermodynamic equilibrium, in accordance with the second law of thermodynamics : : : Under the nonequilibrium conditions provided by illumination, that detailed balancing mechanism is violated and the asymmetry in the elementary processes gives rise to a current : : : The photovoltaic e5ect is a kinetic e5ect and thus has various extensions. Uniform illumination in the absence of a center of symmetry may produce not only an electrical current but also =uxes of other quantities: heat (photothermal e5ect), neutral particles, spin, etc. On the other hand, light beams do not exhaust the nonequilibrium sources. : : : The disequilibrium may be not only due to light but to sound or to colliding or isotropic particle =uxes etc.

71

Practically synonyms are “photorefractive e5ect” and “photogalvanic e5ect”. Most of these systems exhibit a spatial periodicity, but this is de facto not an indispensable prerequisite in this context. 73 Examples are monocrystalline piezoelectric materials, such as ferroelectric ceramics, or liquids and gases showing natural optical activity due to a chirality of their constituent molecules. More recent systems are provided by asymmetric semiconductor superlattices and heterostructures [375]. 74 In the following we are quoting from [29], but most of these statements can be found already in [28]. 72

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Another point already recognized in various studies during the 1970s and reviewed in [28,29] is the fact that the photovoltaic e5ect is a non-linear e1ect in the irradiation 8eld amplitude, no current arises within the realm of linear response (cf. Section 5.5.1). Furthermore, current reversals upon changing the frequency or polarization of the irradiation [159,160] and upon changing the temperature [160] have been observed in this context. The microscopic theoretical analysis is conducted in terms of electron scattering processes in solids [28,29] and goes beyond our present scope. Though such an approach has little in common with our present working model (5.1), it is remarkable that veritable one-dimensional e1ective ratchet potentials exactly like in Fig. 2.1 are appearing in the discussion along these lines. We mention that it is not immediately obvious whether the e5ects of the irradiation, treated on an adequate quantum mechanical level, should be associated with a =uctuating force or rather with a rocking ratchet scheme: On the one hand, besides the direct interaction with the electrons, there may also be non-negligible e5ects of the irradiation on the host material, giving typically rise to a =uctuating potential ratchet mechanism [13]. On the other hand, the naive viewpoint that a signal, which is typically a monochromatic electrical wave, induces an electrical current suggests that the classiFcation as a rocking ratchet—as adopted in the following—may be justiFed. The photovoltaic e5ect is practically exploited in holography, beam ampliFcation and correction, wavefront reversal, etc. [29]. Basic research activity has somewhat decreased in recent years, focusing, e.g. on the so-called mesoscopic photovoltaic e1ect, where random impurities in conductors or microjunctions imitate local symmetry breaking [376,377], on X-ray-induced giant photovoltaic e5ects [378], and on photovoltaic e5ects in asymmetric semiconductor heterostructures and superlattices [375]. Another variation of the photovoltaic e5ect has been theoretically studied in [379,380]. Namely, in a mesoscopic, disordered normal-metal ring, a breaking of the inversion symmetry can be achieved by a static magnetic =ux threatening the ring, which survives even after averaging over the quenched disorder of the individual samples. As theoretically predicted in [379,380], in such a setup the non-linear response to an additional high-frequency electromagnetic Feld is a directed ring-current. While somewhat similar “persistent currents” may also exist at thermal equilibrium, i.e. in the absence of the high-frequency Feld, only away from equilibrium these currents can be exploited to do work, i.e. we are dealing with a veritable ratchet e5ect. Note that the basic ingredients are remarkably similar to the SQUID ratchet systems from Sections 5.7.3 and 5.10, but the detailed physical mechanisms are completely di5erent. Finally, worth mentioning in this context is also the generation of directed photocurrents in undoped, bulk semiconductors with an intact centrosymmetry by adjusting the relative phases of two optical beams at frequencies ! and 2!, see [381–383,780,781] and further references therein and also the discussion at the end of Section 8.3 below. Such a modiFed photovoltaic e5ect leads us beyond the realm of the rocking ratchet scheme and will be treated in more detail under the label asymmetrically tilting ratchets in Sections 5.12.1 and 8.3. 5.7. Rocking ratchets In this section we address the tilting ratchets dynamics (5.1) with an L-periodic, asymmetric potential V (x) and a T-periodic, symmetric external driving force y(t).

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5.7.1. Fast rocking limit In contrast to the slow rocking limit (adiabatic approximation), the regime of very high frequencies has turned out to be rather obstinate against analytical approximations or intuitive explanations. Attempts have been made [228,268,269] but cannot be considered as fully satisfactory. Numerical results, on the other hand, show [42] as a quite remarkable feature that in the fast rocking regime, the “natural” current direction (i.e. the one realized for “simple” potentials V (x) suCciently similar to the asymmetric saw-tooth potential from Fig. 4.1) is just opposite to the one for slow rocking. In order to Fnally conclude this issue, we sketch in the following the main steps of an analytical solution of the fast rocking asymptotics (details of these calculations will be presented in [384]). Under the assumption that the T-periodic function y(t) in (5.1) is of the form (5.4), the asymptotics of the current x ˙ in (5.1) for small T can be in principle determined along the same lines as in Appendix C. In practice, the calculations become extremely tedious since, as we will see, to obtain the Frst non-trivial contribution to the current, one has to go up to the fourth order in T. Things can be simpliFed a lot by mapping (5.1) onto an equivalent improper traveling potential ratchet dynamics (cf. (4.32)) as follows: With the deFnition X (t) := x(t) −

T yˆ (t=T) ;

1

(5.25)

where (cf. (5.4)) ˆ = y(hT) ; yˆ 0 (h) := y(h) yˆ i (h) :=

0

h

(5.26)

dsyˆ i−1 (s) +

0

1

ds syˆ i−1 (s) ;

i = 1; 2; : : : ;

(5.27)

one readily Fnds from (5.1) that T ˙

X (t) = −V X (t) + yˆ 1 (t=T) + (t) : (5.28)

1 Since the relations yˆ i (h + 1) = yˆ i (h) and 0 dh yˆ i (h) = 0 are fulFlled for i = 0; it follows with (5.27) by induction that the same relations are respected for i = 1; 2; : : : . Using the self-averaging property (3.5) of the particle current, we can thus infer from (5.25) that x ˙ = X˙ :

(5.29)

After expanding on the right-hand side of (5.28) ∞ V (k+1) (X (t)) Tyˆ 1 (t=T) k T V X (t) + yˆ 1 (t=T) = ;

k!

(5.30)

k=0

one sees that in comparison with (5.1) we have “gained” one order of T, the “perturbation” in (5.30) is of leading-order T only. Due to this simpliFcation, the approach from Appendix C is now

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applicable with a reasonable e5ort and yields the leading-T result [384] L 2T4 LY 0 d x V (x)[V (x)]2 x ˙ = ; L L

5 0 d x eV (x)=kB T 0 d x e−V (x)=kB T 1 Y := dh[yˆ 2 (h)]2 : 0

(5.31) (5.32)

Here, we have exploited that y(t) in (5.1) is symmetric (cf. (3.17)), otherwise additional terms of order T4 would appear in (5.31), see Section 5.12.1 below. Our Frst conclusion from (5.31) is that the fast rocking ratchet is exceedingly reluctant to produce a current, all contribution up to the order T3 are zero. This fact suggest that also a simple intuitive explanation of the current direction may be very diCcult to Fgure out. Second, for suCciently simple (saw-tooth-like but smooth) potentials V (x), the sign of the current in (5.31) is dictated by that of the steeper slope of V (x), and this independently of any further details of the driving y(t). Our result (5.31) thus correctly reproduces the numerical observation [42] that the “natural” current direction of the fast and slow rocking ratchets are opposite. In other words, a current inversion upon variation of T is typical (“natural”) in rocking ratchet systems at Fnite temperatures T ¿0. We Fnally remark that—much like in the approximation (2.58) for the temperature ratchet—the limits T → 0 and T → ∞ do not commute, i.e. (5.31) is not valid for a Fxed (however small) T if one lets T → ∞, cf. Section 3.7. In the special case of a fast sinusoidal driving y(t) with asymptotically small amplitude our result (5.31) reproduces the one from [269]. Also worth noting is that (5.31) is strictly quadratic in the driving amplitude (see (5.32)). Deviations from this strictly quadratic behavior are expected only in the next-to-leading-order T contributions that have been neglected in (5.31). For this reason, the limit of asymptotically large driving amplitudes can once again not be interchanged with the fast driving limit T → 0. 5.7.2. General qualitative features A Frst remarkable feature of a periodically rocked ratchet dynamics (5.1) occurs if in the deterministic limit (T → 0). Namely, the current x ˙ as a function of the rocking amplitude y(t), but also as a function of other parameters, displays a complex structure of constant “plateaux” which are separated by discontinuous jumps [11,39,42,51,228,354,368]. For a qualitative explanation we Frst note that the current x, ˙ understood as a long-time average (3.5), is independent of the initial condition 75 x(t0 ) [39]. The emergence of the current-plateaux can be analytically understood in detail for a saw-tooth potential V (x) and a driving which periodically jumps between a few discrete values [39,228], while in more complicated cases numerical solutions must be invoked [11,42,354]. Very loosely speaking, the deterministic dynamics (5.1) with periodic y(t) and (t) ≡ 0 is equivalent to a two-dimensional autonomous dynamics and thus admits as attractors generalized Fxed points and periodic orbits, where the word “generalized” refers to the fact that we identify x and x + L as far as the attracting set is concerned. Thus, in the long time limit, that is, after transient e5ects have died out, the particle is displaced by some multiple m of the spatial period L after a certain multiple 75

This property readily follows from the fact that x(t0 ) and x(t0 ) + L obviously lead to the same x ˙ and that di5erent trajectories x(t) cannot cross each other [39].

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n of the time period T, i.e. x ˙ = (L=T)(m=n) :

(5.33)

Remarkably, though x ˙ is independent of the initial condition, several generalized periodic attractors (with the same m=n) may still coexist [42]. The structural stability of these attractors implies that as a function of various model parameters, the ratio m=n and thus x ˙ jumps only at discrete points and is constant in between. In other words, a kind of locking mechanism is at work, closely related to the one responsible for the Shapiro steps in symmetric potentials with an extra tilt F on the right-hand side of (5.1) [385,386]. Further intriguing features, like the appearance of Devil’s staircases of current-plateaux or current reversals of x ˙ as a function of the driving amplitude y(t), are discussed in detail in [39,42,51,228]. Upon including the thermal noise in (5.1), the details of the complex behavior of x ˙ as a function of various model parameters is washed out. While for simple, saw-tooth-type potentials V (x) like in Figs. 2.2 and 4.1 and not too large rocking amplitudes, the deterministic (T = 0) current x ˙ is known [11,39,42,228] to always exhibit the same direction, a current inversion for suCciently fast driving sets in as soon as a Fnite amount of thermal noise (T ¿0) is added, as conFrmed by our perturbative result (5.31). If the deterministic current (T = 0) vanishes, then for weak thermal noise (small T ) an activated barrier crossing problem arises which can be reduced to an escape rate problem via (3.55). In general, analytical progress requires technically sophisticated path-intergal and WKB-type singular perturbation methods which are beyond our present scope, see also Section 3.8. Both, in the limits of small and large driving amplitudes one can readily see that the current approaches zero. Hence, there must be an “optimal” amplitude in between for which the current is maximized. Typically, the dependence of x ˙ upon the amplitude is roughly speaking of a single-humped shape [39,42], onto which, however, the previously described (non-monotonic) Fne-structure for small or zero thermal noise intensity is superimposed. 5.7.3. Applications An experimental realization of a rocking ratchet system has been reported in [387]: a mercury drop in a capillary with a periodically but asymmetrically varying diameter is subjected to an oscillating external electrical force of electrocapillary nature. While thermal =uctuations are negligible and the experimental situation is at most qualitatively captured by the one-dimensional model dynamics (5.1), besides the directed transport itself also the “resonance-like” dependence of the current x ˙ upon the rocking amplitude, as predicted theoretically, has indeed been observed in the experiment. Several further experimental realizations of the rocking ratchet scheme have been proposed: In [354] it has been demonstrated that the phase across an asymmetric SQUID threatened by a magnetic =ux may be modeled by a rocking ratchet dynamics. For more details we refer to Section 5.10 below. A second realization of the rocking ratchet scheme has been suggested in [388]: The proposed system consists of a one-dimensional 76 parallel array of Josephson junctions with alternating critical currents and junction areas in the overdamped limit, see also Section 9.1. In such a system, it can be shown that the relevant soliton-type solutions (also referred to as kinks, vortices, or =uxons) are approximately governed by a one-dimensional overdamped dynamics in an e5ective pinning potential 76

Practically, a closed-loop topology can replace the straight periodic setup of inFnite length.

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which can be chosen ratchet-shaped. In other words, such a Josephson kink can be considered as quasi-particle (endowed with e5ective mass, velocity, interaction with other kinks, and other particle-like properties) moving in an e5ective one-dimensional ratchet potential along the array and can be observed by measuring the time- and space-resolved DC voltage along the array. Taking into account an external periodic driving and thermal =uctuations, a rocking ratchet setup is thus recovered. The technical details of the problem are rather involved and Fnally require a numerical evaluation, see [388] for more details. SigniFcant experimental progress towards a realization of the ratchet e5ect in such sorts of Josephson junction arrays has been accomplished in [389]. A modiFcation, based on a continuous, one-dimensional long Josephson junction (of annular shape), has been put forward in [287]. An e5ective ratchet potential for the kink dynamics emerges either by applying an external magnetic Feld and choosing a properly deformed shape of the annular Josephson junction or by modulating its width. A further option is to deposit a suitably shaped “control line” on top of the junction in order to modulate the magnetic =ux through it [390]. In either way, not only rocking ratchets—as in [388,389]—but also 9uctuating potential ratchets (not necessarily overdamped) can be realized [287,390]. Further theoretical as well as experimental studies along closely related lines by several groups are currently in progress, see also Sections 5.6 and 8.4. As a third realization of the rocking ratchet scheme, it has been proposed in [391] that the application of an alternating current to a superconductor, patterned with an asymmetric pinning potential, can induce a systematic directed vortex motion. Thus, by an appropriate choice of the ratchet-shaped pinning potential, the rocking ratchet scheme can be exploited to continuously remove unwanted trapped magnetic =ux lines out of the bulk of superconducting materials. Quantitative estimates [391] show that thermal =uctuations are practically negligible in this application of the rocking ratchet model (5.1). For a two-dimensional version [392] of the same idea see Section 5.9. Finally, it has been predicted [237] within a simpliFed hopping-model (activated barrier crossing limit) for a crystalline surface, consisting of atomically =at terraces and monoatomic steps, that by application of an AC-Feld a surface-smoothening can be achieved due to an underlying rocking ratchet mechanism. First experimental Fndings which can be attributed this theoretically predictied e5ect are due to [393]. For additional experimental realizations see also Section 8.4. 5.8. In9uence of inertia and Hamiltonian ratchets The rocking ratchet dynamics (5.1) supplemented by a Fnite inertia term m x(t) M on the right-hand side is not only of experimental interest (cf. the asymmetric SQUID model in Section 5.10 below) but exhibits also interesting new theoretical aspects. 77 Without the noise (t), the periodically driven deterministic dynamics is equivalent to a three-dimensional autonomous dynamics and thus in general admits chaotic attractors in the “generalized” sense speciFed at the beginning of this section. Numerical simulations [170,230,231,394,395] show that a chaotic behavior is indeed realized in certain parameter regions of the model. As another crucial di5erence in comparison with the overdamped case, the current in the long time average (3.55) in general still depends on the initial conditions [170,221]. 77

Regarding the issue of Fnite inertia in traveling potential ratchets, Seebeck ratchets, =uctuating force ratchets, and quantum ratchets see Sections 4.4.1, 6.1, 5.5.2, and 8:1, respectively.

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149

As a function of various model parameters, the current shows a still much more complex behavior than in the overdamped case, including multiple inversions even for a “simple” potential-proFle like in Fig. 2.2. For very weak damping, the sign of the current is in fact predominantly opposite to that in the overdamped limit [170]. These general features of x ˙ are expected to be robust also against a certain amount of noise. The same is not expected for further interesting details of the deterministic dynamics reported in [170,230,231,394 –396], some of them strikingly reminiscent of previous Fndings in the context of deterministic di5usion in symmetric one-dimensional maps [397– 403]. Of signiFcant conceptual interest is the noiseless case in the limit of vanishing dissipation, i.e. a conservative (Hamiltonian) deterministic rocking 78 ratchet dynamics m x(t) M = −V (x(t)) + y(t) :

(5.34)

The salient di5erence in comparison with a dissipative system is the time-inversion invariance provided the time-periodic driving y(t) = y(t + T) satisFes (after an irrelevant shift of the time origin) the symmetry condition [251,221] y(−t) = y(t) ;

(5.35)

see also below 79 Eq. (3.41). Another basic feature is the generic appearance of (Hamiltonian) chaos with its complicated hierarchical Fne structure of disjoint stochastic (chaotic) layers, islands, KAM-tori, etc. [405 – 407]. As a consequence, the behavior of the system depends in general on the initial conditions 80 unless one is in the limiting case of strong (hyperbolic) Hamiltonian chaos [165,404,408,409]. Strictly speaking, this case is not generic but it is often adopted as an approximation for suCciently strong perturbations of an integrable system with initial conditions in that stochastic layer which contains x-values ˙ of either sign. While di1usive transport with its intriguing anomalous features (e.g. so-called L]evy =ights) has been analyzed in great detail [405 – 407], our understanding of directed transport in such a system with broken symmetry is considerably less well developed. Under the assumptions that the symmetry (5.35) is respected it has been predicted in [221] that x ˙ = 0 provided the initial condition x(0); x(0) ˙ is part of a stochastic layer which also contains an initial condition with x(0)=0. ˙ Especially, this prediction is independent of whether the potential V (x) is asymmetric or not. The basic reason is that such a trajectory x(t), due to ergodicity reasons, gives on the one hand, rise to the same average current (3.5) as its time-inverted counterpart z(t) := x(−t), i.e. x ˙ = z. ˙ On the other hand, one also concludes that z(t) ˙ = −x(−t) ˙ and thus x ˙ = −z. ˙ As a consequence, it follows [221] that x=0. ˙ A similar conclusion holds [221] if the symmetry conditions from Section 3.2 are respected by the potential V (x) and the periodic driving y(t) (cf. Eqs. (3.16) and (3.17)). Accordingly, the symmetry condition (5.35) may be considered in some sense as the Hamiltonian counterpart of the supersymmetry concept for overdamped systems (see Section 3.5.4). These di5erent symmetries have been explored in quantitative detail in [215] by means of a kinetic 78

A Hamiltonian generalized traveling potential ratchet model has been considered in [404]. Note that in the present context of Hamiltonian ratchets the word “rocking ratchet”—unlike in the rest of this review— is not necessarily reserved for symmetric drivings y(t), i.e. y(t) is T-periodic but need not satisfy (3.17). 80 The dependence of the current (3.5) on the initial conditions x(t0 ); x(t ˙ 0 ), and especially on the “initial phase” t0 in y(t0 ) is obvious in the special case that V (x) ≡ 0 in (5.34). Though this special case is untypical in that it does not exhibit chaos, it still captures some of the essential physics of the general case. 79

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Boltzmann-equation approach with special consideration of the weak and strong damping regimes. Returning to the limit of a Hamiltonian rocking ratchet, we can conclude that if neither of the above-mentioned symmetry conditions is satisFed then the occurrence of a Fnite current x ˙ (ratchet e5ects) is generically expected (and numerically observed) [166,215,221,782]. On the other hand, if the initial condition x(0); x(0) ˙ is not part of a stochastic layer which also contains an initial condition with x(0) ˙ = 0 then generically x ˙ = 0 even if the symmetry conditions (5.35) or (3.16) and (3.17) are respected. Examples with a Fnite current in spite of the symmetry property (5.35) are discussed in [165,408,409] (see also the previous footnote 80). Though it may be diCcult in practice, in principle the entire phase space of the Hamiltonian dynamics (5.34) can be decomposed into its di5erent ergodic components, 81 each of them characterized by its own particle current x. ˙ Next, we observe [408,409] that the “fully averaged particle” current according to the uniform (microcanonical) phase space density can be written as T L T L ∞ p0 9H ; (5.36) dt dx dp x˙ = lim dt dx dp p0 → ∞ 0 9p 0 0 0 − p0 −∞ cf. Sections 2.4 and 3.1. Since the Hamiltonian of the dynamics (5.34) is H =p2 =2m+V (x)−xy(t) it follows that the microcanonically weighted average velocity over all ergodic components in (5.36) is equal to zero [408,409]. An immediate implication of this “sum rule” is that a necessary requirement for directed transport is a mixed phase space since the microcanonical distribution is the unique invariant (reduced) density in this case and is always approached in the long time limit. In other words, even in the absence of the above-mentioned symmetries, systems with strong (hyperbolic) chaos do not admit a ratchet e5ect [408,409]. While in [221,782] the above mentioned L]evy =ights are proposed as the main reason for directed transport in Hamiltonian ratchets, the emphasis in [408,409] is put on the picture that transport in the chaotic layers has its origin in the “unbalanced” currents within the regular islands. The situation in systems with a more than two-dimensional phase space (bringing along Arnold di5usion) has so far not been considered at all. 5.9. Two-dimensional systems and entropic ratchets By explicitly keeping the dynamics that governs the driving y(t) or f(t) in the basic ratchet model dynamics (3.1) –independently of whether a back coupling is absent (see Section 3.4.2) or present (see Section 7.3.1)—one trivially ends up with a two-dimensional system. In this section, however, genuine vectorial generalizations of the basic model (3.1) are considered. The simplest case of such a two-dimensional ratchet system consists of two completely independent equations of the form (5.1), one for each spatial dimension x1 and x2 . Pro forma, one may then deFne a common total potential V (x1 ; x2 ) as the sum of the two individual potentials. Such a system o5ers the possibility to separate particles with di5erent ratios x˙1 =x˙2 according to their traveling direction in the x1 –x2 -plane. 81

In the typical case, some of them are regular and some of them are chaotic. Furthermore, the borderlines between them are the intact KAM tori. Their number is inFnite and they are arranged in a very complicated hierarchical pattern [405 – 407].

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A more complicated situation arises if the dynamics involves a non-trivial common potential V (x1 ; x2 ), periodic and=or asymmetric in only one or both arguments. An example of this kind (periodic in one component only) has been treated already in the context of Feynman’s ratchet in Eqs. (6.4) – (6.7), see also [410]. Another example (periodic in both components) which, instead of the usual linear directed transport, leads to a permanent circular motion of particles, has been worked out in [411], see also [412,413]. In fact, by giving up the requirement of a simple periodicity of the system along any straight spatial direction, it should be in principle possible to direct particles along arbitrarily pre-assigned pathways in properly designed two-dimensional systems [414,415], possibly even along di5erent routes for di5erent species of particles with identical seeds. In [116] a two-dimensional potential landscape V (x1 ; x2 ) was considered which consists of one straight “valley” along the x1 -axis and periodically repeated “side valleys” of Fnite length (dead ends). If the angle between those side-valleys and the x1 -axis is di5erent from ± =2 then the spatial symmetry along the x1 -direction is broken and a time-periodic rocking force generically induces a Fnite current x˙1 . Since this ratchet e5ect will occur even if there are no potential barriers along the x1 -axis, i.e. V (x1 ; x2 = 0) = const., the name entropic ratchet has been coined for this system [116]. Moreover, if an additional bias F is applied along the x1 -axis, a non-monotonic behavior of x˙1 as a function of F may result [116]. This so-called negative di1erential resistance has also been previously observed in the closely related context of networks with dead-ends, see [416] and further references therein. Very similar two-dimensional entropic ratchet schemes have been proposed in [417] for the purpose of separating DNA molecules (see also [418– 421] and Section 5.12.1), in [392] for the purpose of pumping, dispersing, and concentrating =uxons in superconductors by electrical AC-currents (cf. Section 5.7.3), and in [422] for the purpose of rectifying electronic currents with the help of the Coulomb blockade e5ect, see also [423]. Another two-dimensional rocking ratchet scheme is obtained by choosing a potential V (x1 ; x2 ) which has basically the e5ect of a two-dimensional, periodic array of obstacles (“scatterers”). The spatial symmetry is broken by the shape of the single obstacles, in the simplest case a triangle. In its simplest form, such a setup can be imagined as a Galton-board-type device with a broken spatial (“left–right-”) symmetry. This basic idea has been put forward already in the context of the photovoltaic e5ect in non-centrosymmetric materials, see Section 5.5. For the purpose of separating macromolecules such as DNA, two-dimensional arrays of obstacles (“sieves”) have been proposed and quantitatively analyzed in [146,282,413,424 – 426]. The technological feasibility of such sieves— however with symmetric obstacles—has been demonstrated already before these works in [427]. An experimental implementation of the same basic concept has been realized in [428] for the purpose of transporting and separating phospholipid molecules in a two-dimensional =uid bilayer. In contrast to other standard separation methods, such a rocking ratchet system is re-usable and enables continuous operation. Experimentally, transport of electrons in two-dimensional periodic arrays of triangular antidot scatterers under far-infrared irradiation has been demonstrated in [429]. With an approximative classical description of the system being justiFed in the considered parameter regime, essentially a two-dimensional rocking ratchet scheme is thus recovered. A further two-dimesional SQUID ratchet system will be treated in Section 5.10 below. Also the experimental ratchet devices described in Section 4.2.1 and at the end of Sections 4.4.2 and 8.4 –though admitting suggestive and rather faithful e5ective one-dimensional descriptions—are strictly

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speaking of two-dimensional character. A three-dimensional ratchet dynamics is discussed in Section 6.6. Further models with two degrees of freedom are treated in Sections 6.5 and 7.6. 5.10. Rocking ratchets in SQUIDs In the theoretical work [354] it has been demonstrated that the phase across an asymmetric SQUID (superconducting quantum interference device) threatened by a magnetic =ux may be modeled by a one-dimensional rocking ratchet dynamics (cf. Section 5.7.3). The starting point is the standard RSJ model (resistively shunted junction model, also called Steward–McCumber model) for the phase di5erence ’ of the macroscopic quantum mechanical wave function across a conventional Josephson junction <0 C <0 ’(t) M + ’(t) ˙ + Ic sin ’(t) = I (t) + (t) ; (5.37) 2 2 R where C, R, and Ic are the capacitance, resistance, and critical current of the junctions, I (t) is the electrical current =owing through the junction, and <0 := h=2e is the =ux quantum. Thermal =uctuations are modeled by unbiased Gaussian white noise (t) of strength 2kB T=R. For the total phase di5erence across a series of two identical such Josephson junctions one recovers [354] the same Eq. (5.37) except that ’(t) is replaced by ’(t)=2 and the noise strength 2kB T=R by kB T=R. Next, one considers a SQUID with the usual “loop”-geometry, formed by two conducting “arms” in parallel, but with two identical Josephson junctions in series in one “arm”, and a third junction with characteristics C ; R , and Ic in the other “arm”. The total current Itot through the conducting loop follows by adding the currents through both arms. Under the assumption that the loop inductance is much smaller than <0 =(Ic + Ic + Itot (t)) (see also the discussion below (5.51)), the total phase di5erence ’ across the loop is then governed by the equation [354] 1 <0 1 <0 C + C ’(t) + M + ’(t) ˙ = −V (’(t)) + Itot (t) + tot (t) ; (5.38) 2 2 2 2R R Ic V (’) := − cos(’=2) − Ic cos(’ + 2 <=<0 ) ; (5.39) 2 where < is the total magnetic =ux threatening the loop and where tot (t) is a Gaussian white noise with correlation 1 1 + (t − s) : (5.40) tot (t)tot (s) = 2kB T 2R R The noise- and time-averaged “phase current” ’ ˙ is connected to the averaged voltage U across the loop according to the second Josephson equation [354] <0 ’ ˙ (5.41) U= 2 and thus directly accessible to an experimental measurement. In other words, for appropriately chosen external currents Itot (t) and (static) magnetic Felds, a rocking ratchet dynamics is recovered from (5.38), which is in particular of the overdamped form (5.1) if Ic R2 C; Ic R2 C <0 , cf. Section A.4 in Appendix A. Potentials (5.39) with additional Fourier modes may be obtained by more complicated SQUIDs with additional “arms” in parallel.

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Next, we turn to one of the 8rst systems for which a ratchet e1ect has been theoretically described and experimentally measured [25,26]. While these early works focus on the realm of adiabatically slow rocking, the extension beyond this regime has been realized experimentally very recently in [182,183]. The setup consists of the following two-dimensional modiFcation of the above described rocking ratchet SQUID system (5.37) – (5.41): The starting point is a SQUID with the usual “loop”-geometry, consisting of one Josephson junction in each of the two parallel “arms” of the loop. The phase across the junctions in the left (index “l”) and right (index “r”) arm are thus both governed by an equation of the form (5.37). The di5erence between the two phases due to the vector potential of the enclosed magnetic =ux is governed by the “=ux quantization” relation ’l − ’r = 2
(5.42)

The enclosed =ux
(5.43)

where Ll; r are the inductances of the two junctions. Under the simplifying assumptions that Cl = Cr =: C;

Rl = Rr =: R

(5.44)

and with the deFnitions Ic :=

Ic; l + Ic; r ; 2

Ll + L r ; LW := 2 ’ :=

’l + ’ r ; 2

0I :=

Ic; l − Ic; r ; Ic; l + Ic; r

(5.45)

0L :=

Ll − Lr ; Ll + L r

(5.46)

:=

’l − ’r ; 2

(5.47)

it follows by adding and subtracting the two equations of the form (5.37) with indices “l” and “r” that [182,183] <0 C <0 9V (’(t); (t); t) Itot (t) ’(t) M + ’(t) ˙ =− + + 1 (t) ; 2 2 R 9’ 2

(5.48)

9V (’(t); (t); t) <0 C M <0 ˙ (t) = − + 2 (t) ; (t) + 2 2 R 9

(5.49)

V (’; ; t) := − Ic [cos ’ cos

− 0I sin ’ sin ] + 4<0 LW

2

<0 W L Itot (t) − < + L0

:

(5.50)

Here, Itot (t) := Il (t)+Ir (t) is the total electrical current =owing through the SQUID and (t) (i =1; 2) are two unbiased Gaussian white noises with correlation i (t)j (s) =

kB T ij (t − s) : R

Finally, the time-averaged voltage U across the loop is again given by (5.41).

(5.51)

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W Our Frst observation is that for L< <=<0 and 0 =(Ic + Itot (t)) it follows from (5.50) that we are left with an e5ective one-dimensional problem (5.48). The same type of approximation has been made in the derivation of (5.38) – (5.40). In any case, the potential (5.50) is periodic in the variable ’, while the -dependence is conFned by the quadratic term on the right-hand side. On condition that < is not a multiple of <0 =2 and that either 0I = 0 or 0L = 0, the potential (5.50) is neither inversion symmetric under (’; ) → (−’; ) nor (’; ) → (−’; − ), thus a ratchet e5ect is theoretically predicted and has been experimentally observed [182,183]. Especially, a non-vanishing externally applied magnetic Feld is necessary, since otherwise < = 0. Given that the above conditions (2<=<0 not an integer and 0I = 0 or 0L = 0) are fulFlled, it is instructive to rewrite (5.48) – (5.50) in the form <0 C <0 9V˜ (’(t); (t)) Itot (t) (5.52) ’(t) M + ’(t) ˙ =− + + 1 (t) ; 2 2 R 9’ 2 <0 ˙ 9V˜ (’(t); (t)) 0L Itot (t) <0 C M (t) + (t) = − (5.53) − + 2 (t) ; 2 2 R 9 2 < 2 <0 ˜ − : (5.54) V (’; ) := − Ic [cos ’ cos − 0I sin ’ sin ] + <0 4 LW In other words, a two-dimensional rocking ratchet scheme is recovered, with a “rocking force” which acts along the ’-direction if 0L = 0 and points into a more general direction in the ’– -plane if 0L = 0. Further studies on related Josephson ratchet systems are addressed in Sections 5.7.3 and 9.1, see also Section 5.6. 5.11. Giant enhancement of di1usion In this section we return to the overdamped, one-dimensional tilting ratchet scheme (5.1), however, with the e5ective di1usion coe
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y(t) y0 L /2

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(b)

L/2 V0 L

_y 0

Fig. 5.1. (a) Symmetric saw-tooth potential V (x) with period L and barrier height V0 . (b) Time-periodic, piecewise constant driving force y(t) with model parameters y0 (“tilt”), tt (“tilting-time”), and tw (“waiting-period”).

e5ective variation of the setup from [386,438] which enables a controlled selective enhancement of di5usion that in principle can be made arbitrarily strong. We focus on the simplest case of a symmetric sawtooth potential V (x) with period L and barrier height V0 (Fig. 5.1a) and a time-periodic driving force y(t) with three states y0 ; 0, and −y0 . As illustrated in Fig. 5.1b, time-segments of length tt with a constant tilt y(t) = ±y0 are separated by “waiting-periods” tw with vanishing y(t). Further, we henceforth restrict ourselves to weak thermal noise (t), i.e. kB T V0 . We assume that y0 ¿2V0 =L and that the initial particle distribution at time t = 0 consists of a very narrow peak at a minimum of the potential V (x), say at x = 0. As long as t 6 tt we have y(t) ≡ y0 , so the peak moves to the right under the action of the deterministic forces and also broadens slightly due to the weak thermal noise in (5.1). The deterministic time tn at which the peak crosses the nth maximum of V (x) at x = (n − 1=2)L while y(t) = y0 is acting, can be readily Fgured out explicitly [228]. If now tt just matches one of those times tn , then the original single peak is split into two equal parts and if the subsequent “waiting-interval” tw with y(t) ≡ 0 is suCciently long the two parts will proceed towards the respective nearest minimum of V (x) at x = (n − 1)L and x = nL. The result consists in two very sharp peaks after half a period t = tt + tw of the driving force y(t). Similarly, after a full period & := 2(tt + tw ) one obtains three narrow peaks at x = −L; 0; L with weights 14 ; 12 ; 14 , respectively. For the variance x2 (t) − x(t)2 one thus obtains the result L2 =2. In the same way one sees that after n periods the variance amounts to nL2 =2, yielding for the e5ective di5usion coeCcient (3.56) the expression De5 = L2 =8(tt + tw ) :

(5.55)

In the case that tt does not match any of the times tn , the initial single peak is split after half a period t = tt + tw into two peaks with unequal weights. If tt is suCciently di5erent from any tn and the thermal =uctuations are suCciently weak, one of those two peaks has negligible weight. Consequently, after a full period almost all particles will return to x = 0. The e5ective di5usion coeCcient De5 is therefore very small, in particular much smaller than for free thermal di5usion (2.11). An example of the e5ective di5usion coeCcient De5 as a function of tt is depicted in Fig. 5.2a. As usual (cf. Section 3.6) such a multi-peak-structure of De5 is not only expected upon variation of tt but also by keeping tt Fxed and varying for instance the friction coeCcient , corresponding to

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Fig. 5.2. (a) E5ective di5usion coeCcient (3.56) in units of the “bare” D from (2.11) versus scaled “tilting-time” t˜t := tt V0 = L2 from numerical simulations of the stochastic dynamics (5.1). The wiggles re=ect the statistical uncertainty due to the Fnite though extensive number of realizations. The relevant dimensionless parameters in Fig. 5.1 are kB T=V0 = 0:01; tw V0 = L2 = 0:375, and y0 L=V0 = 3. Theoretical predictions for the height of the peaks from (5.55) are indicated by stars. In addition, the theoretical estimate for the peak-widths at half-height from [228] are indicated by arrows. (b) E5ective di5usion coeCcient versus friction coeCcient from simulations of (5.1) with kB T=V0 = 0:005 and y0 L=V0 = 22. The times tt and tw = tt are both kept at Fxed values and also deFne 0 via 0 = (2Ly0 − 4V0 )=L2 tt . Theoretical predictions are indicated analogous to (a).

the situation that di5erent types of particles are moving in the same rocked periodic potential. As Fig. 5.2b demonstrates, the dynamics (5.1) can indeed act as an extremely selective device for separating di5erent types of particles by controlled, giant enhancement of di1usion. Closer inspection shows [228] that the peaks in the e5ective di5usion coeCcient De5 can in fact be made arbitrarily narrow and high by decreasing the temperature or increasing V0 at Fxed T while at the same time keeping y0 L=V0 large. Similarly, as for the friction coeCcient , particles can also be separated, e.g. according to their electrical charge since this implies di5erent values of the “coupling-parameters” V0 and y0 . All these Fndings are obviously robust against various modiFcations of the model as long as one maintains periodicity in space and time and suCciently long “waiting-periods” tt with y(t) ≡ 0 between subsequent “tilting-times” with non-vanishing y(t). A practical realization of such a particle separation device should be rather straightforward. 5.12. Asymmetrically tilting ratchets In this section we consider the ratchet model dynamics (5.1) with a symmetric, L-periodic potential V (x) in combination with a driving y(t) of broken symmetry, either periodic or stochastic. If the characteristic time scale of the driving y(t) is very large, the adiabatic approximation (5.2) for the periodic and (5.9) for the stochastic case can be applied. Exploiting the symmetry of V (x), a straightforward calculation conFrms the expected property that v(y) in (5.3) is an odd function of its argument. In general, the contributions of y and −y in (5.2) or (5.9) will not cancel each other and hence x ˙ will generically be di5erent from zero. However, even though y(t) is asymmetric, prominent examples exist for which the contributions of y and −y do cancel each other, namely those respecting supersymmetry (3.40). Examples are a periodic driving y(t) of the form (3.47) with $1 = 0 and $2 = 0 or the example depicted in Fig. 3.2. In this case, x ˙ → 0 as the characteristic time

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scale of y(t) tends to inFnity, which is a quite exceptional feature within the class of tilting ratchets. Since for fast driving the current approaches zero as well, a qualitative behavior which in fact is reminiscent of a pulsating ratchet arises. It is worth emphasizing, since it may appear counterintuitive at Frst glance, that a symmetric, but not supersymmetric potential V (x) (e.g. in (3.19) with a1 = 0 and a2 = 0) combined with a supersymmetric but not symmetric y(t) (e.g. in Fig. 3.2) generically does lead to a ratchet e5ect 82 x ˙ = 0, see also Fig. 3.3. If moreover Fnite inertia e5ects m x(t) M are included on the right-hand side of (5.1) then supersymmetry does no longer prohibit a current and thus even a pure sinusoidal potential V (x) may be chosen. 5.12.1. Periodic driving The case of slow periodic driving is covered by the adiabatic approximation (5.2). In the opposite case of a very small period T, one Fnds along the same line of reasoning as in Section 5.7.1 the leading-order asymptotics [384] L L T4 L[Y− 0 d x [V (x)]2 + Y+ 0 d x [V (x)]3 =2kB T ] x ˙ = ; (5.56) L L 4 5 0 d x eV (x)=kB T 0 d x e−V (x)=kB T 1 Y± := dh[yˆ 0 (h) ± 2yˆ 2 (h)][yˆ 2 (h)]2 ; (5.57) 0

where yˆ 0 (h) and yˆ 2 (h) are deFned in (5.26) and (5.27). Here we have exploited the symmetry of V (x). In the completely general case, the asymptotic current x ˙ is given by the sum of the contributions in (5.31) and (5.56). that if the potential V (x) is not only symmetric but also supersymmetric then LWe notice (x)]3 = 0 and thus the sign of the current in (5.56) is dictated solely by that of Y . d x [V − 0 On the other hand, for a supersymmetric driving y(t), both coeCcients Y± in (5.57) vanish, that is, x ˙ approaches zero even faster than T4 as T → 0. The possibility that a directed current, or, equivalently, a Fnite voltage under open circuit conditions, may emerge in a symmetric periodic structure when driven by unbiased, asymmetric microwave signals of the form y(t) = $1 cos(2 t=T) + $2 cos(4 t=T + <) ;

(5.58)

has been reported for the Frst time in the experimental work by Seeger and Maurer [30]. From the traditional viewpoint of response theory in this context, the basic mechanism responsible for producing a DC-output by an unbiased AC-input (5.58) then amounts to the so-called harmonic mixing of the two microwaves of frequencies 2 =T and 4 =T in the non-linear response regime. The electrical transport in such quasi-one-dimensional conductors is usually described in terms of pinned charge density waves, which in turn are modeled phenomenologically as an overdamped Brownian particle in a symmetric, periodic “pinning” potential [31,218–220]. The particle couples to the externally applied Feld (5.58) via an e5ective charge, i.e. we recover exactly the asymmetrically 82

To dissolve any remaining doubts, we have veriFed this fact by numerical simulations. A similar prediction has been put forward previously in [439,440] without, however, recognizing the subtleties of supersymmetry in this context.

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tilting ratchet model (5.1). We remark that both in the experimental work [30] and in the subsequent theoretical studies [31,218–220] no emphasis is put on the fact of generating a DC-output by means of an unbiased AC-input per se, and in this sense the ratchet e5ect has been observed only implicitly. Also worth mentioning is that the “pinning”-potential V (x) is usually assumed to be of sinusoidal shape and thus respects supersymmetry. Since the driving (5.58) becomes supersymmetric for <= =2, the current x ˙ will exactly vanish at this point [31,218,219]. This feature does no longer arise for symmetric but not supersymmetric potentials V (x) or if Fnite inertia e5ects [220] become relevant, see also [215,395]. In the context of current generation by photovoltaic e5ects (cf. at the end of Section 5.6) very closely related theoretical and experimental investigations are due to [381–383,780,781]. The same basic idea to produce a directed current by means of the asymmetric tilting ratchet scheme has also been exploited experimentally in a process called zero-integrated Feld gel electrophoresis 83 which uses unbiased pulsed electric Felds to separate chromosomal DNA [417– 421,441– 443]. The ratchet e5ect in a periodically driven, asymmetrically tilting ratchet has been independently re-discovered in [39]. Moreover, the complex structure of the current x ˙ at T = 0, featuring plateaux and Devil’s staircases, similarly as for the rocking ratchet system in Section 5.7.2, has been demonstrated for especially simple examples of asymmetrically tilting ratchet models in [39]. Further variations and extensions of such theoretical models, the details of which go beyond our present scope, can be found in [215,259,356,417,419 – 421,441,444 – 447]. For Hamiltonian (Fnite inertia, vanishing dissipation and thermal noise) and quantum mechanical asymmetrically tilting ratchet systems we refer to Sections 8.3 and 8.4, respectively. 5.12.2. Stochastic, chaotic, and quasiperiodic driving The generation of directed transport in symmetric, periodic potentials V (x) by an asymmetric stochastic driving y(t) of zero average in (5.1) has been for the Frst time exempliFed in [327–330] for the case of Poissonian white shot noise, 84 see also [179,448]. At zero thermal noise (T = 0), a closed analytical solution is available [327,329], while for T ¿0 one has to recourse to asymptotic expansions, piecewise linear potentials, or numerical evaluations [328,330]. Besides the fact of a white-noise-induced directed transport in symmetric potentials per se, the most remarkable Fnding is that the current always points into the same direction as the -spikes of the asymmetric shot noise for any periodic potential (symmetric or not, but di5erent from the trivial case V (x) ≡ 0). We are thus facing one of the rare cases for which our procedure of tailoring current inversions (see Section 3.6) cannot be applied unless an additional systematic bias F is included in (5.1). Leaving aside minor di5erences in the -spikes statistics (cf. footnote 84) the basic reason for this unidirectionality can be readily understood by the mapping onto an improper traveling potential ratchet scheme according to (4.31), (4.32) and our discussion of the corresponding current (4.33), (4.35), (4.38). The generic occurrence of a ratchet e5ect whenever y(t) breaks the symmetry (3.18) has been pointed out in [353] and exempliFed by means of an asymmetric two-state noise y(t) in the adiabatic limit, cf. (5.9). Similar conclusions have been reached in [179,180]. In the case of an asymmetric 83

The gel network in which the DNA moves does not exhibit the usual spatial periodicity but rather acts as a random potential (due to basically static obstacles) in three dimensions. 84 The speciFc shot noise considered in [327–330] is of the form (4.43), (4.44) but with the weights ni in (4.44) not being integers but rather exponentially distributed, positive random numbers, see also (5.13), (5.15).

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dichotomous noise y(t) and without thermal =uctuations (T =0) in (5.1), the exact analytical solution for arbitrary noise characteristics and potentials has been Fgured out and discussed from di5erent viewpoints in [193,355,364,449,450]. Similarly as in (5.17), the above-mentioned shot noise model [327,329] is recovered as a special limit [326] from this analytical solution for dichotomous noise. Approximations and analytically soluble particular cases in the presence of a Fnite amount of thermal noise (T ¿0) have been elaborated in [288,365]. Regarding the asymptotics of fast asymmetric tilting we remark that the expression (5.18) vanishes for symmetric potentials V (x), hence a signiFcantly di5erent structure of the leading-order behavior is expected (compare also the corresponding results (5.31) and (5.56) for periodic y(t)). For asymmetric dichotomous noise such an asymptotics has been derived in [365] within a constant intensity scaling scheme, while for constant variance scaling, as we mainly consider it in our present review, such an asymptotics has not yet been worked out. Turning to applications, it has been argued in [353] that the absence of a priori symmetry reasons and thus the appearance of an asymmetric noise y(t) should be a rather common situation in many systems far from equilibrium, especially in biochemical contexts involving catalytic cycling (cf. Sections 4.6 and 7). SpeciFcally, if y(t) represents a source of unbiased non-equilibrium current =uctuations then an asymmetrically tilting ratchet scheme can be readily realized by means of a Josephson junction [355,449,450], see (5.37). A concrete such source of current =uctuations has been pointed out in [193]. Namely, an asymmetric dichotomous noise may arise intrinsically in point contact devices with a defect which tunnels incoherently between two states [132–138]. A modiFed Josephson junction system with an asymetric total noise composed of two correlated symmetric noise sources has been proposed in [451,452], see also [453– 455]. It is well-known [397–399,456 – 458] that in many situations, a low dimensional dynamical system exhibiting deterministic chaos can induce similar e5ects as a veritable random noise. 85 In the present case of the asymmetrically tilting ratchet scheme, the emergence of directed transport (ratchet e5ect) when the driving y(t) is generated by a low dimensional chaotic dynamics has been demonstrated in [36], see also [179,180,459]. Another interesting intermediate between a stochastic and a periodic driving is represented by the case of a quasiperiodic driving y(t), bringing along the possibility of a strange non-chaotic attractor [460]. Asymmetrically tilting ratchets of this type have been studied in [181].

6. Sundry extensions In this section we address various signiFcant modiFcations and extensions of the pulsating and tilting ratchet schemes from Sections 4 and 5 as well as an additional important observable in the context of Brownian motors, namely their eCciency. Remarkably, while most of those generalizations are conceptually very di5erent from a pulsating or tilting ratchet in the original sense, an approximate or even exact mathematical equivalence can be established in several cases. In other cases, both the physics and the mathematics are fundamentally di5erent. 85

In fact, we may consider a noise (stochastic process) as generated by a chaotic deterministic dynamics in the limit of inFnitely many dimensions. The close similarity between deterministic chaos and noise is also exploited in any numerical pseudo-random number generator.

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6.1. Seebeck ratchets In this section we consider periodic systems under the in=uence of thermal =uctuations, the intensity of which exhibits a spatial variation with the same periodicity as the relevant potential, while no other non-equilibrium perturbations are acting. In a closed circuit composed of two dissimilar conductors (or two dissimilarly doped semiconductors) a permanent electric current arises when their junctions are kept at di5erent temperatures [461]. This constitutes a thermoelectric circuit that converts thermal energy into electrical energy. The e5ect has been discovered in 1822 by Seebeck and has been exploited, e.g. to provide electrical power for satellites. In essence, the Seebeck e5ect has the following microscopic origin: Due to the di5erent Fermi-levels prevailing in each of the conductors, a kind of e5ective potential ramp for the electrons arises at the junction. 86 Moving along the circuit in a deFnite direction, the electrons will encounter at one junction an increasing potential ramp and at the other junction a decreasing counterpart. When looping in the opposite direction, the roles of the ramps is exchanged. While sliding down a decreasing ramp is “for free”, climbing up an increasing ramp requires thermal activation. Therefore, if one junction is kept at a higher temperature than the other, the looping of electrons in one direction is more likely than in the other. Expanding the circular motion through the closed circuit to the real axis yields a periodic e5ective potential V (x) and a periodic temperature proFle T (x). Both have the same spatial period and each of them is typically symmetric under spatial inversion. The spatial symmetry of the system is broken in that the two periodic functions V (x) and T (x) are out of phase. 87 The simplest model for the electron motion consists in an overdamped dynamics like in (4.17) with g(x) = [kB T (x)= ]1=2 :

(6.1)

This model has been studied by BMuttiker [33] and independently by van Kampen [463], and has been further discussed by Landauer [464]. Later, similar models, either derived from a microscopic description of the environment in terms of harmonic oscillators (cf. Section 8.1), or based on a phenomenological approach have been considered in [190,191,465,466] and [292,293,467,468], respectively, see also Section 6.4. Though the physical systems behind this Seebeck ratchet model and the one in (4.17) are quite di5erent, the mathematics is practically the same and in this sense the Seebeck ratchet is closely related to a 9uctuating potential ratchet [51]. One di5erence is that in one case the potential V (x) is asymmetric and the =uctuations of this potential of course “in phase” with the “unperturbed” (average) potential, while in the other case the symmetry is broken due to a phase shift between V (x) and T (x). A second possible di5erence is that after the white noise limit & → 0 in (4.19) the adequate treatment of the multiplicative noise in (4.17) may not always be in the sense of Stratonovich. For instance, if the dynamics (4.17) arises as limiting case with negligible inertia e5ects (white noise limit & → 0 in (4.19) before the limit of vanishing mass) then [291,465,466,469,470] a white noise (t) in the sense of Ito [63,99] arises in (4.17). As a consequence, the second summand in (4.25) 86

Within this very elementary picture we neglect electron–electron interaction e5ects in the form of screening by inhomogeneous charge densities around these potential ramps [462]. 87 A ratchet e5ect also arises for asymmetric V (x) and=or T (x) in phase, however, typically in a quite di5erent physical context, see Section 4.3.2.

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takes the modiFed form 9g2 (x)=9x and the integrand in (4.26) acquires an extra factor g(y)=g(x). A still di5erent treatment of the thermal noise (t) in (4.17) may be necessary in physical contexts without an inertia term right from the beginning, see [463,464,471] and further references therein. Independent of these details, the main conclusion is that x ˙ = 0 if and only if L V (x) d x = 0 (6.2) 0 T (x) provided that both, V (x) and T (x) are L-periodic. 88 One readily veriFes that the two “systematic” conditions implying x ˙ = 0 are indeed the symmetry and supersymmetry criteria from (3.49) and (3.50), respectively. Though the Seebeck ratchet thus exhibits striking similarities with a =uctuating potential ratchet, the equivalence is not exact. However, such an exact equivalence can be readily established with respect to the more general class of pulsating ratchet models by choosing (t) ≡ 0 and 89

V (x; f(t)) = V (x) + 2 kB T (x) f(t) ; (6.3) with f(t) being a -correlated Gaussian noise. The basic physical picture underlying this mathematical equivalence is rather simple: Thermal =uctuations with a spatially periodic variation of their strength (temperature) may equivalently be viewed as (very fast) potential =uctuations (cf. Section 4.3.2). We furthermore remark that by Frst applying the transformation (6.3) to a pulsating ratchet, and then considering the symmetry and supersymmetry criteria (3.16) and (3.39) for such a pulsating ratchet model, one indeed recovers the corresponding original criteria for Seebeck ratchets in (3.49) and (3.50), respectively. Besides the Seebeck e5ect itself, another application of the model may be the electron motion in a superlattice irradiated by light through a mask of the same period but shifted with respect to the superlattice [33]. In such a case, it may no longer be justiFed to neglect inertia e5ects in the stochastic dynamics (4.17). The so-called underdamped regime of such a dynamics, i.e. friction e5ects are weak in comparison to the inertia e1ects, has been analytically treated in [472] by generalizing the methods developed in [473]. There are several well-known phenomena which may in fact be considered a close relatives of the Seebeck e5ect and thus as further instances of the corresponding ratchet scheme: First, we may augment our closed circuit, composed of two di5erently doped semiconductors, by a piece of a metal wire. In other words, we are dealing with an electrical circuit that contains a semiconductor diode (n–p junction). Again, an electrical current results if the diode is kept at a temperature di5erent from the rest of the circuit (thermogenerator), see also Sections 2.9 and 8.4. Second, the same device can also act as a photodiode or photoelement by exposing the n–p junction to a source of light. Especially, in the case of black-body irradiation, one basically recovers the previous situation with two simultaneous heat baths at di5erent temperatures. Third, one may replace the semiconductor 88

Similarly as in Eq. (4.27), the sign of the current x ˙ is found to be opposite to the sign of the intergal on the left-hand side of (6.2). Therefore, a current inversions upon variation, e.g. of is not possible in this model. 89 Strictly speaking, (6.3) does still not respect the L-periodicity (3.3). To remedy this =aw, one has to multiply the square-root in (6.3) by a factor @(x), deFned as @(x):= 1 for x ∈ [0; x0 ); @(x) := − 1 for x ∈ [x0 ; L), and @(x + L) := @(x). L The reference position x0 is then chosen such that 0 @(x)[T (x)]1=2 d x = 0 with the result that (3.3) is indeed satisFed. Note that this extra factor @(x) in (6.3) does not a5ect the stochastic dynamics (3.1) in any noticeable way.

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Fig. 6.1. Same as Fig. 2.1 but with the ratchet and pawl kept at a di5erent temperature than the paddles and its surrounding gas.

diode by a tube diode. In this context, the two above mentioned ways of generating an electrical current are then closely related to the so-called Richardson-e5ect and photoe5ect, respectively. 6.2. Feynman ratchets Throughout the discussion of Smoluchowski and Feynman’s Gedankenexperiment in Section 2.1.1, we have assumed that the entire gadget in Fig. 2.1 is surrounded by a gas at thermal equilibrium. In his lectures [2], Feynman also goes one step further in considering the case that the gas around the paddles is in a box at temperature T1 , while the ratchet and pawl are in contact with a di5erent bath (e.g. another gas in a box) at temperature T2 = T1 , see Fig. 6.1. While Feynman’s discussion [2] focuses on a thermodynamic analysis of this nonequilibrium system and apparently contains a misconception [110,111,474,475], here we concentrate on its microscopic modeling in terms of a stochastic process. Our Frst observation is that there are essentially two relevant (slow) collective coordinates: One is an angle, which characterizes the relative position of the pawl and an arbitrary reference point on the circumference of the ratchet in Fig. 6.1 and which we will henceforth consider as expanded to the entire real axis and denoted as x(t). As we have seen in Section 2.1.1, the possibility that the pawl spontaneously (due to thermal =uctuations) lifts itself up so that the ratchet can freely rotate underneath, is a crucial feature of the system. Therefore, another relevant collective coordinate is the “height” h(t) of the pawl, i.e. its position in the direction perpendicular to x (the “radial” direction in Fig. 6.1). The next modeling step consists in taking into account the thermal environment of the paddles, governing the state variable x(t), and the second heat bath, governing the dynamics h(t) of the pawl. A realistic description both of the impacts of the gas molecules on the paddles (e.g. by means of a Boltzmann equation [215]) and of the thermal =uctuations of the pawl on a microscoping footing is very involved. Along the general spirit of Section 2.1, a phenomenological modeling is the only realistically practicable modeling approach. In a Frst approximation [111,474], these environmental

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e5ects may be modeled by an overdamped dynamics for both x(t) and h(t), i.e. 9V (x(t); h(t))

1 x(t) ˙ =− + 1 (t) ; 9x ˙ = − 9V (x(t); h(t)) + 2 (t)

2 h(t) 9h with two independent white (thermal) Gaussian noises i (t)j (s) = 2 i kB Ti ij (t − s) ;

163

(6.4) (6.5) (6.6)

at temperatures T1 and T2 , respectively [110]. The interaction between x(t) and h(t) arises through the common potential V (x; h) which incorporates the fact that the pawl is (weakly) pressed against the ratchet (e.g. by a spring or due to its own elasticity) and the constraint that the pawl cannot penetrate the ratchet. The latter, non-holonomous constraint can be included by appropriate “potential walls” into V (x; h). An explicit example [111] is ; (6.7) V (x; h) = Ah + h − H (x) where A is the “spring constant” of the pawl, H (x) is the geometrical proFle of the ratchet, and is a parameter characterizing the “steepness” of the potential walls which account for the constraint h¿H (x). Note that (6.4) seems in fact to perfectly 8t into the general framework of a 9uctuating potential ratchet scheme (4.1). However, it actually goes somewhat beyond this scheme in that our usual assumption of the “potential =uctuations” h(t) being independent of the system x(t) is no longer respected, there is a “back-coupling” in (6.5). In spite of the various so far made approximations, the model is still only tractable by means of numerical simulations. Detailed quantitative results of such simulations can be found in [111,474]. Here, we proceed with the additional approximation that the pawl remains permanently in contact with the ratchet, i.e. the constraint h¿H (x) is replaced by h = H (x). Physical realizations of such a modiFed system with a Fxed, one-dimensional “track” (x; H (x)) of the pawl can be readily Fgured out. Moreover, it is clear that in those regions of the track with a small slope H (x), the noise acting on x(t) dominates, while the noise acting on h(t) dominates for large slopes H (x). In other words, an e5ective one-dimensional ratchet dynamics with a state-dependent e5ective temperature T (x) is recovered [111,475 – 477]: The Feynman ratchet can be approximately reduced to a Seebeck ratchet model. The main results of such a simpliFed one-dimensional description are qualitatively the same as for the more complicated two-dimensional original model (6.4) – (6.7) [111,478]: If the paddles experience a higher temperature than the pawl (T1 ¿T2 ) then the rotation is in the direction naively expected already in Fig. 2.1. Remarkably, for T1 ¡T2 the direction is inverted, i.e. the pawl preferably climbs up the steep slope of the ratchet proFle H (x). Experimental realizations of the above Feynman ratchet and pawl gadget are not known. In order that thermal =uctuations will play any signiFcant role, such an experiment has to be carried out on a very small scale. Quantitative estimates in [111] indicate that the necessary temperature di5erences in order to achieve an appreciable ratchet e5ect are probably not experimentally feasible. However, modiFed two-dimensional settings of the general form (6.4) – (6.7), e.g. with 2 (t) consisting of a thermal noise at the same temperature as 1 (t) and a superimposed external driving, may well be

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experimentally realizable, see Section 5.9. Finally, a Feynman ratchet-type model for a molecular motor (cf. Section 7) has been proposed in [5,6], though this model was later proven unrealistic by more detailed quantitative considerations [142,143]. 6.3. Temperature ratchets The properties and possible applications of the temperature ratchet [118] with time-periodic temperature variations (2.6), (2.47) have been discussed in detail in Sections 2.6, 2.10 and 2.11. A modiFed model in which the temperature changes T (t) are governed by a dichotomous random process (cf. Eqs. (3.11) – (3.13)) has been studied in [126,127,479,480]. The resulting, so-called composite noise (t) gives rise to a “minimal” ratchet model in (2.6) in the sense that (t) is a stationary, unbiased, white noise with correlation (t)(s) = 2 kB T (1 + #2 )(t − s) :

(6.8)

The noise is, however, not a thermal noise (e.g. it is not Gaussian distributed), thus the generic appearance of the ratchet e5ect is not in contradiction to the second law of thermodynamics [126,127,479,480]. Next, we consider again the general case that T (t) may be either a periodic function or a random process, satisfying T (t) ¿ 0 for all t. Introducing the auxiliary time [118,481] t tˆ(t) := dt T (t)= TW ; (6.9) 0

1 TW := lim t→∞ t

0

t

dt T (t) ;

(6.10)

it follows that the temperature ratchet dynamics (2.48) can be rewritten in terms of y(tˆ) := x(t(tˆ)) in the form W tˆ) ;

y( ˙ tˆ) = −V (y(tˆ))[1 + f(tˆ)] + ( f(tˆ) :=

dt(tˆ) − 1; d tˆ

(6.11) (6.12)

W tˆ) is a Gaussian white noise where t(tˆ) is the inverse of (6.9) (which obviously exists) and where ( with correlation W tˆ)( W s) ( ˆ = 2 kB TW (tˆ − s) ˆ ;

(6.13)

which is moreover statistically independent of f(tˆ). Exploiting (6.9), (6.10) one can furthermore show that f(tˆ) is unbiased. In general, if T (t) is a stochastic process then the relation between properties of y(tˆ) and x(t) is not obvious, since the time-transformation (6.9) is di5erent for each realization of T (t). However, with respect to the steady state current we can infer from the self-averaging property (3.5) in combination with (6.9) that y ˙ = x ˙ :

(6.14)

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If T (t) is a periodic function of t then the very same conclusion follows immediately. In other words, from (4.11), (6.11), (6.14) we can conclude that, at least with respect to the particle current, the temperature ratchet (2.6), (2.47) is exactly equivalent to a 9uctuating potential ratchet (4.11), independently of whether the time variations of T (t) are given by a periodic function or a stochastic process. On the other hand, a =uctuating potential ratchet can be mapped onto a temperature ratchet, provided f(t)¿ − 1 for all t in (4.11). Especially, from the asymptotics (4.10) for fast stochastic potential =uctuations the corresponding result [126,479,480] for a temperature ratchet is recovered. Likewise, from the prediction (2.58) for a periodically modulated temperature ratchet we can read o5 the asymptotics for ratchets with fast, periodically =uctuating potentials. For similar reasons, the qualitative analysis of the temperature ratchet for slow dichotomous temperature variations in Fig. 2.6 is practically the same as for the on–o5 ratchet scheme [34]. The basic physical picture behind this equivalence of a temperature ratchet and a =uctuating potential ratchet is as follows: Very loosely speaking, one may mimic temperature modulations by potential modulations since, under many circumstances, it is mainly the ratio of potential and temperature which plays the dominant role in transport phenomena (cf. Fig. 2.6). We Fnally recall that, apart from “accidental” cases, the “systematic” conditions implying x ˙ =0 are the symmetry and supersymmetry criteria from (3.51) and (3.52), respectively. Not surprisingly, these are practically the same as the corresponding criteria of symmetry (3.16) and supersymmetry (3.39) for a =uctuating potential ratchet V (x; f(t)) = V (x)[1 + f(t)].

6.4. Inhomogeneous, pulsating, and memory friction 6.4.1. A no-go theorem In the preceding sections we have discussed modiFcations of the Smoluchowski–Feynman ratchet model (2.6) with either a spatial or a temporal variation of the temperature T in (2.5). In the generic case, a Fnite particle current x ˙ results in such a model, as expected from Curie’s principle. In the following, we discuss an apparently rather similar modiFcation of the Smoluchowski–Feynman ratchet model (2.5), (2.6), namely spatial and=or temporal variations of the friction coeCcient , with the rather unexpected result that the average particle current in the steady state is always zero. In the case of a non-constant friction coe

V (x) + (x; t)v +

(x; t)kB T 9 m 9v

P;

(6.15)

ˆ v; t) (cf. (2.22)), which is periodic in x but where v := x. ˙ Going over to the reduced density P(x; still satisFes (6.15), one readily veriFes that the Boltzmann distribution st Pˆ (x; v) = Z −1 exp{−[mv2 =2 + V (x)]=kB T }

(6.16)

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is a steady state solution (cf. (2.31)). Under the suCcient (but not necessary) condition that (x; t)¿0 for all x and t (and that T ¿0) this long time asymptotics can be proven to be furthermore unique [82,83,100,108,109]. The remarkable feature of the steady state distribution (6.16) is that the friction coeCcient (x; t) does not appear at all. For = const: we are dealing with an equilibrium system and the second law of thermodynamics implies the result x ˙ =0

(6.17)

for similar reasons as in Section 2.1. Considering that (6.16) does not depend on the friction coefFcient, it is quite plausible that the result (6.17) carries over to arbitrary (x; t). The same conclusion is corroborated [191,465,466,468] by a more detailed calculation similarly as in Sections 2.3 and 2.4. The basic physical reason behind the result (6.17) is that the model (2.1), (2.5) describes an equilibrium system for arbitrary (x; t): In fact, we have noticed below Eq. (2.5) that the friction coeCcient can also be considered as the coupling strength between the system and its thermal environment. In the absence of other perturbations, the model (2.1) thus continues to describe an equilibrium system even for a non-constant coupling (x; t). Since an equilibrium system reaches an equilibrium state in the long time limit, the second law of thermodynamics can be invoked and (6.17) follows. (Only the transient dynamics depends on the details of (x; t).) Thus, there is no contradiction to Curie’s principle: The current-prohibiting symmetry, which is easily overlooked at Frst glance, is in fact once again the detailed balance symmetry. Returning Fnally to the overdamped limit m → 0, we only state here the outcome of a more rigorous analysis [463,465,466,482,483], namely that this limit cannot be consistently carried out in the stochastic dynamics (2.1) itself but only on the level of the Fokker–Planck equation (6.15), with the result of a probability current in (2.17) of the form like in (2.21) with (x; t) in place of . The conclusion (6.17) then follows along the same line of reasoning as in Section 2.4. 6.4.2. Inhomogeneous and pulsating friction The microscopic origin of a time-independent inhomogeneous friction mechanism (x) has been discussed in Section 3.4.1, namely a broken translation invariance of the thermal environment with respect to the relevant (slow) state variable(s) of interest. Physical examples are the Brownian motion near geometrical conFnements of the =uid due to deviations from Stokes friction [373,468,484 – 486], phase-dependent dissipation in Josephson junctions due to the interference of pair and quasiparticle tunneling currents [487], generic chemical reactions [488,489] (cf. Section 3.4.1), and protein friction in molecular motors, see Section 7.3. In the following, we restrict ourselves to the most important case that (x) is strictly positive and exhibits the same periodicity L as the potential V (x). As mentioned in the preceding subsection, the overdamped limit in the presence of an inhomogeneous friction amounts [463,465,466,482,483] to replacing by (x) in (2.17), (2.21). By means of the transformation x

x(x) W := d x (x )= W ; (6.18)

W :=

0

0

L

2

d x

(x ) L

;

(6.19)

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the Fokker–Planck equation for P(x; W t) takes exactly the constant friction form (2.14) if one replaces x by xW and by . W Including the “perturbations” f(t), y(t), and F (cf. (3.1)), the transformed equivalent Langevin equation takes the form

Wx(t) W˙ = −VW (x(t); W f(t)) +

= ( W x)[y(t) W + F] + (t) ;

(6.20)

where (x) W := (x(x)) W and W ) W : VW (x; W f(t)) := V (x(x); W f(t)) + (kB T=2) ln( (x)=

(6.21)

With (6.18), (6.19) one sees that VW (x) W and (x) W exhibit again the same periodicity L as V (x) and (x). In other words, we have mapped the original overdamped ratchet dynamics with inhomogeneous friction to our standard working model (3.1) with the only exception that the homogeneous external perturbation [y(t) + F] acquires a spatially periodic multiplicative factor. Namely, an originally pure tilting ratchet now picks up some pulsating potential admixture, while a static force F is now accompanied by a modiFcation of the static part of the periodic potential proFle. As a consequence, the basic qualitative features of such inhomogeneous friction ratchet models can be readily understood on the basis of our previously discussed results. For instance, a ratchet e5ect may now arise even if both V (x; f(t)) and (x) are symmetric according to (3.16) but each with a di5erent Ux-value, i.e. they are out of phase, since this gives rise to a genuine e5ective ratchet potential with broken symmetry in (6.21). Regarding various interesting quantitative results for several speciFc models we refer to [190,191,465 – 467,490 – 493]. An additional time-dependence of the friction (x) may arise under certain temporal variations of the system-plus-environment which are suCciently slow in comparison with the characteristic relaxation time of the environment in order to always maintain an (approximate) accompanying equilibrium state of the bath. In such a case, the time dependence of (x; t) may be absorbed into the potential and the forces appearing on the right-hand side of the properly rewritten original stochastic dynamics (3.1) similarly as in Section 6.3. Afterwards, the remaining x-dependence can again be transformed away as in (6.20). The special instance of a pulsating potential V (x; f(t)) in combination with a pulsating friction coe
90

A trivial example is: if f(t) is in state 1 then for x ∈ [0; L] the friction (x) is non-zero only within [0; 3L=4] and the transition probability into state 2 only within [L=2; 3L=4]; if f(t) is in state 2 then everything is shifted by L=2.

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6.4.3. Memory friction and correlated thermal noise Instead of forcing an unbiased (F = 0) system of the general form (3.1) by means of the perturbations f(t) or y(t) away from thermal equilibrium, one may as well consider a modiFcation of the friction term x(t) ˙ (while f(t) = y(t) = 0). Much like in the previous subsection, the overdamped limit becomes then rather subtle and one better keeps a Fnite mass m in the original description (2.1). The simplest such generalization [66,79 –81,84,89,92–96] includes a so-called linear memory friction of the form t m x(t) M + V (x(t)) = −

(t ˆ − t ) x(t ˙ ) dt + (t) ; (6.22) −∞

see also Sections 3.4.1 and 8.1 (the lower integration limit 0 in (8.4) is recovered from (6.22) by observing that x(t) ˙ ≡ 0 for times smaller than the initial time t = 0). The proper generalization (cf. (2.5), (3.2)) of the 9uctuation–dissipation relation then reads [66,79 –81,84,89,92–96] (t)(s) = (|t ˆ − s|)kB T ;

(6.23)

see also (3.37), (3.38). Unless (t) is a stationary Gaussian process with zero mean and correlation (6.23), the environment responsible for the dissipation and =uctuations in (6.22) cannot be a thermal equilibrium bath [97] and therefore a ratchet e5ect is expected generically (and indeed observed), as exempliFed in [119]. Especially, the fact that some noise (Gaussian or not) is uncorrelated (white) does not necessarily imply that its origin is a thermal equilibrium environment nor does a correlated noise exclude thermal equilibrium. 6.5. Ratchet models with an internal degree of freedom In this section we brie=y review Brownian motors which posses—in addition to the mechanical coordinate x—an “internal degree of freedom” analogous to the chemical state variable of molecular motors (cf. Sections 4.6 and 7), but without the main intention of representing a faithful modeling of such intracellular transport processes. Another closely related model class are the two-dimensional tilting ratchet systems from Section 5.9. The so-called active Brownian particles [228,494,495] with an “energy depot” as additional internal variable have been considered in [496,497] under the in=uence of a static ratchet potential. The internal energy depot models the capability to take up energy from the environment, store it, and (partially) convert it into directed motion. While the original, phenomenological model dynamics from [496,497] does not Ft into the generalized pulsating ratchet scheme from (7.3), it is possible to transform it into an equivalent form closely related to (7.3), namely a combined =uctuating potential and temperature ratchet with a back-coupling mechanism. Upon variation of the noise strength or of the energy supply, a remarkably rich behavior of the particle current x, ˙ both in magnitude and sign, is recovered [496,497]. A di5erent type of “active Brownian particles”, namely a reaction–di1usion system with one species of particles possessing a =uctuating potential ratchet type internal degree of freedom (chemical reaction cycle), has been demonstrated in [498] to induce a pattern forming process. Note also the connection of this setup with the collective ratchet models from Section 9.

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A precursor of a two-headed motor enzyme model is the “elastic dumb-bell” from [41], consisting of two point-like Brownian particles which are linked by a (passive) elastic spring, and which move in either the same or two di5erent (shifted) on–o5 ratchet potentials, see also [499] and [500] for the cases of =uctuating and traveling potential ratchet schemes, respectively. The corresponding “rocking-ratchet” situation, i.e. a static ratchet potential but a periodically varying external driving force, has been studied in [501] and extended in [502] and [184] to the cases when each of the two particles moves in a two-dimensional ratchet potential and in a one-dimensional symmetric periodic potential, respectively. The analogous “=uctuating force ratchet” in the limit of a “rigid dumb-bell” has been considered in [368,499,503]. Note that there exists a close connection to the models for single molecular motors in Section 7.6, especially those in [445,504 –506]. A Frst experimental realization of such a two-head-like system with an active, spring-like element was reported as early as 1992 [507]: a curved strip of gel with periodically varying curvature (by externally applied electric Felds) moves in a worm-like fashion with its two ends (“heads”) along a ratchet-shaped substrate. A second experimental ratchet system with an “internal degree of freedom” was presented in [508]: a water droplet in oil is positioned on a ratchet shaped surface and its shape (internal degree of freedom) is periodically changed by means of externally applied electric Felds. With the shape also the contact angles between the droplet and the surface change, with the result of a systematic directed motion. Since the droplet covers several periods of the ratchet, the rough picture is a somewhat similar worm-like motion as before, though the actual systems and their possible applications are of course completely di5erent. A Brownian particle in a periodic electric potential with an autonomously rotating “internal electric dipole” has been theoretically analyzed in [509]. Since the direction of this rotation breaks the spatial symmetry, the periodic potential may be chosen symmetric in this model. While there exists a close formal analogy with the traveling potential ratchet scheme from Section 4.4.1, the physical picture is di5erent [300]. 6.6. Drift ratchet In this section we discuss in some detail the so-called drift ratchet scheme [175] which resembles a rocking ratchet but at the same time goes substantially beyond our original tilting ratchet model from (5.1). We will outline the theoretical framework of a particle separation device based on this drift ratchet scheme, presently under construction [510] in the laboratories of the Max-Planck-Institut in Halle (Germany). The system basically consists of a piece of silicon—a so-called silicon wafer—pierced by a huge number of identical pores with a ratchet-shaped (periodic but asymmetric) variation of the diameter along the pore-axis [510], see Figs. 6.2 and 6.3. The pores are Flled with a liquid (e.g. water) which is periodically pumped back and forth in an unbiased fashion, i.e. such that no net motion of the liquid is produced on the average. Suspended into the liquid are particles of micrometer size and the objective is to separate them according to their size. For a theoretical description of the particle motion we consider a single, inFnitely long pore under the idealizing assumptions that the particles have spherical shape, that the suspension is suCciently diluted such that particle interaction e5ects are negligible, and that the interaction with the pore walls can be captured by perfectly re=ecting boundary conditions. For the typical parameter values of the real experiment, buoyancy e5ects due to the in=uence of gravitation as well as inertia e5ects of the

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7 5

x [µm]

3 1 _1 _3 _5 _7

0

2

4

6

8

10

12

14

16

18

z [µm] Fig. 6.2. Scanning-electron-microscope picture of a silicon wafer, pierced by a huge number of practically identical pores with pore-distances of 1:5 m and pore-diameters of about 1 m. Fig. 6.3. Schematic cross-section (x–z-plane) through a single pore with an experimentally realistic, ratchet-shaped variation of the diameter along the pore axis (z axis).

particle are negligibly small, i.e., the particle dynamics in the viscous liquid is strongly overdamped. Assuming that the three-dimensional time-dependent velocity Feld ˜v(˜x; t) of the liquid is known, the particle ˜x(t) is governed by the deterministic dynamics ˜x˙(t) = ˜v(˜x(t); t). Here, ˜v(˜x; t) is, strictly speaking, not the velocity Feld of the =uid alone but rather the speed with which a spherical particle with center at ˜x(t) and a small but Fnite radius is carried along by the surrounding liquid. This deterministic dynamics induced by the streaming liquid has to be complemented by the di5usion of the micrometer sized particle due to random thermal =uctuations ˜(t), which are caused by the impacts of the surrounding liquid molecules, and which we model in the usual way as Gaussian white noise. We thus end up with the following stochastic dynamics for the trajectory ˜x(t) of a microsphere inside a single pore: ˜x˙(t) = ˜v(˜x(t); t) + ˜(t) : (6.24) The vector components i (t); i = 1; 2; 3, of the noise ˜(t) are unbiased Gaussian processes with correlation 2kB T ij (t − s) : i (t)j (s) = (6.25)

The friction coeCcient is in very good approximation given by Stokes law 6 RC, where R is the particle radius and C the viscosity of the liquid. In view of the external, time-periodic pumping of the liquid through the pores, the above so-called drift-ratchet scheme has a certain similarity to a rocking ratchet system. On the other hand, it also reminds one of the hydrodynamic ratcheting mechanism based on the so-called Stokes drift

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171

Particle current [µm /s]

0.6 0.3 0.0 _ 0.3 _ 0.6 _ 0.9 _1.2

40 Hertz, νR=1.0

_ 1.5

40 Hertz, νR=0.5 100 Hertz, νR=0.5

_1.8

0.0

0.2

0.4 0.6 0.8 1.0 Particle diameter [µm]

1.2

1.4

Fig. 6.4. Numerical simulation of the stochastic dynamics (6.24), (6.25) for a pore shape as depicted in Fig. 6.3, at room temperature (T = 293 K). The friction coeCcient in (6.25) is given by Stokes law 6 RC, where R is the particle radius and C = CR Cwater the viscosity of the liquid in units of the viscosity Cwater of water. The velocity Feld in (6.24) has been obtained numerically with a sinusoidal pumping of the liquid at a frequency of 40 and 100 Hz. The pumping amplitude A is chosen as A = 2L, where L = 6 m is the period of the ratchet-shaped pore in Fig. 6.3. Depicted is the timeand ensemble-averaged particle current z ˙ along the pore axis (z-axis) versus the particle diameter for various driving frequencies and viscosities.

[296 –298,300,301] as discussed in the context of traveling potential ratchets in Sections 4.4 and 4.5. However, in contrast to both, the rocking as well as the traveling potential ratchet paradigms, in the present case (6.24) no “ratchet-potential” is involved. 91 Furthermore, the dynamics within a single pore is still a complicated three-dimensional problem that cannot be reduced in a straightforward manner to an e5ective one-dimensional model. After one period of driving, the liquid in the pore returns to the same position from where it started out. Why should we not expect the same null-e5ect for the suspended particles? The basic reason is as usual the far from equilibrium situation, created in the present case by the periodic pumping, in combination with Curie’s principle, which predicts the generic appearance of a preferential direction of the stochastic particle dynamics with broken spatial symmetry (6.24). The physical mechanism for the emergence of such a non-vanishing net particle current are the thermal di5usion between “liquid layers” of di5erent speed and the collisions with the pore walls: Through the asymmetry of the pore-proFle, an asymmetry between pumping forth and back arises for both the thermal inter-layer di5usion and the collisions with the pore-walls, resulting in a non-vanishing particle displacement on average after one driving period. The fact that the excursions of the particles during one driving period are typically much larger than the net displacement after one period (see Fig. 6.4) motivates the name “drift ratchet”. The calculation of the velocity Feld ˜v in (6.24) is a rather involved hydrodynamic problem in itself. For details of the necessary approximations (and their justiFcation) in order to make the problem tractable at least by numerical methods we refer to [175]. Once such an approximation for ˜ ·˜v = 0, the velocity Feld ˜v appearing in (6.24) can be written Under the assumption of an incompressible =uid, i.e. ∇ as the curl of some vector potential, but never as the gradient of a scalar potential. 91

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1 hour

10000 1000

10 min

100

1.5

1.0

P2 / P1

Particle density

2.0

1 min 5 sec

1

0.5

0.0

_120 _ 90

10

0.1

_ 60

_ 30

0.01 0

z [µm]

30

60

90

0.01

0.1

1 10 100 Pumping time [min]

1000

Fig. 6.5. Time evolution of the particle density (within the liquid-plus-particle suspension) along the z-axis starting with a homogeneous initial distribution (normalized to unity). The pore length (along the z-axis) is 126 m and the extension Uz of each of the two adjacent basins along the z-axis is 24 m. Other details are like in Fig. 6.4 with particle radius R = 0:36 m, pumping frequency 100 Hz, pumping amplitude A = L, and relative viscosity CR = 0:5. Fig. 6.6. Ratio P2 =P1 of particle densities for two types of particles versus time t. The setup is the same as in Fig. 6.5 but with a pumping amplitude A = 2L and with radii of the two types of particles R1 = 0:36 m and R2 = 0:7 m (corresponding to opposite current directions in Fig. 6.4). The ratio of the densities P2 =P1 refers to the border of the right basin at z = 87 m (=126=2 m + 24 m). Solid line: Overall homogeneous initial densities. Dashed line: initially homogeneous densities in the pore region and vanishing densities in the two basin regions.

˜v is available, the numerical simulation of the stochastic dynamics is straightforward. Typical results for realistic parameter values are depicted in Fig. 6.4, demonstrating that the direction of the particle current depends very sensitively on the size of the particles. While, according to Section 3.6, such current inversions are a rather common phenomenon, the distinguishing feature of our present device is its highly parallel architecture: 92 a typical silicon waver contains about one million pores per square centimeter. On the other hand, the pores in a real silicon wafer are not of inFnite length—as so far assumed—but rather the wafer is connected at both ends to basins of the liquid-plus-particle suspension and the actual pumping device. For practical applications, not the steady state current in an inFnite pore is of main importance, but rather the time needed to achieve reasonably large concentration di5erences between the two basins (see also the discussion below equation (3.7)). We now focus on the case of two identical basins, each of an extension Uz along the z-axis and of the same cross section as the wafer (perpendicular to the z-axis). The typical time evolution of the particle density for such a setup is depicted in Figs. 6.5 and 6.6. These calculations predict a remarkable theoretical separating power of the device. Its experimental realization—presently under construction [510]—thus appears to be a promising new particle separation device, possibly superior to existing methods for particles sizes on the micrometer scale. 92

We remark that also the experimental systems from [38,273,276] (discussed in Section 4.2.1 and at the end of Section 4.4.1) include a parallelization in two dimensions, while in the present case three dimensions are exploited.

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6.7. Spatially discrete models and Parrondo’s game The spatially discretized counterpart of our working model (3.1) arises when the state variable x(t) is restricted to a set of discrete values xi . In the simplest case, the time evolution is given by a so-called Markov-chain dynamics, i.e. transitions are only possible between neighboring states xi and xi±1 , and they are governed by transition rates ki → i±1 (t), which in general may still depend on time. As a consequence, the probability distribution Pi := (xi − x(t)) evolves in time according to a master equation of the form P˙ i (t) = −[ki → i+1 (t) + ki → i−1 (t)]Pi (t) + ki+1 → i (t)Pi+1 (t) + ki−1 → i (t)Pi−1 (t) :

(6.26)

The spatial periodicity of the system implies that there is an integer l with the properties that xi+l = xi + L ;

(6.27)

ki+l → j+l (t) = ki → j (t)

(6.28)

for all i and j. A periodic Markov-chain model (6.26) – (6.28) may arise in several di5erent contexts. The most prominent is the activated barrier crossing limit as discussed in Section 3.8, i.e. the spatially continuous dynamics (3.1) is characterized by rare transition events between metastable states xi . In the simplest (and most common) case l = 1, i.e. there is only one metastable state xi per spatial period L and the rates are—possibly after temporal coarse graining (see Section 3.8)—independent of time. While the actual calculation of those rates k± := ki → i±1 is in general highly non-trivial, once they are given, the determination of the current and the di5usion coeCcient is straightforward, see (3.55), (3.56) and the footnote 48. The problem of calculating the rates ki → i±1 (t) simpliFes a lot if the characteristic time scale of the driving f(t) and=or y(t) in (3.1) is much larger than the intrawell relaxation time within any metastable state (but not necessarily larger than the characteristic interwell transition times 1=ki → i±1 (t) themselves). Under these circumstances, an adiabatic approximation like in Section 2.10 can be adopted, with the result that at any given time t, the rates ki → i±1 (t) are given by a Kramers– Smoluchowski-type expression analogous to (2.45). Comparing (3.1) with (2.34), we see that in those rate expressions not only the instantaneous e5ective potential Ve5 (x; t) = V (x; f(t)) − xy(t) − xF (cf. (2.35)) depends on f(t) and=or y(t), but also the locations xmin = xi of the metastable states (local minima) and of the activated states (local maxima) xmax . Besides the slow variations of f(t) and=or y(t), the implicit assumptions of this approximation are that the number of metastable states within one spatial period L is the same for all times t, that their position changes in the course of time continuously or with not too big jumps, and that the potential barriers between any two of them is much larger than the thermal energy kB T . Within these restrictions, any spatially continuous class of ratchets from Section 3.3 immediately entails a spatially discretized counterpart. Especially, we note that the characteristic features of the di5usion ratchet scheme will be a time-dependent temperature T (t) in the Kramers–Smoluchowski rates (2.45), while for a Seebeck ratchet (Section 6.1), the e5ective barriers UVe5 and pre-exponential factors in (2.45) have to be calculated along the lines of Section 4.3.2.

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Along this general ideology, the spatially discretized on–o5 ratchet scheme (see Section 4.2) has been worked out in [172,511], while a modiFed on–o5 description of a Feynman ratchet (see Section 6.2) is due to [478]. As another variation, an asynchronously pulsating on–o5 model (cf. Section 3.4.2) has been put forward in [197]. In such a model, (6.28) is no longer satisFed and instead within each spatial period L the potential switches independently between its onand an o5-state. If these switching events within neighboring periods are correlated or anticorrelated, the current is enhanced as compared to the completely uncorrelated case [197]. Related, spatially continuous, asynchronously pulsating ratchet models have been studied in [198–202]. Spatially discretized pulsating ratchet models have been addressed in [129,512], temperature ratchets in [128,129], traveling potential ratchets in [164], and rocking ratchets in [129,172,512], see also [233,234,236,237,513] for the case of extremely slow rocking. For biological intracellular transport processes (cf. Sections 4.6 and 7), spatially discretized descriptions arise naturally and have been analyzed in detail e.g. in [8,9,16,22,186,187,514 –520]. In all those works, the above-mentioned approximation of the rates ki → i±1 (t) by instantaneous Kramers–Smoluchowski-type expressions (2.45) have been exploited. The advantage of such an approach is that closed analytical solutions can often be obtained, especially if the driving f(t), y(t), and=or T (t) jumps (either periodically or randomly) between only a few di5erent values. Since the main qualitative Fndings are very similar as for the spatially continuous case (see Sections 4 and 5) we do not discuss these features in any further detail at this place. We only remark that if the spatially continuous model leads to a vanishing current in the adiabatically slow driving case (e.g. for =uctuating potential and temperature ratchets), then at least two metastable states xi per period L (i.e. l ¿ 2) are required for a ratchet e5ect in the spatially discrete counterpart [511]. In any other case, one metastable state xi per period L (i.e. l = 1) is suCcient. Such spatially discretized, adiabatically driven models with a minimal number l of states per period are sometimes called minimal ratchets in view of their mathematical and conceptual simplicity. We emphasize again that while discrete models are usually easier to analyze than their spatially continuous counterparts, the actual hard problem has now been shifted to justifying such a discretized modeling and to determine the rates (“phenomenological model parameters”) either from a more detailed (usually continuous) description (cf. Section 3.8) or from experimental observations. A second context in which a spatially discretized dynamics (6.26) arises is the numerical method for solving the originally continuous problem (3.1) which has been introduced in [358] and applied to various speciFc models in [162,250,366,521,522]. Choosing the rates ki → i±1 (t) according to the recipe from [358], this numerical scheme approximates the solution of the continuous system better and better as the number of states l per period increases. Conversely, the often analytically solvable models with only very few states xi per period L may be still considered as a Frst rough approximation of the spatially continuous problem. Another cute application of the discretized on–o5 ratchet scheme has been invented by Parrondo [523–529]. Namely, the spatially discretized random dynamics for both the on- and the o5-conFgurations of the potential are re-interpreted as games, and by construction each of these two games in itself is fair (unbiased). The astonishing phenomenon of the ratchet e5ect then translates into the surprising observation that by randomly switching between two fair games one ends up with a game which is no longer fair. This so-called Parrondo paradox is thus in some sense the game theoretic transFguration of Brillouin’s paradox from Section 2.9. Generalizations are obvious: For instance, by switching between two games, each (weakly) biased into the same direction, the

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resulting game may be biased just in the opposite direction. Another option is to take as starting point for the translation into a game a ratchet model di5erent from the on–o5 scheme [530,531], and so on. 6.8. In9uence of disorder In this section we brie=y review some basic e5ects which arise if the periodicity of the potential (3.3) is modiFed by a certain amount of quenched spatial disorder. Our starting point is an “unperturbed”, strictly periodic system in the activated barrier crossing limit as discussed in Section 3.8, i.e. transitions between neighboring spatial “cells” of length L can be described by “hopping”-rates k+ and k− to the right and left, respectively. Without loss of generality we furthermore assume that the unperturbed current x ˙ in (3.55) is positive, i.e. k+ ¿k− :

(6.29)

In the simplest case we may now introduce a quenched randomness as follows: For each pair of neighboring cells we interchange with a certain probability p the original transition rates k+ and k− to the right and left. For instance, in a piecewise linear “saw-tooth potential” as depicted in Fig. 4.1, such an interchange of the transition rates can be realized by randomly inverting the orientation of each single saw-tooth with probability p independently of each other. Without loss of generality we can restrict ourselves to probabilities 0 6 p 6 12 :

(6.30)

The following basic e5ects have been unraveled by Derrida and Pomeau [233,234]: Upon increasing p the particle current x ˙ monotonically decreases from its initial value (3.55) and vanishes for p ¿ p1 , where k− : (6.31) p1 := k+ + k − More precisely, for p ¿ p1 the mean displacement x(t) grows asymptotically slower than linearly with t. The e5ective di5usion coeCcient (3.6) increases monotonically from its unperturbed value (3.56) and diverges at p = p2 , where p2 :=

2 k− : 2 k+2 + k−

For p2 6 p 6 p3 , where

k−

; p3 := √ k+ + k−

(6.32)

(6.33)

a superdi5usive behavior arises (De5 = ∞), i.e. the dispersion [x(t) − x(t)]2 grows asymptotically faster than linearly with t, switching over [532] to a subdi5usive behavior (slower than linear growth of the dispersion, i.e. De5 = 0) for p3 ¡p 6 1=2. (Note that 0¡p2 ¡p1 ¡p3 ¡1=2.) At least in the regimes where they are Fnite, the quantities x ˙ and De5 are self-averaging, i.e. the same (Fnite) value is observed with probability 1 for any given realization of the quenched disorder. A simple intuitive explanation of these results does not seem possible, which may not be so surprising in view

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of the above-mentioned self-averaging issue and other subtle problems of commuting limits in this context, see [235,532,533] and references therein. More general, but still uncorrelated randomizations of the transition rates between pairs of neighboring “cells” of length L are given already by Derrida in [234]. A variety of cases with correlated randomization has been discussed in [534] (see also the review [532]) together with several interesting physical applications. A bold but apparently quite satisfactory approximative extension beyond the activated barrier crossing limit has been proposed in [171]: The basic idea is to evaluate, either analytically or numerically, for the unperturbed (strictly periodic) ratchet dynamics both the current x ˙ and the di5usion coeCcient De5 . Introducing these results for x ˙ and De5 into (3.55) and (3.56) yields formal expressions for the rates k± even though these rates no longer adequately describe the actual transitions between neighboring “cells”. Assuming that a randomization of the ratchet potential can still be captured by a corresponding randomization of the formal forward and backward rates k± , one thus can continue to use Derrida and Pomeau’s formulas [233,234] for an approximative description of such a randomized ratchet dynamics. For the example of an on–o5 ratchet scheme, a fair agreement of this approximative approach with accurate numerical simulations has been reported in [171]. Another, more systematic Frst step beyond the activated barrier crossing limit is due to [535], considering a =uctuating force ratchet with a very general disordered potential V (x) that is (additively) driven by asymptotically weak symmetric Poissonian shot noise (cf. Section 5.5). A deterministic (T = 0) rocking ratchet model with quenched spatial disorder has been addressed in [536]. Similarly as before, the current decreases and the (deterministic) di5usion accelerates with increasing disorder, but apparently these quantities no longer exhibit the experimentally important self-averaging property. Results more in accordance with the above described standard scenario of Derrida and Pomeau are recovered upon including inertia e5ects [396]. A similar overdamped case but with Fnite T and adiabatically slow rocking has been addressed in [537]. 6.9. E
xF ˙ : Pin

(6.34)

Both averages in this equation are meant with respect to all random processes and time-periodicities involved in (3.1) and transients are assumed to have died out. For ergodicity reasons, both averages can then also be rewritten as long time averages for a single realizations of the stochastic dynamics (3.1), cf. (3.5). In order to quantitatively calculate the eCciency (6.34) for the di5erent classes of ratchet models (3.1), a very general and elegant framework has been developed by Sekimoto [321,474,538–540], unifying and putting on Frm grounds the various previously proposed, model-speciFc expressions for Pin in (6.34).

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As pointed out in Section 3.4.2, the origin of a random external driving f(t) and=or y(t) may be conceived as a thermal heat bath, very weakly coupled to the system variable x(t) in order that back-coupling (friction-type) e5ects are negligible, 93 but at a temperature much higher than the temperature T of the thermal noise (t). From the viewpoint of a Carnot machine, the temperature T is thus to be associated with the cooler heat bath and the maximally achievable Carnot e
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an eCciency smaller than the Carnot value, an extended model with two diodes may approach this theoretical upper limit for the eCciency arbitrarily close [548], demonstrating that even a system which is simultaneously in contact with more than one heat bath may still operate reversibly, see also [543]. Universal, i.e. largely model-independent features of eCciencies for ratchet models close to thermal equilibrium (Onsager regime of linear response) have been worked out in [14,478,538,544]. Remarkably, by moving out of the linear response regime into the far from equilibrium realm the eCciency may not necessarily decrease [544]. Similarly, for some ratchet models, the eCciency may even increase upon increasing the temperature T of the thermal heat bath both, for systems near [544] and far from equilibrium [129,522,549], in contrast to what one would expect from a Carnot eCciency point of view. As already mentioned, =uctuating potential ratchets and temperature ratchets cannot reach the maximal Carnot eCciency. SpeciFcally, the on–o5 scenario leads under typical conditions to eCciencies of a few percent [198,511,521,522,544]. However, in the case that many on–o5 ratchet are coupled together (see Section 7.4.4) the eCciency may again reach values of 50% and beyond [14,550]. ECciencies of at most a few percent have also been reported for =uctuating potential ratchets (see Section 4.3) [129], temperature ratchets (see Section 6.3) [128–130], and coupled rocking ratchets [551] (see Eq. (9.34)). Based on experimental measurements of intracellular transport processes, the possibility that the molecular motor kinesin (cf. Section 7) may reach an eCciency as high as 50% or even 80 –95% is discussed in [519] and [515,516], respectively, see also [300,552]. Other deFnitions of eCciencies than in (6.34) have been introduced and discussed in [12,300,544, 553–559]. Related quantities like entropy production, Kolmogorov information entropy, and algorithmic complexity have been explored in [190,478,521,522]. Evidently, with respect to the deFnitions of such alternative eCciency-type quantities it does not make sense to ask whether they are “right” or “wrong” (apart from the trivial requirement that they are “well-deFned” in the mathematical sense). Rather, the crucial question regards their usefulness [544]. For instance, it may be possible to agree on one such quantity as being a particularly appropriate quality measure in a certain context [12]. In many cases this will indeed be the standard “eCciency for generating force” (6.34). However, in other cases, it may be important to accomplish a certain task not only by means of a minimal amount of input energy—as in (6.34)—but in addition within a prescribed, Fnite amount of time. This constrained optimization task is the basis of the alternative “eCciency of transportation” concept from [300,555,556], which has been to some extent anticipated in [553], and which is also closely related to the issue of Fnite-time thermodynamics [560 –562]. For further details, we refer to the above-cited original works, see also at the end of Section 8.4. 7. Molecular motors In this section we exemplify in detail the typical stochastic modeling procedure by elaborating the general scheme from Section 4.6 for a particularly important special case of intracellular transport, namely so-called motor enzymes or molecular motors which are able to travel along polymer Flaments inside a cell. SpeciFcally, we shall focus on molecular motors from a subfamily of the so-called kinesin superfamily, which are capable of operating individually. For the two other main

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superfamilies of motor enzymes (myosin and dynein) many of the basic qualitative modeling ingredients remain the same as for kinesin, while the details are di5erent [563,564]; we will brie=y address the case of molecular motors which only can operate collectively, e.g. the so-called myosin II subfamily, later in this chapter. More involved variants of intracellular transport like rotary mechanochemical energy transducers are treated e.g. in [565 –569]. Also not covered by the present section are “Brownian ratchets”—a notion which has been coined in a rather di5erent context, namely as a possible operating principle for the translocation of proteins across membranes [121–125]. A collection of computer animations which visualize several of these intracellular transport processes is available on the internet under [570]. 7.1. Biological setup The most primitive living cells are the so-called prokaryotes, i.e. cells without a nucleus (mostly bacteria) [343]. Their interior is basically one large soup without any internal partition. Since prokaryotic cells are at the same time very small, the intracellular transport of various substances can be accomplished passively, namely through thermal di5usion. In contrast, eucaryotic cells (the constituents of any multicellular organism) are not only higher organized but also considerably larger so that passive di5usive transport becomes too ineCcient [571]. Their distinguishing features are the existence of a cell nucleus (responsible for the storage and transcription of the genetic material), many other internal compartments, called organelles, and a network of polymer Flaments—the “cytoskeleton”—which organizes and interconnects them. These Flaments radiate from a structure near the nucleus called the centrosome to the periphery of the cell and so support the shape of the cell. Besides several other intracellular functions, which go beyond our present scope, they act as a circulatory system, connecting and feeding distinct regions of the cell. They are paths along which nutrients, wastes, proteins, etc., are transported in packages, called vesicles, by speciFc motor proteins (mechanoenzymes). One major type of such polymer Flaments are Fbers of proteins called microtubuli, with the constituent protein “tubulin”—a dimer of two very similar globular proteins (0-tubulin and E-tubulin) about 4 nm in diameter and 8 nm long [343]. The microtubulus is composed of typically 13 protoFlaments (rows of tubulin-dimers) that run parallel to the axis of the Fber. The emerging shape of the microtubulus resembles that of a hollow, moderately =exible tube with an outer diameter of about 25 nm, and inner diameter of about 17 nm, and an overall length of up to a few m. Due to the asymmetry of the tubulin-dimers, the tube has a polarity, one end exposes only 0-tubulin, and the other only E-tubulin. On top of that, the tube exhibits a deFnite chirality or helicity since the dimer-rows of neighboring parallel protoFlaments are shifted against each other. One speciFc motor enzyme which can travel on a microtubulus and pull along various objects like chromosomes, viruses, or vesicles with chemicals in it, is the protein “kinesin” [343,563]. The necessary energy to move against the viscous drag is supplied by the so-called ATPase, i.e. the exothermic chemical hydrolysis of ATP (adenosine triphosphate) into ADP (adenosine diphosphate) and Pi (inorganic phosphate). The shape of a single kinesin molecule is rather elongated, about 110 nm in length and about 10 nm in the other two spatial directions. One of its ends consists of a bifurcated “tail”, capable of grasping the cargo to be carried, then follows a very long rod-shaped middle segment, the 0-helical coiled-coil stalk, while the other end bifurcates into two identical globular “heads” or “motor domains” [143,572]. In spite of the nomenclature, the functioning of

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the heads is actually quite similar to that of human legs, proceeding along the microtubulus in a “step-by-step” or “hand-over-hand” fashion [573]. We emphasize that the comparison with the walking of a human is common but should not be overstretched: There is evidence that the bound head in fact produces a rotation that “swings” the second head towards its next binding site [143,563]. The reason is that the kinesin as a hole seems to possess an (approximate) axis of rotational symmetry ◦ by 180 , implying that we should think of the two heads not as “right and left feet” but rather as “two left feet” [143]. Each single foot, on the other hand, does not share this (spatial inversion-) symmetry, it has well distinguishable “heel” and “Fngers”. Each head comprises in particular a microtubulus-binding site as well as an ATP-binding site, called the ATP-binding pocket. Accordingly, each head can bind and hydrolyze ATP on its own. The underlying chemical reaction cycle consists of the following four 96 basic steps (and corresponding states) with the result of about 20kB T energy gain per cycle [9]: State 1: The motor domain is interacting with the environment and attached to the microtubulus, but without anything else bound to it. Transition into state 2: The head binds one ATP molecule out of the environment in its ATP binding pocket. Transition into state 3: The ATP is broken up into ADP and Pi —the so-called power-stroke—with the above-mentioned energy gain of about 20kB T . Transition into state 4: The Pi is released from the ATP binding pocket and simultaneously the aCnity to the microtubulus decreases dramatically, so that the head typically detaches. Transition into state 1: The ADP is released, the aCnity to the binding sites (E-tubulin) of the microtubulus becomes again large, with the result that the head will, after some random di5usion, attach to one of them, and we are back in state 1. The “energy factories” of the cell are constantly supplying fresh ATP and removing the used ADP and Pi , thereby keeping the concentration of ATP inside the cell about 6 decades above its thermal equilibrium (detailed balance) value, so that the probability of an inverse (endothermic) chemical cycle, transforming ADP and Pi back into ATP is completely negligible. It is noteworthy that the heads do not hydrolyze ATP at any appreciable rate unless they interact with the microtubulus, indicating that at least part of the chemical cycle is intimately coupled to the binding to a microtubulus [573]. The hydrolyzing step takes place while the head is attached to the microtubulus; the subsequent release of Pi enables the head to release its hold so that it can take another step on its journey along the microtubulus. The key to the energy transduction is thus the large change in aCnity between the heads of the motor protein and the protein Flament on which it walks. A particularly strong aCnity develops between the microtubulus-binding site of a head and the E-tubulin monomers. As a consequence, each tubulin dimer can bind at most one head and thus a single head has to cover the length of two dimers (about 16 nm) during each step of the motor enzyme along the microtubulus. To complete the picture, it should be mentioned that the motor enzyme proceeds along the microtubulus in a straight way, it does not “spiral” around the hollow tube during its journey [143,502,573,575]. Rather it follows with high Fdelity a path parallel to the protoFlaments so that the helicity of the microtubulus most likely plays no essential role; the main origin of the spatial asymmetry as far as the kinesin walk is concerned is that of the constituent dimers of the microtubulus together with that of the binding sites of the single heads. Remarkably, each given species of the kinesin superfamily can travel only in one preferential direction along the 96

Additional intermediate steps can be identiFed [574] but are usually neglected due to their short lifetimes.

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microtubulus, but di5erent species may move in opposite directions though they may be of striking structural similarity [143,572,576 –579]. Kinesin is a so-called processive motor enzyme, that is, it can operate individually. A single kinesin molecule can cover a distance of the order of 1 m before it may loose contact with the microtubulus and di5uses away [563,572,580,581], and this possibly even against an opposing force of up to 5 pN [142,582]. The reason seems to be on the one hand that the time-interval during which a single head is detached from the microtubulus while “stepping forward” is relatively short (one speaks of a high “duty ratio”) and on the other, that the two heads coordinate their actions so that at least one head is always attached [574]. A striking manifestation of this coordination is the fact [143,563] that apparently it is the energy gain out of the power stroke of the “front” head which triggers the “rear” head to make a step forward. For a more detailed exposition of the biophysical basics and experimental Fndings we refer to the excellent recent monograph by Howard [519]. 7.2. Basic modeling-steps 7.2.1. Biochemical framework Our Frst step in modeling a motor enzyme consists in recalling the description of a general biochemical reaction [185,583–589]: In principle, the starting point should be a quantum chemical ab initio treatment of all the electrons and atomic nuclei of the molecules involved in the reaction. Due to the clear cut separation of electron and nuclei masses, the electron dynamics can be adiabatically eliminated for each Fxed geometrical conFguration of the nuclei (Born–Oppenheimer approximation [584,586]) with the result of an e5ective potential energy landscape for the nuclei’s motion alone. In principle, there are many quantum mechanical energy eigenstates of the electrons for any Fxed conFguration of the nuclei, giving rise to a multitude of possible “potential energy surfaces” in the conFguration space of the nuclei [584,586 –588]. We assume that only one of them (the ground state energy of the accompanying electrons) is relevant in our case and especially is always well separated from all the other potential energy surfaces. In other words, the e5ective potential landscape governing the dynamics of the nuclei is single valued and no excitations of the electronic states are involved in the reaction cycle. Since the nuclei are already fairly massive objects, quantum mechanical e5ects will often play only a minor role for their dynamics, and we can focus on an approximate classical treatment. Indeed, while for very simple chemical reactions, a semiclassical or fully quantum chemical treatment may be necessary and still feasible, classical molecular dynamics is the only practically realistic approach in the case of a complex biomolecular system with hundreds or thousands of atoms, as we consider it here. In other words, all the relevant quantum mechanics of the system is assumed to be already encapsulated in the e5ective potential in which the nuclei move. 97 So far our description still comprises both the molecular motor 98 and its environment, typically some aqueous solution containing in particular ATP, ADP, and Pi molecules in certain concentrations. The role of the environment is twofold: On the one hand, it acts as a heat bath, giving rise to 97

Several of the above assumptions are in fact not necessary for the validity of our Fnal reduced description (see below), i.e. after the elimination of the (fast) bath degrees of freedom and the discretization of the chemical state variables. 98 More precisely: the compound motor–Flament system, see below.

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randomly =uctuating forces and to the associated damping (energy dissipation) mechanism in the molecular motor’s dynamics. On the other hand, it represents a source and sink of the reactants (ATP molecules) and products (ADP and Pi ) of the chemical reaction cycle. The in=uence of that part of the environment which acts as thermal heat bath can be taken into account along the same line of reasoning as in Sections 2.1.2 and 3.4.1. The result is a classical stochastic dynamics for the motor enzyme with a certain type of random noise and dissipation term, possibly supplemented by a renormalization of the e5ective potential landscape and the nuclear masses [92,93]. Under the assumption that the typical potential barriers are large compared to the thermal energy kB T , the conFgurations of the motor enzyme (deFned by the coordinates of the nuclei) will be restricted for most of the time to the local minima (metasable states) of the potential landscape and small =uctuations there about, 99 while transitions between di5erent local minima are rare events. In the case of simple chemical reactions, these transitions furthermore occur practically always along the same “most probable escape path”, called also “chemical pathway”, “reaction path”, or “intrinsic reaction coordinate” in this context [583,587,588]. One thus can describe all the essential conFgurations of the reaction in terms of this single intrinsic reaction coordinate and small (thermal) =uctuations there about. The latter can again be taken into account by means of dissipation and =uctuation terms in complete analogy to the above-mentioned modeling of the thermal heat bath [92,93,185]. As a result, a renormalization of the potential, the noise, the dissipation mechanism, etc., in the stochastic dynamics of the “intrinsic reaction coordinate” will arise, but the main point is that Fnally an e5ective description of the entire reaction in terms of a single generalized coordinate (also called collective coordinate, state variable or reaction coordinate) can be achieved, see also Section 3.4.1. In the case of complex biomolecules such as a motor enzyme, di5erent possible paths between the various metastable states may be realized with non-negligible probability [574,589]. In such a case, more than one collective coordinate (state variable) has to be kept in order to admit a faithful representation of all the possible pathways in the reduced description. Moreover, only some of those state variables can be identiFed with chemical reaction coordinates, while others are of a more mechanical or geometrical nature (see below). Finally, these concepts can also be generalized to cases without a clear cut distinction of metastable states and rare transition events, i.e. some of the (non-chemical) state variables may be governed by a predominantly relaxational or di5usive dynamics. Often, an equivalent way to discriminate relevant (generalized) coordinates which should be explicitly kept from “irrelevant noise” which can be savely eliminated is according to their characteristic time scale [93,150,186] (see also Section 3.4.1): On the smallest time scales (femtoseconds) the motion of the molecule consists of fast but small =uctuations, while signiFcant conformational changes will develop only on a much slower time scale of milliseconds. 7.2.2. Mechanical and chemical state variables For realistic systems, the above program—starting with the full quantum mechanical problem and ending with a simple approximate dynamics in terms of a few relevant classical stochastic variables—cannot be practically carried out. Therefore, a phenomenological modeling, roughly based 99

Trivial neutral translational and rotational degrees of freedom are assumed to have been eliminated already.

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on the above intuitive picture and supplemented by experimental evidence is necessary, see also Section 3.4.1. In the case of a motor enzyme like kinesin, the picture one has in mind is the following: The actual chemical conversion of ATP into ADP and Pi takes place in relatively well deFned and small regions of the enzyme—the ATP binding pockets of the two heads. This chemical cycle is captured by a set of chemical reaction coordinates or state variables y. On the other hand, the much larger conformational changes of the enzyme as a whole are represented by a di5erent set of “mechanical” 100 collective coordinates or state variables x. Note that both y and x are ultimately describing nothing else than the geometrical conFguration of the nuclei, but the distinction between chemical and mechanical coordinates are both conventional and suggestive. 101 Particularly diCcult to explicitly derive from Frst principles is the central feature of the enzymatic chemical reaction cycle, namely that reactant and product molecules can be exchanged with the environment. Typically, such events are possible (with non-negligible probability) only in certain speciFc conFgurations of the enzyme and it is assumed that the collective coordinates (x; y) are capable to faithfully monitor such events and, in particular, of whether some reactant=product molecule is presently attached to one of the heads or not. 102 The binding probabilities for both reactants and products depend on their concentrations in the environment of the enzyme. The fact that these dependences should be simply proportional to the respective concentrations is very suggestive and we will take it for granted in the following without any further derivation from a more fundamental description. It is quite plausible that whether or not one or both heads of the kinesin are attached to the microtubulus will have a signiFcant in=uence on both, the chemical reaction process and the mechanical behavior [519,582]. A priori, we should therefore not speak of an isolated kinesin but rather of the compound kinesin–microtubulus system. However, similar as for the previously discussed attachment and detachment of reactants and products, the attachment and detachment of the heads as well as the in=uence of the microtubulus in the attached state can be represented by the relevant collective coordinates (x; y) of the motor enzyme alone, if they have been appropriately chosen. 7.2.3. Discrete chemical states We recall that the “mechanical coordinates” x describe conFgurational changes of the enzyme as a whole, while the actual chemical ATPase is monitored by the “chemical coordinates” y and takes place in the rather restricted spatial regions of the ATP binding pockets. One therefore expects that transitions between di5erent “chemical states” y are accomplished during rather short time intervals in comparison with the typical time scales on which the global geometrical 100

“Mechanical” may refer here either to the fact that x represents the global geometrical shape of the molecule, or to the fact that some mechanical “strains” in the molecule, which have been created by the chemical transitions, may be released through a relaxational dynamics of x. 101 Note that the same applies for the (already eliminated) “irrelevant” degrees of freedom both of the environment and of the molecular motor itself: The dissipation and =uctuation e5ects, to which they give rise, may be either due to “mechanical” processes (vibrations, elastic and=or inelastic collisions, etc.) or due to chemical processes (making and braking of chemical bonds, etc.). 102 It is indeed plausible that the set of possible geometrical shapes of the enzyme while a reactant molecule is bound will be satisfactorily disjoint from the corresponding set in the absence of the reactant, and similarly for the products, provided the coordinates (x; y) have been suitably chosen.

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conFguration x notably changes [519]. As a consequence, one can neglect the details of the transitions between chemical states itself and focus on a discrete number of states, m = 1; 2; : : : ; Mtot , with certain “instantaneous” transition rates km → m (x) between them, which in general still depend on the conFguration x. Similarly, the potential landscape, which x experiences, still depends on the “chemical state” m. Formally, the transition rates km → m (x) are those between the local minima with respect to the y-coordinates at Fxed x, which, however, need not necessarily be local minima in the full x–y-space. In doing so, it is taken for granted that a well-deFned, relatively small number Mtot of discrete “chemical states” exists and that all transitions between them can be described in terms of rates km → m (x). Though such an approach is known to be problematic in other types of proteins due to their general “glass-like” properties and especially for the binding and unbinding processes of reactants and products in “pockets” of the proteins [590,591], in the context of motor enzymes like kinesin it has to our knowledge not been theoretically or experimentally challenged so far and we will therefore follow the general belief in its adequacy. At this point it should be emphasized again that one motor enzyme incorporates two “heads”, each endowed with an ATP-binding pocket and able to loop through its own chemical reaction cycle. Thus the set of “chemical coordinates” (vectors) y is in fact composed of two subsets (scalars), y = (y1 ; y2 ), one for each head, and similarly the discretized states are of the form m = (m1 ; m2 );

m1; 2 ∈ {1; 2; : : : ; M } :

(7.1)

For instance, for the standard model for the ATP reaction cycle consisting of M = 4 distinct states (cf. Section 7.2.1), the compound set of states m will comprise Mtot = M 2 = 16 elements. Note that exactly simultaneous reaction steps in both heads have negligible probability, i.e. only indices m = (m1 ; m2 ) with m = (m1 ; m2 ) or m = (m1 ; m2 ) are possible in km → m (x). In order to further reduce the number of non-trivial transition rates km → m (x), one common and suggestive assumption [519] is that only transitions between “neighboring” states within either of the two chemical cycles occur with non-negligible probability, i.e. km → m (x) = 0

if m ∈ {(m1 ± 1; m2 ); (m1 ; m2 ± 1)} ;

(7.2)

where states m1 which di5er by a multiple of M are identiFed, and similarly for m2 . In other words, each of the two chemical reaction cycles loops through a deFnite sequence of states, bifurcations into di5erent chemical pathways are ruled out. Note that the cooperativity between the two heads, mentioned at the end of Section 7.1, is mediated by the geometrical conFguration x and will manifest itself in the x-dependence of the rates, possibly reducing the number of non-trivial transition rates (7.2) once again. 7.3. Simpli8ed stochastic model While the so far reasoning and approximations have been relatively systematic and microscopically well founded, further possible simpliFcations are necessarily of a more drastic and phenomenological nature.

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In concrete models, the geometrical conFguration of the motor is usually assumed to be characterized by a single 103 relevant state variable x. One convenient choice for x turns out to be the position of the molecular motor along the microtubulus. To be precise, x may for instance be chosen to represent the position of the common center of mass of the two heads. Indeed, knowing that the motor enzyme walks in a step-by-step fashion straight along the E-tubulin sites of one and the same protoFlament, it is suggestive that the geometrical conFguration of each of the two heads can be reconstructed quite faithfully from the knowledge of the position x (the rest of the motor molecule (“tail” and “middle segment”, cf. Section 7.1) does not seem to play a signiFcant role for the actual “motor function”). Once the relevant collective chemical and mechanical state variables have been identiFed along the above line of reasoning, their “thermal environment” consists of two parts (cf. Section 7.2.1): Namely, on the one hand there are the huge number of “irrelevant” degrees of freedom of the liquid which surrounds the protein, and on the other hand there are those of the protein itself and of the microtubulus with which it interacts. Upon eliminating them along the lines of Section 3.4.1, their e5ects on the discretized chemical state variables are captured by the phenomenological rates (7.2). However, their e5ects on the mechanical state variable x are more involved due to the fact that x does not simply represent the cartesian coordinate of a point particle but rather the complicated geometrical conFgurations of the entire motor protein and in this sense is a generalized coordinate. As a consequence, the so-called solvent friction, caused by the eliminated degrees of freedom of the surrounding =uid, comprises not only a Stokes-type viscous friction against straight translational motion but also a damping force against conFgurational changes of the geometrical enzyme structure. Similarly, the so-called protein friction [9,519,592], caused by the eliminated degrees of freedom of the enzyme and the microtubulus, is composed of two analogous partial e5ects: on the one hand, a viscous drag against straight translational motion due to the continuous making and breaking of bonds between the motor and the microtubulus; on the other hand an e5ective “internal” frictional force against changes of the geometrical conFguration. All these friction mechanisms are in general not invariant under arbitrary translations of x and are therefore explicitly x-dependent. 104 The same carries over to the thermal =uctuations which they bring along (“solvent noise” and “protein noise”), see also Section 3.4.1. Since quantitatively the e5ects of protein friction are typically comparable or even more important than those of solvent friction [9,519], a quite signiFcant spatial inhomogeneity of the friction and the thermal noise is expected [553]. We recall that the microscopic origin of both solvent and protein friction is partly of a mechanical (geometrical) nature (mainly collisions and vibrations, respectively) and partly due to the making and breaking of numerous weak chemical bonds, as detailed in Section 7.2.1. On an even more basic level, all these distinctions become again blurred since the ultimate origin of friction is always the “roughness” of some e5ective potential energy landscape.

103 In other words, each “head” has its own (discrete) chemical state variable (cf. Eq. (7.1)), but the geometrical shape of the entire motor (including the two “heads”) is described by a single (continuous) state variable x. 104 Regarding the Stokes-type viscous friction, we recall that x represents not only the position but also the changing geometrical shape of the motor molecule. As mentioned in Section 6.4.2 we furthermore expect corrections of Stokes friction due to the nearby microtubulus, which are again in general x-dependent.

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7.3.1. Stochastic ratchet dynamics On the basis of the above considerations, the simplest working model for the stochastic dynamics governing the mechanical coordinate (position) x(t) is of the form

x(t) ˙ = −Vm (x(t)) + F + (t) ;

(7.3)

where m=m(t) is understood as a stochastic process, with states (7.1) and transition rates km → m (x). The assumption of a Frst order (overdamped) dynamics in time is justiFed as usual by the fact that on these small scales inertia e5ects can be safely neglected [186]. The damping coeCcient and the random noise (t) model the e5ects of the environment and of the eliminated fast degrees of freedom of the molecular motor itself (possibly also of the microtubulus) and both these contributions are treated as a single thermal bath. Under the assumption that the origin of (t) is a very large number of very fast processes (on the time scale of x) we can model those =uctuations as a Gaussian noise of zero mean and negligible correlation time (t)(s) = 2 kB T(t − s) :

(7.4)

In fact, already the very form of the dissipation assumed on the left-hand side of (7.3) leaves no other choice for the noise (t) at equilibrium, see Section 3.4.1. A further assumption implicit in (7.3) is the independence of the coupling to the heat bath (see below (2.5)) from the chemical state m and the geometrical conFguration x. The former simpliFcation is plausible in view of the fact that the chemical processes only involve a very restricted region of the entire motor enzyme. On the other hand, the x-independence of is not obvious in view of our above considerations about solvent and protein friction, but can be justiFed as follows: First, inhomogeneous friction, and in particular protein friction, can be modeled quite well by means of potentials Vm (x) in (7.3) with a suitably chosen “roughness” on a very “Fne” spatial scale. After a spatial coarse graining, only the broader structures of the potential survive while the initially homogeneous “bare” friction is dressed by an inhomogeneous renormalization contribution. A second possibility consists in a change of variable 105 as detailed in Section 6.4.2. Note that accounts for the coupling of the thermal environment (fast degree of freedom) of the molecular motor only. The additional slow variable representing the cargo of the motor can be accounted for [582] via a contribution of the form −x

˙ cargo to the force F in (7.3) under the tacit but apparently realistic assumption that its connection to the motor (via “tail” and middle segment”, cf. Section 7.1) is suCciently elastic [505,506,565]. Although the cargo is typically much bigger than the motor itself, this viscous drag force seems negligibly small [582] in comparison with the intrinsic friction of the motor, modeled by x(t) ˙ in (7.3). The deterministic mechanical forces in (7.3) on the one hand, derive from an e5ective, free-energy like potential Vm (x) and on the other hand, leave room for the possibility of an externally applied extra force F. Originating from the potential energy landscape in which the nuclei of the motor and

105

In this case, the transformed potentials in (7.3) remain periodic but in general pick up an F-dependence, which we neglect for the sake of simplicity (see below).

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its environment move, the e5ective (renormalized) potential Vm (x) in addition accounts for some of the e5ects of the eliminated fast degrees of freedom. The approximate independence of this e5ective potential Vm (x) from the external load F is assumed here for the sake of simplicity 106 [552,593– 595]. On the other hand, the dependence of the potential on the chemical state m is crucial. The latter in conjunction with the x-dependence of the chemical reaction rates km → m (x) is called the mechanochemical coupling mechanism of the model motor enzyme, decisive for the chemical to mechanical energy transduction. 107 The underlying picture is that certain chemical reaction steps take place preferably or even exclusively while the molecular motor has a speciFc geometrical shape x. In turn, certain mechanical relaxations of strains or thermally activated conFgurational transitions may be triggered or made possible only after a certain chemical reaction step has been accomplished. Clearly, the dynamical behavior of the motor enzyme is invariant after a step of one head has been completed if at the same time the chemical states m1 ; m2 of the two heads are exchanged [143,596]. This invariance under a displacement x → x + L and simultaneously (m1 ; m2 ) → (m2 ; m1 ) has to be respected by the potentials Vm (x) and the rates km → m (x), Vm (x + L) = VmW (x) ;

(7.5)

km → m (x + L) = k(mW → m )(x) ;

(7.6)

where the bar denotes the exchange of the vector components: (m1 ; m2 ) := (m2 ; m1 ) ;

(7.7)

and where the spatial period L is given by the length of one tubulin dimer (about 8 nm). Consequently, the functions Vm (x) and km → m (x) are invariant under x → x + 2L without any change of the chemical states. The polarity of the microtubulus, on which the motor walks, re=ects itself in a generic spatial asymmetry of the potential Vm (x) as well as of the rates km → m (x). Note that on top of that, there is also an intrinsic asymmetry of the motor domains (but not of the entire enzyme, see Section 7.1): ◦ If one detaches a motor domain from the microtubulus, turns it around by 180 , and puts it back on the microtubulus, no invariance arises [143,577,578], that is, re=ection symmetry is broken. In other words, the asymmetry of the microtubulus is necessary to make manifest the asymmetry of the motor, while the asymmetry of the compound system is caused and maybe even mutually enhanced by both [17,597]. The stochastic dynamics (7.3) as it stands is a convenient starting point for numerical simulations (cf. Section 2.2) but not for quantitative analytical calculations. Exactly like for the =uctuating potential ratchet model in Eqs. (4.12), (4.13), one obtains the following master equation 106

As far as x describes the center of mass of the molecular motor, the simple F-dependence on the right-hand side of (7.3) is fully justiFed. However, in so far as x at the same time accounts for the geometrical shape of the motor molecule, the relation between position and shape and hence the e5ective potentials Vm (x) are expected to change upon application of a force F. 107 The F-independence of the rates km → m (x) (and a forteriori of the number Mtot of chemical states) is plausible on the basis of the physical picture from Section 7.2.3 (the chemical processes are spatially localized and thus involve negligibly small changes of the geometrical conFguration of the motor molecule).

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(reaction–di5usion equation) equivalent to (7.3): 9 [Vm (x) − F]Pm (x; t) kB T 92 9 Pm (x; t) = + Pm (x; t) 9t 9x

9x2 km → m (x) + Pm (x; t)km → m (x) ; −Pm (x; t) m

(7.8)

m

where Pm (x; t) is the joint probability density that time t the chemical state is m and the motor at enzyme is at the position x, with normalization m d x Pm (x; t)=1. In order to technically simplify matters one deFnes similarly as in (2.22) reduced densities ∞ 1 ˆ {Pm (x + 2nL; t) + PmW (x + 2(n + 1)L; t)} : P m (x; t) := 2 n=−∞

(7.9)

The reduced densities satisfy the same master equation (7.8) but are periodic in x with period 2L and normalization 2L d x Pˆ m (x; t) = 1 : (7.10) m

0

Symmetries (7.5), (7.6) furthermore imply that Pˆ m (x; t) = PmW (x + L; t) = Pˆ m (x + 2L; t) :

(7.11)

Once Pˆ m (x; t) is determined, the average speed of the motor enzyme follows along the same line of reasoning as in Section 2.3 as 108 2L 1 F− x ˙ = d x Vm (x)Pˆ m (x; t) : (7.12)

0 m A further interesting quantity is the rate rATP = rATP (t) of ATP-consumption per time unit, given by 2L ATP rATP = @m; m d x {Pˆ m (x; t) km → m (x) − Pˆ m (x; t)km → m (x)} ; (7.13) m; m

0

ATP where @m; m is the indicator function for ATP-binding transitions m → m . For example, using the labeling of the chemical states from Section 7.2.1 for the standard ATP-hydrolysis cycle with M = 4 states, we have   1 if m = (1; m2 ) and m = (2; m2 ) ;   ATP 1 if m = (m1 ; 1) and m = (m1 ; 2) ; @m; (7.14) m =    0 otherwise :

A comparison of the above model setup with the working model from Section 3.1 very obviously establishes a close connection between our present section about molecular motors and the general 108

We recall that the argument t in x ˙ is omitted (cf. (3.4)) since in most cases one is interested in the steady state st ˆ ˆ behavior with P m (x; t) = P m (x). The same applies for the rate rATP = rATP (t) in (7.13).

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framework for our studies of ratchet models, especially the class of pulsating ratchets 109 according to the classiFcation scheme from 110 Section 3.3. However, there is also one important point in which the present model goes beyond the latter general framework. Namely, there is a back-coupling of the state-variable x(t) to the “potential =uctuations” m(t) through the x-dependence of the transition rates km → m (x). Especially, the statistical properties of the potential =uctuations m(t) can no longer be assumed a priori as stationary. We will show later in Section 7.4.2 that far away from equilibrium an e5ective x-independence of the potential =uctuations m(t) may arise nevertheless, entailing stationarity of their statistical properties in the long-time limit, i.e. a veritable pulsating ratchet scheme is recovered. 7.3.2. Nonequilibrium chemical reaction At thermal equilibrium, the concentrations of ATP, ADP, and Pi are not independent, their ratio 0 0 CATP =CADP CP0i satisFes the so-called mass action law. Especially, the numerical value of this ratio must be independent of whether any motor enzymes (acting as catalyst) are present or not. Since this represents a single constraint for three variables, there still remains a freedom in the choice of 0 0 two out of the three equilibrium concentrations CATP , CADP , and CP0i . We consider an arbitrary but Fxed such choice from now on. Since the system is an equilibrium system, the stochastic dynamics has furthermore to respect the so-called condition of detailed balance [98–101,148–152]. For our speciFc model (7.3) this condition can be readily shown to imply the following relation between the transition rates km → m (x) and the corresponding potentials Vm (x) and Vm (x) for any pair of chemical states m and m : Vm (x) − Vm (x) km → m (x) = exp : (7.15) km → m (x) kB T Thus, one of the two rates in (7.15) can be considered as a free, phenomenological function of the model, while the other rate is then Fxed. Note that the appearance of negligibly small rates in (7.2) as well as the symmetry relations (7.5), (7.6) are still compatible with (7.15). The salient point is now to clarify what is meant by saying that one goes “away from equilibrium” in our present context. Meant is, that as far as the heat bath properties of the environment (random =uctuations and energy dissipation mechanism) are concerned, nothing is changed as compared to the thermal equilibrium case. The only things which change are the concentrations of reactants and=or products [598]. 0 For instance, if the ATP concentration CATP is changed away from its equilibrium value CATP , then all the rates km → m (x) remain unchanged except those which describe the binding of ATP to one of the two heads of the molecular motor. As discussed in Section 7.2.2 these rates simply

109

The driving f(t) of the pulsating potential V (x; f(t)) is denoted here by m(t) and the pulsating potential itself by Vm (x) = Vm(t) (x) (cf. (3.1) and below (7.3), respectively). Moreover, m(t) is here a discrete and—in general— two-dimensional state variable (cf. (7.1)), though in most concrete models (see Sections 7.4 –7.6) again a simpliFed, e5ectively one-dimensional description will be adopted. 110 It may be worth to recall that for traveling potential ratchets and their descendants (Sections 4.4 and 4.5) a broken symmetry of the potential is not necessary for directed transport, though for real molecular motors this symmetry will be typically broken.

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0 acquire an extra multiplicative factor of the form CATP =CATP , i.e. (7.15) is generalized 111 to CATP km → m (x) Vm (x) − Vm (x) ATP = 1+ ; (7.16) − 1 @m; m exp 0 km → m (x) kB T CATP ATP where @m; m is the ATP-binding indicator function from (7.14). Similar modiFcations arise if the concentrations of ADP and Pi are changed. However, in order to describe the real situation one may without loss of generality, assume that these concentrations have already their correct value 0 due to our choice of CADP and CP0i . In doing so, it follows from the quantitative biological Fndings mentioned in Section 7.1 that CATP has to be chosen about six decades beyond its equilibrium value 0 CATP :

CATP 106 : 0 CATP

(7.17)

From the conceptual viewpoint we are thus facing the following interesting setup of a far from equilibrium system: On one hand, the system is in contact with a thermal equilibrium heat reservoir as far as dissipation and =uctuations are concerned. On the other hand, it is in contact with several reservoirs of reactant and products with concentrations which are externally kept far away from equilibrium. All these various reservoirs are physically localized at the same place but the e5ects due to their direct interaction with each other is practically negligible. Only the indirect interaction by way of the motor molecules (catalysts) is relevant. 7.4. Collective one-head models At this stage, the number of free, phenomenological functions in (7.8) is still very large. There is little chance to make a convincing guess for each of them on the basis of our present knowledge about the structure and functioning of the real motor enzyme, while for Ftting the dynamical behavior of the model to experimental curves, the available variety and accuracy of measurements is not suCcient. Our next goal must therefore be to reduce the e5ective number Mtot of relevant chemical states. 7.4.1. A.F. Huxley’s model The most prominent such simpliFcation goes back to Huxley’s 1957 paper [4] and consists in the assumption of one instead of two heads per motor enzyme. In our model (7.3) this means that m is no longer composed of two “substates”, see Eq. (7.1), but rather is a scalar state variable with Mtot = M values m ∈ {1; 2; : : : ; M } :

111

(7.18)

Without discretizing the chemical state variable(s) (or equivalently, assuming that a separation of time-scales exists such that a rate description is justiFed) the proper reformulation of a relation like in (7.16) does not seem possible, see also Section 4.6 and [186].

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Likewise, x now represents the center of mass of a single head. As a consequence, symmetries (7:5), (7:6) become Vm (x + L) = Vm (x);

km → m (x + L) = km → m (x)

(7.19)

and Eqs. (7.9) – (7.11) are replaced by Pˆ m (x; t) :=

∞

Pm (x + nL; t) ;

(7.20)

n=−∞

Pˆ m (x + L; t) = Pˆ m (x; t) ; M m=1

0

L

d x Pˆ m (x; t) = 1 :

Furthermore, Eqs. (7.12) and (7.13) assume the form M L 1 F− d x Vm (x)Pˆ m (x; t) ; x ˙ =

0 m=1 L d x {Pˆ 1 (x; t)k1 → 2 (x) − Pˆ 2 (x; t)k2 → 1 (x)} : rATP = 0

(7.21) (7.22)

(7.23) (7.24)

Finally, all rates km → m (x) with m = m±1 are zero according to (7.2), and for m =m±1 Eq. (7.16) takes the form km → m (x) Vm (x) − Vm (x) CATP ; (7.25) exp − 1 = 1+ m; 1 m ; 2 0 km → m (x) kB T CATP where the ATP binding transition is assumed to be m = 1 → m = 2 and where states which di5er by a multiple of M are identiFed. A second ingredient of Huxley’s model is a “backbone” to which a number N of such single headed motors is permanently attached. The emerging intuitive picture is a centipede, walking along the polymer Flament. The interaction of the single-headed motors is mediated by the common backbone, assumed rigid and moving with constant speed x, ˙ but otherwise they are considered as operating independently of each other. We may then concentrate on any of the single heads and without loss of generality denote the site where this speciFc head is rooted in the backbone by xt. ˙ In physical terms, we are dealing with a mean 8eld model (N → ∞), described by an arbitrary but Fxed reference head according to (7.3), where the potentials Vm (x) and the rates km → m (x) may, in general, acquire an additional dependence on the backbone site xt. ˙ The possible di5erence between the center of mass of the head x and the point xt ˙ where it is attached to the backbone may, for instance, re=ect a variable angle between the head’s length axis and the polymer Flament, similarly to a human leg while walking. Thus, we may also look upon xt ˙ as an additional relevant (slow) mechanical state variable of the motor. However, no extra equation of motion for this coordinate is needed since it already follows in the spirit of a mean Feld approach from the behavior of the other relevant mechanical state variable x. For instance, a term of the form A(x − xt) ˙ 2 in the potentials

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Vm (x) models a harmonic coupling of the head to the uniformly advancing backbone, with spring constant A. As in any mean Feld model, the characteristic feature is the appearance of an a priori unknown “order parameter”, x ˙ in our case, which has to be determined self-consistently in the course of the solution of the model (7.3), (7.12) (for an explicit example see (7.28), (7.29) below). We emphasize that for a rigid backbone, in the limit N → ∞ Huxley’s mean Feld approach is not an approximation but rather an exact description because the interaction between the single motors is of inFnite range. We remark that such a model of N single headed motors with a mean Feld coupling through a rigid “backbone” may even be acceptable as a rough approximation in the case of a single kinesin molecule. Admittedly, the number N = 2 of involved heads makes a mean Feld approximation somewhat questionable. On the other hand, taking into account that a rather “heavy” load is attached to the motor, may render the assumption of an uniformly moving backbone not so bad [582]. On top of that, the cooperativity of the two heads in the real kinesin is at least roughly incorporated into the model through their interaction via the backbone and through the implicit assumption that the motor will not di5use away from the microtubulus even if both heads happen to take a step at the same time. More suggestive is the case when an appreciable number N of single motor molecules truly cooperate. This may be a couple of kinesins which drag a common “big” cargo. More importantly, there exist motor enzymes di5erent from kinesin which indeed are interconnected by a backbone-like structure by nature. Examples are the so-called myosin enzymes, walking on polymer Flament tracks called actin, thereby not carrying loads but rather playing a central role in muscular contraction [343,563,571]. While the quantitative and structural details are di5erent from the kinesin– microtubulus system, the main qualitative features of the myosin–actin system are suCciently similar [577,599,600] such that the same general framework (7.2) – (7.12) is equally appropriate in both cases. 112 Though a single myosin enzyme again consists of two individual motor domains, their cooperativity seems not so highly developed as for kinesin [577,602] and therefore the above-mentioned mean Feld approximation for a large number N of interacting single heads appears indeed quite convincing. In his landmark paper [4], Huxley proposed a model of this type without any knowledge about the structural features of an individual motor enzyme. It is not diCcult to map the slightly di5erent language used in his model to our present framework, but since the details of his setting cannot be upheld in view of later experimental Fndings, we desist from explicitly carrying out this mapping here. While the model is apparently in satisfactory agreement with the main experimental facts available at that time, Huxley himself points out [4] that “there is little doubt that equally good agreement could be reached on very di5erent sets of assumptions, all equally consistent with the structural, physical and chemical data to which this set has been Ftted. The agreement does however show that this type of mechanism deserves to be seriously considered and that it is worth looking for direct evidence of the side pieces”.

112

Intriguingly enough, certain species of the myosin superfamily (e.g. the so-called myosin V subfamily) show again a behavior similar to kinesin [563,601]. In the following we always have in mind collectively operating myosin species (the myosin II subfamily).

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7.4.2. Free choice of chemical reaction rates One speciFc point of Huxley’s model is worth a more detailed discussion since it illustrates a much more general line of reasoning in the construction of such models. In doing so, we Frst recall that we are dealing with a single head motor model described by M = 4 (scalar) chemical states m: (1) the head without anything bound to it; (2) the head with an ATP bound; (3) the head with an ADP and a Pi bound; and (4) the head with an ADP bound. The chemical state variable travels back and forth between neighboring states of this cycle according to the transition rates km → m (x), respecting (7.25) if m = m ± 1 and being zero otherwise. For the case of kinesin, we have discussed at the beginning of Section 7.1 in addition the “aCnity” between head and Flament in each state, which essentially tells us whether the head is attached to the Flament or not in the respective state, and which has to be taken into account in the concrete choice of the respective model potentials Vm (x). We remark that this correspondence between states and aCnity is somewhat di5erent for myosin [9] and again di5erent in Huxley’s model, but will not play any role in the following, since it only regards quantitative, but not qualitative properties of the potential Vm (x). As a Frst simpliFcation, Huxley postulates a 3-state model, in which m = 2 and 3 in our above scheme are treated as a single state, and the question arises of whether and how this can be justiFed, at least in principle. One possible line of reasoning goes as follows: Aiming at a uniFcation of m = 2 and 3 means in particular that we should choose V2 (x)=V3 (x) and thus k2 → 3 (x)=k3 → 2 (x) according to (7.25). Since there are no further a priori restrictions on the choice of these rates, we may take them as independent of x and very large. 113 Thus, as soon as the system reaches either state 2 or 3 it will be practically instantaneously distributed among both states with equal probability. One readily sees that the two states can now be treated as a single “superstate” if the two transition rates out of this state are deFned as half the corresponding original values k2 → 1 (x) and k3 → 4 (x). At Frst glance, it may seem that in this reduced 3-state model, condition (7.25), in the case that m represents the new “superstate”, has to be modiFed by a factor 12 . However, since in the stochastic dynamics (7.3) only the derivative of the potential Vm (x) appears, this factor 12 can be readily absorbed into an additive constant of that potential. Given the reduced model with M =3 states, Huxley furthermore assumes that the 3 “forward” rates km → m+1 (x) can be freely chosen, while the 3 “backward” rates km+1 → m (x) are negligibly small. On the other hand, Eq. (7.25) tells us that the 3 forward rates can indeed be chosen freely, but once they are Fxed, the 3 backwards rates are also Fxed. At this point, one may exploit once again the observation that only the derivatives Vm (x) enter the dynamics (7.3) and therefore we still can add an arbitrary constant to any of the three model potentials Vm (x). Under the additional assumption that exp{[Vm (x)−Vm+1 (x)]=kB T } varies over one spatial period L at most by a factor signiFcantly smaller 0 than (CATP =CATP )1=3 102 (see (7.17)), one readily sees that by adding appropriate constants to the 3 potentials Vm (x) one can make the ratios km → m+1 (x)=km+1 → m (x) rather small for all 3 values of m according to (7.25). Pictorially speaking, by adding proper constants to the potentials Vm (x) one 0 can split and re-distribute the factor CATP =CATP from (7.17) along the entire chemical reaction cycle. In this way, all 3 backward rates km+1 → m (x), though not exactly zero, can indeed be practically neglected. Generalizations to more than 3 states and to the neglection of only some, but not all, backward rates are obvious. 113

Such a choice is obviously admissible within our general modeling framework; how to justify it against experimental Fndings is a di5erent matter [574,603– 605].

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A Fnal important observation concerns the case M = 3 with all the forward rates still at our disposition and all the backward rates approximately neglected. SpeciFcally, one may assume that V2 (x) = V3 (x) and that the corresponding forward rate k2 → 3 is x-independent and very large. The two states m = 2 and 3 can then again be lumped into a single superstate. The result is [16] a model (7.3) with M = 2 e5ective chemical states but with both, the forward and backward rates between these two states, still free to choose. We have thus achieved by way of various simplifying assumptions our goal to substantially reduce the number of free, phenomenological functions in the model (7.3). Still, even for the minimal number M = 2 of chemical states the shape of the two potentials and especially the choice of the two rates [9,514,582,596] are very diCcult to satisfactorily justify on the basis of experimental Fndings. Accordingly, the existing literature does not seem to indicate that a common denominator of how these functions should be realistically chosen is within hands reach. 7.4.3. Generalizations Huxley’s choice of model parameters and functions (7.18), (7.19) in the general setup (7.3) has been subsequently modiFed and extended in various ways in order to maintain agreement with new experimental Fndings. Most of the following works include veriFcations of the theoretical models against measurements, though we will not repeat this fact each time. Moreover, a detailed discussion of the speciFc choices and justiFcations of the free, phenomenological parameters and functions in the general model (7.3) in those various studies goes beyond the scope of our review. Our main focus in this section will be on the character of the mechanochemical coupling (cf. Section 2.7) and the relevance of the thermal noise for the dynamics of the mechanical state variable in (7.3), see also Section 7.7 for a more systematic discussion of these points. With more structural data of the actin–myosin system on the molecular level becoming available, Huxley and Simmons [606] already in 1971 came up with a more realistic modiFcation of the original model, featuring a “fast” (chemical) variable with a small number of discrete states, tightly coupled to a “slow” (mechanical) continuous coordinate. For more recent studies along these lines see also [607– 613] and references therein. A very recent, analytically solvable model, closely resembling Huxley’s original setup and in quantitative agreement with a large body of experimental data, is due to [519]. The issue of the chemical to mechanical coupling has been for the Frst time addressed in detail by Mitsui and Oshima [144], pointing out that deviations from a simple and rigid one-to-one coupling may play an important role. A connection between a model of the Huxley type with Feynman’s ratchet-and-pawl gadget has apparently been realized and worked out for the Frst time by Braxton and Yount [5,6], though their model was later proven unrealistic by more detailed quantitative considerations [142,143]. A similar Feynman-type approach has been independently elaborated by Vale and Oosawa [7]. More importantly, they seem to have been the Frst to bring into play the crucial question of the relative importance of the thermal =uctuations appearing in the dynamics of the mechanical coordinate (7.3) as compared to conformational (relaxational) changes powered by the chemical cycle (that is, ultimately by the power stroke). One extreme possibility is characterized by barriers of the potentials Vm (x) which can be crossed only with the help of the thermal noise (t) in (7.3), independently of how the chemical state

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m evolves in the course of time. An example is the =uctuating potential ratchet (4.11) with f(t) restricted to a discrete number (M ) of possible values, all smaller than unity in modulus. In such a case, the role of the chemical cycle is merely the breaking of the detailed balance, necessary for a manifestation of the ratchet e5ect in the x-dynamics. Moreover, the mechanochemical coupling is typically (i.e. unless the rates km → m (x) exhibit a very special, strong x-dependence, see below) loose, the number of chemical cycles per mechanical cycle randomly varies over a wide range. The opposite possibility is represented by the traveling potential ratchet mechanism, see Section 4.4. Each chemical transition m → m = m + 1 induces a strain in the mechanical coordinate via Vm (x) in (7.3) which then is released while x relaxes towards the closest local minimum of Vm (x). As m proceed through the chemical loop, also the local minima of Vm (x) are shifting forward in suCciently small steps such that x typically advances by one period L after one chemical cycle. In this case, the thermal noise has only an indirect e5ect through the chemical rates km → m (x), but as far as the mechanical dynamics (7.3) is concerned, almost nothing changes in comparison with a purely deterministic ((t) ≡ 0) behavior. In other words, the mechanochemical coupling is very rigid, the mechanical coordinate x is almost exclusively powered by the chemical reaction and its behavior is basically “slaved” by the chemical transitions. The mechanical coordinate x may at most play a role in that the practically deterministic relaxation of x after a chemical transition m → m may delay the occurrence of the next transition m → m until the new local minimum of Vm (x) has been reached. Essentially, the system can thus be described by the chemical reaction cycle alone, possibly augmented by appropriate deterministic refractory periods (waiting times) after each reaction step [515,516,518]. It then does not seem any more appropriate to speak of a noise-induced transport in the closer sense and the ratchet e5ect only enters the picture via the somewhat trivial traveling potential ratchet mechanism from Section 4.4, see also Section 2.7. A situation intermediate between these two extreme cases arises if the potential Vm (x) exhibits (approximately) =at segments, requiring di5usion but no activated barrier crossing for being transversed. An example is the on–o5 ratchet scheme from Section 4.2. Another compromise between the two extremes consists in the following scenario: thermally activated barrier crossing is unavoidable for the advancement of x. Yet, due to the choice of the rates km → m (x), the next chemical step becomes only possible after the respective barrier crossing has been accomplished. In other words, though thermal noise e5ects are an indispensable ingredient for the working of the motor enzyme model, the stepping of x and m is tightly coupled. Since the thermal activation processes can be considered as rate processes, such a model can be mapped in very good approximation to an augmented reaction cycle, with some mechanical states added to the chemical ones. The proper notion for such a situation seem to be “mechanochemical reaction cycle”. For a more systematic treatment of such issues see Section 7.7. The conclusion of Vale and Oosawa [7] is that, within their Huxley-type model (7.3), thermal noise in the mechanical coordinate x plays an important role; speciFcally, a mechanism similar to a =uctuating potential ratchet (Section 4.3), a temperature ratchet model (Sections 2.6 and 6.3), or a combination of both is postulated (later criticized as being unrealistic in [9,142,143]). While the latter conclusion is mainly of a qualitative nature, a more quantitative investigation of the same question is due to Cordova et al. [10], with the result that for cooperating motor enzymes like myosin, thermal activation processes are—within their choice of model parameters and functions in (7.3)—crucial in the dynamics of the mechanical variable x, while for kinesin such processes may be of somewhat less importance. In deriving the latter conclusion, these authors go

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one step beyond Huxley’s framework in that they also analyze the motion of a single head (no backbone), and especially of two heads without invoking a mean Feld approximation for the motion of the backbone, which, in this context should then rather be viewed as a “hinge” connecting the two heads. A further reFned variation of Huxley’s model has been worked out by Leibler and Huse [8,9], together with a few-head model (beyond mean Feld) in a general spirit similar to that of Cordova et al. [10]. In this model, however, a tight mechanochemical coupling is built-in from the beginning, namely the choice of the parameters and functions in (7.3) is such that thermal noise e5ects on the mechanical coordinate x play a minor role by construction. Furthermore, all the transition rates km → m are assumed to be independent of x. Within such a model, it is shown that at least M = 4 chemical states are required to avoid incompatibilities with known experimental Fndings. The main achievement of these studies [8,9] is a uniFed description of “porter” motor proteins, e.g. kinesin, operating individually and spending a relatively short time detached from the polymer Flament (moderate-to-large duty ratio), and of “rowers”, e.g. myosin, which operate collectively and are characterized by a small duty ratio. Thus, “porters” are essentially processive, and “rowers” non-processive motor enzymes. A reFned model similar in spirit has been put forward in [514]. 7.4.4. Julicher–Prost model One of the most striking statistical mechanical features of interacting many body systems, both at and far away from thermal equilibrium, is the possibility of spontaneous ergodicity-breaking, entailing phase transitions, the coexistence of di5erent (meta-) stable phases, and a hysteretic behavior in response to the variation of appropriate parameters. There is no reason why such genuine collective e5ects should not be expected also in Huxley-type mean Feld models, but it was not before 1995 that JMulicher and Prost [550] explicitly demonstrated the occurrence of those phenomena in such a model, see also [14,596,614 – 616]. SpeciFcally, they focused on the dependence of the average velocity x ˙ upon (parametric) variations of the external force F in (7.3), henceforth called x-versus-F ˙ characteristics. As already mentioned, formally the crucial point in such a mean Feld approach is the appearance of a self-consistency equation for the “order parameter” x. ˙ Typically, this equation is non-linear 114 and the existence of multiple (stable) solutions signals the breaking of ergodicity. After having observed such a situation in their model, JMulicher and Prost pointed out in a subsequent work [617] the following remarkable consequence of the hysteretic x-versus-F ˙ characteristics: if the rigid backbone is coupled to a spring, then an e5ective external force F depending on the position of the backbone arises. If the spring is suCciently soft, then the changes of F are suCciently slow such that the parametric x-versus-F ˙ characteristics can be used. If this relation furthermore exhibits a hysteresis loop with the two x-versus-F ˙ branches conFned to either side of x ˙ = 0, then a permanent periodic back-and-forth motion of the backbone is the result. Remarkably, strong indications for both, spontaneous ergodicity breaking (dynamical phase transition in the velocity–force relationship) as well as spontaneous oscillations can indeed be observed in motility assays [618] and in muscle cells under suitable conditions [14,596,614 – 617,619 – 621], respectively. We recall that spontaneous breaking of ergodicity with its above-mentioned consequences is a common phenomenon already at equilibrium. In contrast, a Fnite current x ˙ = 0 at F = 0 as well 114

For an example, see Eqs. (7.28) and (7.29) below.

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as spontaneous oscillations [622– 626] represent genuine collective non-equilibrium e5ects which are excluded at thermal equilibrium by the second law of thermodynamics. Both, from the conceptual viewpoint and with regard to the mechanochemical coupling issue, the JMulicher–Prost model exhibits a couple of noteworthy features. A Frst crucial assumption of the model is that not only the backbone itself but also the positions of the N individual motors with respect to the backbone are perfectly rigid. Since the backbone moves with a speed x ˙ it follows that for any single motor x˙ = x ˙ :

(7.26)

Much like any intensive state variable in equilibrium thermodynamics, the “order parameter” x ˙ within such a mean Feld approach is a macroscopic state variable and is not any more subject to any kind of random =uctuations in the thermodynamic limit N → ∞. In other words, the stochastic equation for the single (uncoupled) motors (7.3) simpliFes to an equivalent deterministic (noise-free) dynamics (7.26) for every single motor in the presence of a mean Feld (perfectly rigid all-to-all) coupling. Here x ˙ plays the role of a formal (not yet explicitly known) deterministic force and—as already pointed out in Section 7.4.1—our next goal must now be to derive a self consistency equation for this order parameter x ˙ if we wish to determine its explicit value. To this end we Frst notice that working with (7.26) instead of (7.3) is tantamount to setting T = 0 and −Vm (x) + F = x ˙ in (7.3). Accordingly, the Frst two terms on the right-hand side of the master equation (7.8) may be replaced by the equivalent simpliFed expression −x9P ˙ m (x; t)=9x. A second essential assumption is that the N individual motor enzymes are rooted in the backbone either at random positions or—biologically more realistic—with a constant spacing which is incommensurate with the period L of the polymer Flament. As a consequence, the reduced spatial distribution of particles m Pˆ m (x; t) approaches an x- and t-independent constant value for N → ∞. As a Fnal assumption, a one-head description of the individual motor enzymes with M =2 chemical states is adopted. 115 Exploiting the above mentioned fact that Pˆ 1 (x; t) + Pˆ 2 (x; t) is a constant and normalized on [0; L] according to (7.22), one can eliminate Pˆ 2 (x; t) from the master equation (7.8), yielding in the steady state 116 (superscript st) the ordinary Frst order equation [550] x ˙

d ˆ st st st P (x) = −Pˆ 1 (x)k1 → 2 (x) + [1=L − Pˆ 1 (x)]k2 → 1 (x) ; dx 1

(7.27)

st st supplemented by the periodic boundary condition 117 Pˆ 1 (x + L) = Pˆ 1 (x). The unique solution is y x+L 2 → 1 (z) dy k2 → 1 (y) exp{ x d z k1 → 2 (z)+k } st x x˙ ˆ P 1 (x) = : (7.28) L 2 → 1 (z) Lx[exp{ ˙ d z k1 → 2 (z)+k } − 1] 0 x˙

115

According to Section 7.4.1, this model may equally well be viewed as a M = 2 state model of enzymes with two highly coordinated heads. See also Section 7.4.2 for references and more details regarding such a model. 116 The convergence towards a steady state in the long time limit is tacitly taken for granted. A partial justiFcation of this ansatz can be given a posteriori by showing that such a solution indeed exists and moreover satisFes certain stability conditions against perturbations. Especially, the task to prove that no additional (non-stationary) long time solutions co-exist is a delicate issue. In practice, the only viable way consists in a direct numerical simulation of a large number N of coupled stochastic equations. st 117 There is no normalization condition for Pˆ 1 (x) alone.

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st Note that the non-negativity of Pˆ 1 (x) is guaranteed if k1 → 2 (x) ¿ 0 and k2 → 1 (x) ¿ 0 for all x. Finally, by eliminating in the same way Pˆ 2 (x; t) in the self-consistency equation (7.23) for x ˙ one Fnds that L 1 st F− d x (V1 (x) − V2 (x))Pˆ 1 (x) : (7.29) x ˙ =

0

By introducing (7.28) into (7.29) a closed (transcendental) self-consistency equation for the order parameter x ˙ is obtained. Much like in the elementary mean Feld theory (Weiss theory) for a ferromagnet, the occurrence of multiple solutions will signal the breaking of ergodicity and thus a phase transition. Apart from the need of solving a transcendental equation at the very end, the above model is one of the very rare special cases (cf. Section 4.3.1) of an analytically exactly tractable =uctuating potential ratchet. We Fnally recall that by interpreting the M = 2 state model as a reduced M = 4 state description, both rates k1 → 2 (x) and k2 → 1 (x) are still at our disposition (see Section 7.4.2). Besides the tremendous technical simpliFcation of the problem, the most remarkable feature of the JMulicher–Prost model (7.28), (7.29) is that only the di5erence V1 (x) − V2 (x) of the two potentials counts (one may thus choose one of them identically zero without loss of generality). It follows that the emerging qualitative results for a generic (L-periodic and asymmetric) choice of V1 (x) − V2 (x) will be valid independently of whether the mechanochemical coupling is loose (e.g. a dichotomously =uctuating potential ratchet with V2 (x) ˙ V1 (x), see Section 4.3) or tight (e.g. a traveling two-state ratchet with V2 (x) = V1 (x + L=2), see Section 4.4.2). Whether this feature should be considered as a virtue (robustness) or shortcoming (oversimpliFcation) of the model is not clear. ModiFed Huxley–JMulicher–Prost type models have been explored by Vilfan et al. [627,628]. Their basis is a M =2 state description of the single motors with a built-in tight mechanochemical coupling through the choice of the rates and potentials, but, at variance with JMulicher and Prost, without a completely rigid shape of the motors with respect to the backbone: unlike in (7.26), the center of mass of an individual motor may di5er from the position where it is rooted in the backbone, say xt. ˙ Similarly as in Huxley’s original work, the possibility of “strain”-dependent (i.e. x − xt-dependent) ˙ rates km → m (x) plays an important role. With a rigid backbone, a mean Feld approach is still exact for N → ∞ but technically more involved than in the JMulicher–Prost model, while resulting in qualitative similar collective phenomena [628]. In contrast, by admitting an elastic instead of a rigid backbone, 118 the interaction between the motors is no longer of inFnite range and corrections to a mean Feld approximation may become relevant under certain experimental conditions [627]. Another reFned version of the Huxley–JMulicher–Prost setup, taking into account an extended number of biological Fndings, is due to Derenyi and Vicsek [630]. While M = 4 chemical states are included, only two di5erent potential shapes Vm (x) are proposed, one of them being identically zero, and a tight mechanochemical coupling is built in through the choice of the rates km → m (x). While a very good agreement with di5erent experimentally measured curves is obtained, the issue of genuine collective phenomena is not speciFcally addressed. Further studies of collective e5ects in coupled Brownian motors will be discussed in Section 9.

118

A computer animation (Java applet) which graphically visualizes the e5ect is available on the internet under [629].

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7.5. Coordinated two-head model In this subsection we return to the description of a single motor enzyme with two heads within the general modeling framework (7.1) – (7.17). Especially, we recall that this model respects an invariance under a spatial displacement by one period if simultaneously the chemical states of the two heads are exchanged, see (7.5), (7.6). We furthermore recall that for a processive motor enzyme, i.e., one which can operate individually (for instance kinesin), the two heads need to coordinate their actions in order that at least one of them is always attached to the polymer Flament. Our goal is to approximately boil down the two-dimensional chemical state vector m = (m1 ; m2 ) into an e5ective one-dimensional (scalar) description. To this end, we make the assumption that the two heads are so strongly coordinated that between subsequent steps there exists a time instant at which not only both heads are attached to the Flament, but on top of that, the heads are in the same chemical state, m1 = m2 . Taking such a conFguration as reference state, one of the two heads will be the Frst to make a chemical transition into another state. This may be, with certain, generically unequal probabilities, either the front or the rear head, and the chemical state may, again with typically unequal probabilities (cf. (7.16), (7.17)), either go one step forward or backward in its reaction cycle as time goes on. 119 Our central assumption is now, that once one of the heads has left the reference state m1 = m2 , the other head will not change its chemical state until the Frst one has returned into the reference state. We are not aware of experimental observations which indicate that such a property is strictly fulFlled, but it appears to be an acceptable approximation, especially in view of the great simpliFcation of the model it entails. Moreover, if one starts with a reduced description of the chemical cycle in each head in terms of only two e5ective states, based on a similar line of reasoning as in the preceding Section 7.4.2, then necessarily one of these two states must correspond to the head being attached to the polymer Flament and the other to the detached situation. Since both heads cannot be detached simultaneously, our assumption is thus automatically fulFlled in such a two-state description for each head. If one makes the additional simplifying assumption that, starting from the reference state with both heads attached, only the rear head is allowed to detach, then an e5ective one-dimensional description of the chemical states of the two heads is straightforward: after the rear head has returned into the reference state, it either will have attached at the same binding site (E-tubulin) from which it started out or it will have advanced to the next free binding site at a distance 2L. In the former case, it is again the same head which will make the next chemical reaction out of the reference state, while the other head continues to be stuck. In the latter case, the rear head has completed a step, 120 x → x + L, and is now the new front head. If we additionally exchange the chemical labels m1 and m2 of the two heads then we are back in the original situation due to the symmetry of system (7.5), (7.6). In other words, we have obtained an e5ective description in which only one of the chemical state variables, say m1 , can change, while m2 is stuck all the time. Dropping the index of m1 , one readily sees that the e5ective potentials Vm (x) and rates km → m (x) in this new description are now indeed L-periodic in x and satisfy (7.25). 119

There seems to be no general agreement upon whether such inverse processes are possible with Fnite (however small) probability [18] or not [143]. 120 We recall that the stepping head itself advances by 2L, but the center of mass x of the two heads only by L.

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The more general case that out of the reference state m1 =m2 both, the front and the rear head may detach from the Flament with certain probabilities, can only be captured approximately by means of an e5ective one-dimensional chemical state variable: 121 Namely, one has to assume that if the rear head detaches, then x can only take values larger than the initial reference position xref (but smaller than xref + L). Likewise, if the front head detaches, x is restricted to [xref − L; xref ]. These two possibilities can be imitated by “splitting probabilities” with which x(t) in (7.3) will evolve into the positive or negative direction after detachment by way of an appropriate choice of the potentials Vm (x). Especially, these potentials have to be chosen such that a recrossing of xref after detachment is practically impossible. 122 For the rest, the mapping to an e5ective one-dimensional chemical state variable m with L-periodic potentials Vm (x) and rates km → m (x) satisfying (7.25) can be accomplished exactly like before. Two noteworthy features which can be described within such a general modeling framework are thus (i) the possibility not only of “forward” but also of “backward” steps and (ii) the possibility that a head re-attaches to the same binding site from where it started out. Both these possibilities may be realized only with a small probability 123 under “normal” conditions [582,603,631] but could become increasingly important [18,504,582,632,633] as the load force F in (7.3) approaches the “stopping force” or “stall load”, characterized by zero net motion x ˙ = 0 (cf. Section 2.6.2). 7.6. Further models for a single motor enzyme The above interpretation of (7.8), (7.18) – (7.25) as a model for a single motor enzyme with two highly cooperative heads has, to our knowledge, not been pointed out and derived in detail before. 124 However, practically the same model dynamics (7.3) has been used to describe the somewhat artiFcial scenario 125 of a single head moving along a polymer Flament [9,10]. By changing the interpretation, such results can immediately be translated into our two-head setting. As mentioned in Section 7.4.3, models with two completely independently operating heads, except that they are connected by a “hinge”, have been brie=y addressed numerically in [10]. A reFned model of this type has later been put forward and analyzed by Vilfan et al. [628] exhibiting good agreement with a variety of experimental curves and structural results.

121 The problem is that now the information about which of the two heads is chemically active (detached) must be uniquely encapsulated in x in addition to the position of the center of mass. 122 To be speciFc, we may model the chemical reference vector-state mref with both heads attached by a potential Vmref (x) with a very deep and narrow minimum at xref . If mref goes over into one of the “neighboring” states, say m , then Vm (x) should have pronounced maxima on either side of xref such that x(t) will proceed rather quickly and irrevocably away from xref , either to the right or left. The actual direction into which x(t) disappears decides a posteriori whether it was the front or the rear head which has detached. 123 We remark that there are also models which rule out such backward steps a priori [143]. 124 Somewhat similar ideas can be found in [18,544,552,596]. 125 Whether or not manipulated, single-headed kinesin can travel over appreciable distances on a microtubulus seems to be still controversial [18,143,564,633– 637]. Remarkably, the experimental data from [637] could be Ftted very well by an on–o5 ratchet model (A video illustrating motility data can be viewed on the internet under [638].). There seems to be evidence [572] that single-headed motion is fundamentally di5erent from two-headed motion.

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Valuable contribution to the general conceptual framework [11] of single motor modeling and especially of the mechanochemical coupling [12] are due to Magnasco, see also [186]. At variance with our present setup, the chemical processes within the entire motor enzyme are described from the beginning by a single, continuous chemical state variable [12] (see also Sections 4.6 and 7.3.2). Published practically at the same time, models similar in spirit, but with only two discrete chemical states have been proposed by Astumian and Bier [15], by Prost et al. [13], and by Peskin et al. [17]. The underlying picture is that, essentially, the motor enzyme as a whole is either “attached” to or “detached” from the protein Flament. The emphasis in all these works [11–13,15,17] (see also [639]) is put on the fundamental aspects and generic properties of motion generation in such systems; apart from the general features of spatial periodicity and broken symmetry, no contact with any further biological “details” is established. Yet, by using reasonable parameter values in a =uctuating potential saw-tooth ratchet model, measured data for the average speed x ˙ and the rate of ATP-consumption (cf. (7.23) and (7.24)) could be reproduced within an order of magnitude [15,16]. On the other hand, it was demonstrated in [515,516] that even within the simplest two-state models for a single motor (M = 2 in (7.18)), a large variety of even qualitatively contrasting results can be produced upon varying the model parameters. Not only a realistic choice of the model parameters but also of the details of the model itself is therefore indispensable. A biologically well founded description of a motor enzyme with two cooperating heads, similar to our present setup with M = 2 chemical states, has been introduced by Peskin and Oster [18]. A central point in this study is once more the relative roles of the thermal =uctuations and the relaxational processes due to release of chemically generated strain in the dynamics of the spatial coordinate x in (7.3). Another important feature of the model is that, besides regular forward steps of the heads, also backward steps after detachment of the front head are admitted with a certain probability. The result, after Ftting the model parameters to the experiment, is that—within this speciFc model—thermal =uctuations play a minor role. Furthermore, it is found that backward steps are about 20 times less probable than forward steps. 126 The model by Derenyi and Vicsek [504] is to some extent similar in spirit to the one by Peskin and Oster. Especially, backward steps are admitted and the two heads act highly cooperatively. The built-in mechanochemical coupling is a compromise in that thermal activation is indispensable but the rates km → m (x) are tailored such that the next chemical step can only occur after x has crossed the respective barrier and is basically undergoing a purely mechanical relaxation. The model can be mapped almost exactly to a M = 2 state model from Section 7.2.3, though the original formulation [504] in terms of two rigid heads, coupled by a hinge and an “active” spring with variable rest length is admittedly more natural in this speciFc instance. In either case, the model can be described in very good approximation by an augmented reaction cycle with mechanical states properly added to the chemical states. The distinguishing feature of the model, the experimental justiFcation of which remains unclear [563,564,640,641] is that the two heads cannot pass each other: The distance between the front and the rear head (in other words, of the spring) can change but never become zero so that the heads never exchange their roles. The virtue of the model is its ability to Ft very well various measured curves. The limiting case of a very strong “active” spring, such that thermal activation is no longer important, has been explored in [642]. A somewhat related model with two e5ectively asymmetric heads is due to [445], see also Section 6.5. 126

See also the discussion at the end of the previous subsection.

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The conceptual framework [15] of Astumian and Bier for modeling single molecular motors has been further developed and reFned in a remarkable series of works [16,54,55,187,349,553,643,644]. Various aspects and results of their central study [16] have been repeatedly referred to already in the outline of our general modeling framework. The chief points in [16] are a comprehensive discussion of the mechanochemical coupling problem and the conclusion that many experimental indications and theoretical arguments seem to be compatible with a rather loose coupling, especially when a suitably augmented cooperative two-head model is invoked [643,644]. A complementary discussion along a closely related spirit is given in [639]. Especially worth mentioning is that the =uctuational analysis of measured single motor protein trajectories in [604] is incompatible with a certain class of very simple (=uctuating potential) ratchet models but have been demonstrated in [643,644] to be perfectly reproducible by means of a more elaborated and reFned description. Non-cooperative discrete-state models with a built-in tight mechanochemical coupling in the spirit of [8,9,164,514] (see at the end of Section 7.4.3) have been addressed in [186,515 –520], especially with respect to their behavior under the in=uence of an external load F. Notwithstanding the conclusion in [8,9] that at least four states are necessary for a realistic model, the agreement of the two- and three-state models proposed in [515,516,520] and [519], respectively, with experimental observations is quite good. Various generalizations of these “mechanochemical reaction cycle models” (cf. Sections 7.4.3 and 7.7) are due to [517], while the extension to general waiting time distributions has been addressed in [515,516,518], admitting in addition to thermally activated mechanochemical rate processes e.g. also the description of mechanical relaxation of strain. Their drawback is a large number of additional phenomenological model parameters. The viewpoint [13] of Prost and collaborators with respect to modeling single motor enzymes has been further elaborated in [14,614] and especially in [544,594 –596]. While the general framework has much in common with that of Astumian and Bier, these workers put special emphasis on the possible relevance of “active sites”, i.e. a pronounced dependence of the transition rates km → m (x) on the mechanical state x, such that transitions are practically excluded outside of certain small x-regions. They furthermore leave room to the possibility that a traveling potential ratchet mechanism may dominate over a possibly coexisting =uctuating potential ratchet mechanism, in which case the mechanochemical coupling might be rather tight. An explicit modeling of cooperative two-headed motor enzymes along somewhat similar lines as in Section 7.5 is brie=y mentioned in [544,596]. The resulting description with M = 2 e5ective chemical states associates each state with one of the heads being bound to the Flament and the other detached. One thus recovers the traveling potential ratchet model from [40], advancing in discrete steps of L=2 as detailed in Section 4.4.2. The in=uence of an external load F on velocity and processivity (detachment rate of the molecular motor from the microtubulus) has been addressed in [594], see also [517]. A related study due to [595] suggests a loose mechanochemical coupling at least under heavy load. The case that the load is not an externally imposed constant force but rather is due to the “cargo”, modeled as additional relevant dynamical variable that interacts with the motor via an elastic coupling, has been addressed in [505,506], see also below Eq. (7.4). A detailed analysis of a somewhat extended model class with pronounced “active sites” and a strong traveling potential component has been carried out in [552,593,645]. In agreement with experiments, these models reproduce a “saturation” of the current x ˙ as a function of the ATP concentration [13,14,614], captured by a Michaelis–Menten relation for a large class of moderately and strongly cooperative models under zero lead F [552,593,645], while for Fnite load a somewhat

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modiFed quantitative behavior is expected [552,593,595,641]. We remark that while for cooperative two-head models with only M = 2 chemical (or “internal”) states per head, the assumption of “active sites” is indeed indispensable for such a saturation of the current, the same is no longer true as soon as M ¿2. 7.7. Summary and discussion We close with some general remarks regarding the modeling of molecular motors as reviewed in this chapter. Previously introduced notions and facts are freely used without explaining them or citing the original literature again. To some extent, this discussion continues and makes more precise those from Sections 4.6 and 7.4.3. The general importance of asymmetry induced rectiFcation, thermal =uctuations, and the coupling of non-equilibrium enzymatic reactions to mechanical currents according to Curie’s principle for intracellular transport processes is long known [23,24]. The present framework has the virtue that it is based on a quantitative microscopic modeling and as such is not restricted to the linear response regime close to thermal equilibrium. Within this general framework, roughly speaking two approaches of modeling molecular motors may be distinguished: The Frst, “traditional” one is a bottom-up-type strategy, starting with a certain set of biological facts (measurements and more or less “basic” conclusions therefrom) and then constructing an “ad hoc” model on this basis. The second is the top-down-type approach, followed to some extent in more recent works based on the “ratchet paradigm” and elaborated in full detail in our present chapter. Our Frst main conclusion is that all models known to this author are compatible (possibly after some mapping or transformation of state variables) with the basic framework from Section 7.2, and most of them also with the simpliFed description in terms of a single mechanical state variable x and the corresponding model dynamics (7.3), identiFed below (7.14) as a (generalized) pulsating ratchet scheme. In other words, such an approach is not in contradiction with “traditional” biological models, but may well o5er a fresh and more systematic (top-down) view of things [12,186,571,643,644,646]. Within this still very general class of models (7.3) the most realistic choice of model parameters and model functions is still under debate and certainly also depends on the speciFc type of molecular motor under consideration (especially whether it is of processive (individually acting) or non-processive (collectively acting) nature or even consists of a single motor domain (head) only). Conversely, it is remarkable that all these di5erent species can be treated within one general framework. Three basic questions in this respect, which are not always suCciently clearly separated from each other, regard: (i) The possibility of an (approximate) description in terms of a single (e1ective) chemical state variable. (ii) The relative importance of the thermal 9uctuations appearing in the dynamics of the mechanical coordinate (7.3) as compared to conformational changes powered by the chemical cycle. (iii) The character (loose or tight) of the mechanochemical coupling. The answer to these questions may not only depend on the type of molecular motor under consideration (see above) but also on whether an external load F is acting and possibly on still other external conditions. For example, it may well be that, as the load F increases, the relative importance in (7.3) of thermal activated barrier crossing and deterministic relaxation processes (i.e. the answer to question (ii) above) considerably changes. The reason is that, as the force F increases, existing

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e5ective potential barriers in the stochastic dynamics (7.3) may disappear and new ones appear. 127 Similarly, the external load F may also signiFcantly change the character of the mechanochemical coupling 128 [16,504,519,552,582,593–595,632,647,648] (i.e. the answer to question (iii) above). The answer to the Frst of the above questions depends on the cooperativity of the two heads: If they act completely independently of each other, they can obviously be described individually, and a single (scalar) chemical state variable m for each head is then suCcient. That the same may be possible for a very strong coordination of the heads has been demonstrated in Section 7.5. On the other hand, if the cooperativity is loose but non-negligible, then a reduction of the two-dimensional chemical state space (cf. (7.1)) is impossible. The second question is sometimes also discussed under the label of power-stroke versus motordi5usion modeling strategy [143,519,648]. In the Frst case, the chemical cycle “slaves” the mechanical cycle by creating a sequence of strong mechanical strains (power strokes) that are released by concomitant, basically deterministic changes of the mechanical state (geometrical shape). Typically, there is little back reaction of the mechanical to the chemical coordinate, and we are thus essentially dealing with a genuine traveling potential ratchet scheme. In the second case, thermal =uctuations play a major role in the dynamics of the mechanical state variable (7.3). The Frst model of this type goes once again back to Huxley [4] and the apparent lack of strong experimental support for the power stroke concept [648] has served as a motivation for various other such models ever since. Especially, this controversy has a long history already within the realm of “traditional” biological modeling and the gain of new insight in this respect from an approach based on the “ratchet paradigm” may be limited. Also, we may emphasize once more that in either case thermal noise plays a crucial role with respect to the chemical reaction cycle—in this sense any model of a molecular motor (not only those of the motor-di5usion type) “rectiFes” thermal =uctuations. We further remark that also within a motor-di5usion modeling, the mechanochemical coupling may still be either tight (e.g. Huxley’s model) or loose (e.g. the on–o5 ratchet). On the other hand, a power-stroke model always implies a tight mechanochemical coupling. Another related question within a motor-di5usion modeling is whether the thermal =uctuations acting on the mechanical state variable can be treated within the activated barrier crossing limit (see Section 3.8) or whether free di5usion-like behavior plays a signiFcant role. Only in the former case, a description of the mechanical state variable in terms of discrete states and transition rates between them is admissible, see Section 6.7 and [515,516,518,520]. Note that both options are still compatible with either a tight or a loose mechanochemical coupling. In the case of a tight coupling in combination with an activated barrier crossing description, a so-called “mechanochemical reaction cycle” arises (cf. Section 7.4.3). Since from a fundamental viewpoint, the distinction between chemical and mechanical state variables is somewhat arbitrary anyway (see Section 7.2.2), we are then basically recovering an e5ective power-stroke model. We Fnally come to the question (iii) of the mechanochemical coupling (see also Section 2.7). We Frst remark that a tight coupling not necessarily means that the chemical state variable always “slaves” the mechanical one (genuine power-stroke model) but that one variable “slaves” the other at each stage of the mechanochemical reaction cycle (the chemical reaction may be blocked—due to active sites, i.e. strongly x-dependent rates km → m (x)—until some mechanical transition between 127 128

The “total” or “e5ective” potential in (7.3) is given by Vm (x) − xF. Especially, F may change the shape of the potentials Vm (x), as discussed below Eq. (7.4).

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di5erent geometrical shapes of the motor has been accomplished, e.g. in the above e5ective powerstoke model). Restricting ourselves to the simplest case of a one-dimensional chemical state variable, a tight mechanochemical coupling means that a description in terms of a single e1ective state variable is possible, and a loose coupling means that such a description is impossible. In other words, the state space is either essentially one- or two-dimensional. In the Frst case, there is a unique “pathway” in the x–m-space, in the second case bifurcations exist. Examples are genuine traveling potential ratchets and =uctuating potential ratchets, respectively. We, however, emphasize that the conclusion suggested by the latter example, namely that a loose mechanochemical coupling implies that thermal =uctuations play an essential role in the dynamics of the mechanical coordinate (7.3), can be easily demonstrated as incorrect by counterexamples. In other words, the thermally induced randomness of the chemical reactions suCces to produce bifurcations in the “pathway” through the full x–m-space. The possibility of a loose mechanochemical coupling is widely considered as one of the main conceptually new aspect of the “ratchet paradigm” as compared to “traditional” biological models. 129 However, in its simplest and most pronounced form, namely the =uctuating potential ratchet scheme from Section 4.3 (i.e. with x-independent rates km → m ) it is apparently incompatible with the =uctuational analysis of single (two-headed) motor protein trajectories [604]. On the other hand, the experimental data for single-headed kinesin from [637] could be Ftted very well to an on–o5 ratchet model. The currently prevailing opinion seems to be that a loose coupling is unlikely for processive motors like two-headed kinesin but a realistic option in the case of non-processive (cooperative or single-headed) motors [519,563,564,600,601,603,604,613,631,637,641,648– 651]. However, room for the possibility of a loose coupling even in the case of kinesin is still left e.g. in [16,582,632,652]. If one considers the concept of a loose mechanochemical coupling as the only substantial new contribution of the “ratchet paradigm” to the modeling of molecular motors, then—in the so far absence of striking experimental indications of such a coupling—the merits of this paradigm may still be considered as questionable. However, such a viewpoint may not do due justice to other noteworthy achievements like the prediction of new collective e5ects from Section 7.4.4 or the uniFed new view and working model. 8. Quantum ratchets For many of the so far discussed ratchet systems, especially those for which thermal =uctuations play any signiFcant role, the characteristic length-, energy-, etc., scales are very small and it is thus just one more natural step forward to also take into account quantum mechanical e5ects. Before we enter the actual discussion of such e5ects, two remarks are in place: Frst, we have encountered in Sections 5.7.3 and 5.10 theoretical models and experimental realizations of Josephson and SQUID ratchet systems. Since the basic state variables in such devices are phases of macroscopic quantum mechanical wave functions, it is tempting to classify them as quantum ratchet systems. Our present viewpoint, however, is that the decisive criterion should be the classical or quantum mechanical character of the e5ective dynamics governing the relevant state variables of a system, independently of whether the microscopic basis of this e5ective dynamics is of classical or quantum 129

Sometimes, also the possibility of a motor-di5usion modeling approach is considered as such.

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mechanical nature, see also [94]. For instance, the existence of stable atoms, molecules, and solids is clearly a genuine quantum mechanical phenomenon, yet a classical theory of gases, liquids, and solids can be established. From this viewpoint, the Josephson and SQUID ratchet systems as discussed in Sections 5.7.3 and 5.10 are thus classical ratchets. The realization of a full-=edged quantum mechanical ratchet according to our present understanding in SQUID systems will be discussed later in Section 8.4. As a second remark we mention that the proper quantum mechanical treatment, e.g. of the Seebeck e5ect (Section 6.1) or the photovoltaic e5ects (Section 5.6), may arguably be considered as very early quantum ratchet studies of considerable practical relevance. However, in the present section we put our main emphasis not on a faithful quantum mechanical modeling of such speciFc systems but rather on the exploration of the basic features of much simpler models. Namely, our main focus will be on the interplay between tunneling and the e1ects induced by the thermal environment (i.e. dissipation and thermal noise) in the quantum mechanical counterparts of the classical tilting ratchet dynamics (5.1). 8.1. Model In the case of classical Brownian motion, we have introduced in Section 2.1.2 a model which takes into account the in=uence of the thermal environment along a rather heuristic line of reasoning, see also Sections 2.9 and 3.4.1. In contrast, on a quantum mechanical level, such a heuristic modeling of dissipation and thermal noise, e.g. on the level of the SchrModinger equation, is much more problematic and liable to subtle inconsistencies for instance with the second law of thermodynamics or some basic principles of quantum mechanics, see [100,243,653– 658] and further references therein. To avoid such problems, we follow here the common route [66,84,94 –96,189, 659 – 662] to describe both the system and its thermal environment within a common Hamiltonian framework, with the heat bath being modeled by an inFnite set of harmonic oscillators. Especially, within a quantum mechanical approach, keeping a Fnite mass of the system is unavoidable, i.e. a quantum ratchet is by nature endowed with 8nite inertia. If one insists in considering the overdamped limit m → 0 then this limit usually has to be postponed to the very end of the calculations. Similarly as in Section 3.4.1, our starting point is a one-dimensional quantum particle with mass m in an asymmetric, periodic ratchet potential V (x) of period L in the presence of a tilting force Feld y(t) that is unbiased on average. This bare system is furthermore coupled via coupling strengths cj to a model heat bath of in8nitely many harmonic oscillators with masses mj and frequencies !j (!j ¿0 without loss of generality) yielding the compound (system-plus-environment) Hamiltonian p2 + V (x) − xy(t) + HB ; 2m 2 ∞ pj2 x c 1 j HB := + mj !j2 xj − : 2m 2 mj !j2 j j=1

H(t) =

(8.1) (8.2)

Here, x and p are the one-dimensional coordinate and momentum operators of the quantum Brownian particle of interest, while xj and pj are those of the bath oscillators. As initial condition at time t = 0 we assume that the bath is at thermal equilibrium and is decoupled from the system. The inFnite number of oscillators guarantees an inFnite heat capacity and thus a reasonable model of a heat bath

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that keeps its initial temperatures for all later times 130 t¿0. For the rest, it turns out [66,95,96, 659 – 662] that the e5ect of the environment on the system is completely Fxed by the frequencies !j and the ratios cj2 =mj , or equivalently, by the so-called spectral density J (!) :=

∞ cj2 (! − !j ) : 2 j=1 mj !j

(8.3)

By way of integrating out the bath degrees of freedom in (8.1) one obtains [66,95,96,659 – 662] the following one-dimensional Heisenberg equation for the position operator x(t): t M + V (x(t)) − y(t) = − ˙ ) dt + (t) : mx(t)

(t ˆ − t )x(t (8.4) 0

Like in (2.1), the left-hand side can be associated to the bare system dynamics, while the right-hand side accounts for the in=uence of the environment through the damping kernel 2 ∞ d! !−1 J (!) cos(!t) (8.5)

(t) ˆ := 0 and the operator valued quantum noise ∞ pj (0) cj x(0) (t) = cos(!j t) ; cj sin(!j t) + xj (0) − mj !j mj !j2 j=1

(8.6)

containing the initial conditions of the bath and of the particle’s position. Exploiting the assumed thermal distribution of the bath HB at t = 0 one Fnds [66,95,96,659 – 662] that (t) becomes a stationary Gaussian noise with mean value zero. Moreover, one recovers the usual connection (via J (!)) between the random and the friction e5ects of the bath on the right-hand side of (8.4) in the form of the =uctuation–dissipation relation ˝ ∞ ˝! (t + &)(t) = cos(!&) − i sin(!&) ; (8.7) d! J (!) coth 0 2kB T √ where · indicates the thermal average (quantum statistical mechanical expectation value), i := −1, and & ¿ 0. In the following, we will focus on a so-called ohmic bath, characterized by a linear initial growth of the spectral sensity J (!), a “cuto5” frequency !c , and a “coupling parameter” : J (!) = ! exp{−!=!c } :

(8.8)

The cuto5 !c is introduced in order to avoid unphysical ultraviolet divergences but will always be chosen much larger than any other relevant characteristic frequency of the model. The special role of such an Ohmic heat bath becomes apparent by observing that the corresponding damping 130

Further shortcomings of a heat bath with a Fnite number of oscillators are: (i) Both the memory kernel (8.5) and the noise-correlation (8.7) do not decay to zero for large times, rather they are quasi-periodic. (ii) The future behavior of the “noise” (8.6) becomes predictable from its past, at least in the classical limit, see Section 11-5 in [663].

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kernel (8.5) approaches

(t) ˆ = 2 (t) ;

(8.9)

when the cuto5 !c goes to inFnity. The integral in (8.4) thus boils down to the memory-less viscous ˙ friction − x(t). In other words, in (8.8) has the meaning of a damping coeCcient due to viscous friction. In the classical limit, i.e. for ˝=kB T much smaller than any other characteristic time scale of the noiseless system (8.4), the correlation (8.7) with (8.5) correctly approaches the classical =uctuation– dissipation relation from (6.23). Furthermore, in this limit all quantum =uctuations vanish, so that q-numbers go over into c-numbers and (8.4) reproduces (for y(t) ≡ 0) the classical model (6.22) of a real valued stochastic process x(t) in the presence of Gaussian noise (t) and (2.1) in the special case of a memoryless damping (8.9), see also Section 3.4.1. For later purposes, it is useful to distinguish between two di5erent variants of the classical limit: The Frst one, which we call formal classical limit, consists in letting ˝ → 0, i.e. quantum e1ects are simply ignored within such a description, independent of how relevant they are in the true system under study. This limit is formal in so far as in reality ˝ is a natural constant. A second possibility, which we call physical classical limit, consists in focusing on large temperatures T such that ˝=kB T is suCciently small and thus quantum e1ects become indeed negligible in the real system. As suggested by the above mentioned Fndings in the classical limit, the harmonic oscillator model for the thermal environment (8.2), (8.3), (8.8), provides a rather satisfactory description in a large variety of real situations [66,94 –96,660,661,664,665], even though for many complex systems, one does not have a very clear understanding of the actual microscopic origin of the damping and =uctuation e5ects. In fact, it seems to be widely believed that once the dissipation mechanism is known to be of the general form appearing on the right-hand side of Eq. (8.4), i.e. to be a linear functional of the system velocity, then for a heat bath at thermal equilibrium all the statistical properties of the quantum noise (t) in (8.4) are uniquely Fxed, i.e. independent of any further microscopic details of the thermal heat bath. Arguments in favor of this conjecture have been given, e.g. in [81,94,95,189,666], but a veritable proof does not seem to exist yet, see also Sections 2.1.2, 3.4.1 and [80,92,93,97] for the classical limit. Under the assumption that the conjecture holds, it can be inferred [95] that any dissipative dynamics of the form (8.4) which is in contact to an equilibrium heat bath can be represented by a harmonic oscillator model (8.1), (8.2). This does not mean that in every such physical system the actual bath is a harmonic oscillator bath, but only that one cannot tell the di5erence as far as the behavior of the system x(t) is concerned [95]. We Fnally remark that the damping kernel (8.5) does not change in the classical limit, it is the same for both a quantum mechanical or classical treatment of the system dynamics. In other words, the knowledge of the dissipation term in the classical limit appears to be suCcient to completely Fx the quantum mechanical stochastic dynamics. Historically, the harmonic oscillator model has apparently been invoked for the Frst time by Einstein and Hopf [78] for the description of an oscillating electrical dipole under blackbody irradiation and subjected to radiation damping. 131 A classical model with a harmonic oscillator potential 131

A preliminary toy-model, somewhat related to the problem considered by Einstein and Hopf is due to Lamb [667]. It can be mapped onto a harmonic oscillator model [95] but does not involve =uctuations of any kind. The same proviso applies for further related early works, like e.g. [668,669].

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3

2

U (x)

V ( x ) / V0

1

V (x)

0

-1

+ U (x)

-2

-3 -1

-0.5

0

0.5

1

1.5

2

x/L

Fig. 8.1. Solid: The ratchet potential V (x) = V0 [sin(2 x=L) − 0:22 sin(4 x=L)]. Note that this potential is almost identical to the spatially inverted potential from (2.3), see also Fig. 2.2. Dashed and dotted: “tilted washboard potentials” U ± (x) in (8.10) with Fl = 0:1V0 ; l = L=2 .

V (x), but otherwise exactly like in (8.1) – (8.3) has been put forward by Bogolyubov [670], however, without explicitly working out the statistical properties of the =uctuations (t), especially their Gaussian character and the classical counterpart (6.22) of the =uctuation–dissipation relation (8.5), (8.7). The latter issues, together with a quantum mechanical transcription of the model, has been accomplished by Magalinskii [84]. Subsequent re-inventions, reFnements, and generalizations of the model have been worked out, e.g. in [85,88,89,94,95,189,659]. 8.2. Adiabatically tilting quantum ratchet For general driving y(t), Eq. (8.4) gives rise to a very complicated non-equilibrium quantum dynamics. To simplify matters [161,671,672], we restrict ourselves to very slowly varying tilting forces y(t) such that the system can always adiabatically adjust to the instantaneous thermal equilibrium state (accompanying equilibrium). We furthermore assume that y(t) is basically restricted to the values ±F, i.e. the transitions between ±F occur on a time scale of negligible duration in comparison with the time the particle in (8.4) is exposed to either of the “tilted washboard” potentials U ± (x) := V (x) ∓ Fx ;

(8.10) U ± (x)

still cf. Fig. 8.1. As a Fnal assumption we require a positive but not too large F, such that display a local maximum and minimum within each period L. Apart from this, the tilting force y(t) may still be either of stochastic or of deterministic nature. Within the so deFned model, we are essentially left with six model parameters, 132 namely the particle mass m, the “potential parameters” V0 , L, and F (see Fig. 8.1), and the “thermal environment 132

Throughout this section the cuto5 !c in (8.8) is chosen much larger than any other characteristic frequency of the system and therefore does not appear any more in the following.

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parameters” and T . We now make the assumption that these parameters are chosen such that a classical particle which starts at rest close to any local maximum of U ± (x) will deterministically slide down the corresponding slope but will not be able to subsequently surmount any further potential barrier and so is bound to end in the next local minimum. Di5erently speaking, a moderate-to-strong friction dynamics is considered and deterministically “running solutions” are excluded. We further assume weak thermal noise, that is, any potential barrier appearing in (8.10) is much larger than the thermal energy, i.e. UU min kB T ;

(8.11)

where UU min denotes the smallest of those potential barriers. As a consequence, we are dealing with a barrier crossing problem (see Section 3.8) and thus the average particle current in either of the two potentials U ± (x) can be expressed in terms of two rates according to (3.55), see also (5.6). Moreover, the assumption of rare jumps of y(t) between the two values ±F makes it possible to express the net current by way of an adiabatic limit argument analogous to (5.2), (5.9) in terms of these two partial currents. In this way, one Fnally arrives at the following expression for the averaged net particle current in terms of two rates: L x ˙ = (1 − e−FL=kB T ) (kr+ − kl− ) : 2

(8.12)

Here, kr+ indicates the escape rate from one local minimum of U + (x) to its neighboring local minimum to the right, and similarly kl− denotes the rate to the left in the potential U − (x). We also recall that the average on the left-hand side of (8.12) indicates a thermal averaging (quantum statistical mechanical expectation value) together with an averaging over the driving y(t). Within a purely classical treatment of the problem, i.e. within the formal classical limit ˝ → 0, any of the two rates k in (8.12) describe thermally activated transitions “over” a certain potential barrier UU between neighboring local minima of the corresponding potential. Due to the weak noise condition (8.11), such a rate k is given in very good approximation by the well-known Kramers-rate expression [66]

U (x0 ) −UU=kB T k= e ; (8.13) 2 |U (xb )|

2 + 4m|U (xb )| −

; (8.14) := 2m where xb and x0 denote the above mentioned local potential-maximum and -minimum, respectively, and where indices r, l, and ± have been dropped. Note that in the overdamped limit m → 0, the Kramers–Smoluchowski rate-expression from (2.45) is recovered. Turning to a quantum mechanical treatment of the problem, the rates in (8.12) in addition have to account for quantum tunneling “through” the potential barriers. Especially, due to our assumption that moderate-to-strong friction is acting, the tunneling dynamics is incoherent and a quantum rate description of the current (8.12) is valid. To evaluate these rates, a sophisticated line of reasoning has been elaborated [66,96,660]. Starting with the Hamiltonian system-plus-reservoir model (8.1) and adopting the so-called “imaginary free energy method” [66,673] or, equivalently, the

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211

“multidimensional quantum transition state theory” [66,674,675], it is possible to express the escape rate k in terms of functional path integrals. After integration over the bath modes and a steepest descent approximation, one obtains the semiclassical form k = Ae−S=˝ :

(8.15)

Here, the exponentially dominating contribution S is deFned via the non-local action ∞ ˝=kB T ) 2

q(&) − q(& mq˙2 (&) + U (q(&)) + Sb [q] := : d& d& 2 4 & − & 0 −∞

(8.16)

This action has to be extremized for paths q(&) under the constraints that q(& + ˝=kB T ) = q(&) for all &, and that there exists a &-value such that q(&) = xb . A trivial such extremizing q(&) is always q(&) ≡ xb . Among this and the possibly existing further extrema one selects the one that minimizes Sb [q], say qb (&), to obtain S := Sb [qb ] − ˝EU (x0 ) :

(8.17)

The pre-exponential factor A in (8.15) accounts for =uctuations about the semiclassically dominating path qb (&). For a numerical exempliFcation [161,671,672] we use T as control parameter and Fx the Fve remaining model parameters m, , V0 , F, and l := L=2 . Without specifying a particular unit system this can be achieved by prescribing the following Fve dimensionless numbers: First we Fx V0 , F, l and thus U ± (x) through Fl=V0 = 0:2, UU min =V0 = 1:423, and |U + (xb )| l2 =V0 = 1:330 corresponding to the situation depicted in Fig. 8.1. Next we choose =mJ0 = 1 with J0 := [V0 =l2 m]1=2 , meaning a moderate damping as compared to inertia e5ects. To see this we notice that J0 approximates rather well the true ground state frequency !0+ := [U + (x0 )=m]1=2 in the potential U + (x), !0+ = 1:153 J0 , and similarly for U − (x). In particular, =mJ0 =1 strongly forbids deterministically running solutions. In order to specify our last dimensionless number we remark that within the weak noise assumption (8.11) it can be shown [66] that in the potential U + (x) genuine quantum tunneling events “through” the potential barrier are rare above a so-called crossover temperature Tc+ =

˝+ ; 2 kB

(8.18)

while for T ¡Tc+ tunneling yields the dominant contribution to the transition rates. An analogous crossover temperature Tc− arises for the potential U − (x) which is typically not identical but rather close to Tc+ . With the deFnitions Tcmax = max{Tc+ ; Tc− };

Tcmin = min{Tc+ ; Tc− } ;

(8.19)

we now Fx our last dimensionless quantity through UU min =kB Tcmax = 10. In this way, the weak noise condition (8.11) is safely fulFlled for T 6 2Tcmax , i.e. up to temperatures well above both Tc+ and Tc− . At the same time, the so-called semiclassical condition [66] UU min kB Tcmax ;

(8.20)

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10 10

〈x〉 /Ω 0 L

10

.

10 10 10 10 10 10

-4

-5

.

+ 〈x〉qm

-6

-7

. _ 〈x〉

qm

-8

.

+ 〈x〉cl

-9

-10

-11

T

max c

/ T

min

c

-12

0.5

1

1.5

2

2.5

T

max c

3

3.5

4

4.5

/ T

Fig. 8.2. The classical steady state current x ˙ cl and its quantum mechanical counterpart x ˙ qm for the ratchet potential from Fig. 8.1 in dimensionless units x =LJ ˙ 0 . Note that in the present Arrhenius plot (logarithmic ordinate) the observed behavior of the quantum current near Tcmax =T = 3:5 is not the signature of a singularity but rather of a current inversion. Further worth mentioning features are the non-monotonicity of x ˙ qm and that x ˙ qm tends towards a Fnite limit when T → 0.

can be taken for granted when evaluating the quantum mechanical transition rates (8.15) for all T 6 2Tcmax . SpeciFcally, the prefactor A appearing in (8.15) can be evaluated within a saddle point approximation scheme [66] if the semiclassical condition (8.20) holds. Moreover, the implicit assumption in (8.12) that not only thermally activated barrier crossings are rare (see (8.11)) but also tunneling probabilities are small, is self-consistently fulFlled if (8.20) holds. For more details regarding the actual numerical calculation of those rates we refer to [161]. Representative results [671,672] for the above speciFed quantum ratchet model are depicted in Fig. 8.2. Shown are the current x ˙ qm following from (8.12) within the above sketched quantum mechanical treatment of the rates according to (8.15) together with the result x ˙ cl that one would obtain by means of a purely classical calculation (formal classical limit ˝ → 0) according to (8.13). The small dashed part in x ˙ qm in a close vicinity of the crossover temperatures Tcmax and Tcmin from (8.19) signiFes an increased uncertainty of the semiclassical rate theory in this temperature domain. Our Frst observation is that even above Tcmax , quantum e5ects may enhance the classical transport by more than a decade. They become negligible, that is, the physical classical limit is approached, only beyond several Tcmax . In other words, signiFcant quantum corrections of the classically predicted particle current set in already well above the crossover temperature Tcmax , where tunneling processes are still rare. (They can be associated to quantum e5ects other than genuine tunneling “through” a potential barrier.) With decreasing temperature, T ¡Tcmin , quantum transport is even much more enhanced in comparison with the classical results. The most remarkable feature caused by the intriguing interplay between thermal noise and quantum tunneling is the inversion of the quantum current direction at very low temperatures [161,414,415,671,672,676 – 678]. Working within a formal a classical limit (˝ → 0), such a reversal for adiabatically slow driving is ruled out. Finally, x ˙ qm approaches

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213

a Fnite (negative) limit when T → 0, implying a Fnite (positive) stopping force 133 also at T = 0. In contrast, the classical prediction x ˙ cl remains positive but becomes arbitrarily small with decreasing T . A curious detail in Fig. 8.2 is the non-monotonicity of x ˙ qm around Tcmax =T 2:5, caused via (8.12) by a similar resonance-like T -dependence in the prefactor A of one of the underlying quantum mechanical transition rates (8.15). 8.2.1. Tunneling-induced current inversion The most remarkable result of the preceding subsection (see also Fig. 8.2) is the inversion of the current upon decreasing the temperature. On the other hand, within the formal classical limit (˝ → 0) the current never changes its direction. Since at high temperatures the physical classical limit is approached, i.e. the formal classical limit provides a more and more accurate approximation for the true physical system, the temperature controlled current inversion represents a new signature of genuine quantum mechanical e1ects. In the following we provide a simple heuristic explanation of this Fnding [161,415,671]. As a Frst simpliFcation, we exclusively focus in the exponentially leading contribution in the semiclassical rate expression (8.15), i.e. the sign of the current in (8.12) is given by that of Sl− −Sr+ . For suCciently large temperatures, quantum mechanical e5ects become negligible and the exponentially leading part in (8.15) goes over into that of (8.13). Indeed, one can show [66,96] that for T ¿Tcmax only the trivial extremizing paths q(&) ≡ xb in (8.16) exist for both potentials U ± (x), and thus we recover with (8.17) that Sr+ =˝ = UUr+ =kB T and Sl− =˝ = UUl− =kB T . In other words, the lower of the two barriers UUr+ and UUl− determines the direction of the current. A second case for which the extremization of the action (8.16) can be readily carried out is the combined limit T → 0 and → 0 (no heat bath), resulting in the familiar Gamow formula for the exponentially leading tunneling contribution in (8.17), namely x1 √ 1=2 S = 2 2m (8.21) dq [U (q) − U (x0 )] : x0

As before, x0 denotes a local minimum of U (x) and x1 is the Frst point beyond the considered potential barrier with the property that U (x1 ) = U (x0 ). The absolute value in (8.21) is needed since x1 ¡x0 for the escapes to the left, i.e. across UUl− . Thus, the smaller of the two Gamow-factors Sr+ and Sl− determines the direction of the current. Strictly speaking, by letting → 0 we of course violate our previously made assumption that deterministically running solutions should be ruled out. However, it is plausible that small but Fnite will exist for which our qualitative arguments can be adapted self-consistently. From Fig. 8.1 one can see by naked eye that the activation energy barrier UUr+ to proceed in the potential U + (x) from one local minimum to the neighboring local minimum to the right is smaller than the corresponding barrier UUl− . Hence the current is positive for suCciently large temperatures. In contrast, the fact that Sr+ is larger than Sl− cannot deFnitely be read o5 by eye directly from Fig. 8.1 since the two quantities are rather similar, but it can be readily veriFed numerically. In other words, for very small T indeed a negative current is predicted. A change of sign of the current at some intermediate temperature is thus a necessary consequence. 133

Recalling the deFnition from Section 2.6.2, the stopping force is that external force F in (3.1) which leads to a cancellation of the ratchet e5ect, i.e. x ˙ = 0.

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V(x) (a)

V

0

x L +

U (x) (b)

V

0

∆U

+

r

x

λ

+

L -

U (x) (c)

V

0

∆U

-

l

λ

L

x

−

Fig. 8.3. (a) Stylized saw-tooth ratchet potential V (x) with spatial period L and barrier height V0 . (b) The tilted ratchet potential U + (x) = V (x) − Fx from (8.10) with the “tunneling length” 3+ and the potential barrier UUr+ , relevant for the tunneling rate kr+ from the local minimum x0 = 0 to its neighboring local minimum to the right. (c) Same for the potential U − (x) = V (x) + Fx. [The depicted F-value is V0 =3L.]

Things become even more obvious by considering instead of the smooth potential from Fig. 8.1 a stilized saw-tooth proFle 134 as sketched in Fig. 8.3. Focusing on the local minimum x0 = 0, the fact that UUr+ ¡UUl− is read o5 immediately from Fig. 8.3. Denoting by 3 := |x1 − x0 | the 134

For such a singular potential shape the -factor in the crossover temperature (8.18) is no longer given by (8.14). Instead of changing the deFnition of , one may also slightly smoothen out the singularities of the potential.

P. Reimann / Physics Reports 361 (2002) 57 – 265

“tunneling-length”, the Gamow-factor (8.21) takes the simple form √ S = 43 2mUU 3 ;

215

(8.22)

where indices r, l, and ± have been omitted as usual. From Fig. 8.3 one reads o5 that 3+ = L, 3− = L=(1 + LF=V0 ), UUl− = V0 , and UUr+ = V0 (1 − LF=V0 ), where we have assumed without loss of generality, that 0 6 F 6 V0 =L. It readily follows that for small-to-moderate tilting forces F ∈ [0; 0:618V0 =L] we have that Sr+ ¿Sl− and the current is therefore negative. In conclusion, the basic physical mechanism behind the opposite sign of the current at high and low temperatures is apparently rather simple and robust, suggesting that this feature should be very common in tilting quantum ratchet systems. Since the decrease of temperature is accompanied by a transition from thermally activated to tunneling dominated transport, the concomitant change of the transport direction may be considered as tunneling induced current inversion, see Section 8.4. 8.3. Beyond the adiabatic limit For an non-adiabatic tilting force y(t) in (8.1) the determination of the average particle current is in general very diCcult. An approximative analytical approach becomes possible within a so-called tight-binding model description. The starting point consists in the observation that the Frst two terms on the right-hand side of (8.1) deFne a time-independent particle dynamics in a periodic potential and can thus be treated within the standard Bloch-theory for independent (quasi-)particles in a one-dimensional lattice [462]. Under the assumptions that both the external tilting force y(t) and the thermal =uctuations of the environment, entering through the last two terms in (8.1), are suCciently weak, one can focus on a single-band truncation of the problem, i.e. the Hilbert-space accessible to the particle is spanned solely by the Bloch-states of the lowest energy-band. Especially, both the thermal energy kB T and the energy ˝!c associated to the cuto5 in (8.8) have to be restricted to values much smaller than the excitation energy into the second band (or the continuum). Upon going over from these Bloch-states of the lowest band to a new basis {|n}∞ n=−∞ of the so-called localized-or Wannier-states [462], the truncated model Hamiltonian (8.1) takes the standard single-band tight-binding form [679] H(t) = −

∞ ˝ (|nn + 1| + |n + 1n|) − xy(t) + HB ; 2 n=−∞

(8.23)

where both in (8.23) and (8.2) the operator x is deFned as x := L

∞

n|nn| :

(8.24)

n=−∞

The quantity ˝ in (8.23) is the so-called tunneling coupling energy between neighboring potential minima. In principle, its explicit value can be determined from the Bloch-states and the potential V (x) [462]. Alternatively, the tunneling coupling energy may be considered as an adjustable model parameter. An additional approximation implicit in (8.23) is the assumption that only tunneling between neighboring potential minima of V (x) plays an appreciable role. In other words, the so-called coherent tunneling (co-tunneling) is neglected.

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By construction, the single-band tight-binding model (8.23) cannot capture thermally activated transport across the energy barriers between neighboring minima of V (x); its validity is restricted to quantum mechanical tunneling processes at low energies and temperatures. Furthermore, the model does not exhibit any traces of a possible asymmetry in the periodic potential V (x). One is therefore restricted to e5ectively symmetric potentials V (x) and a ratchet e5ect may only be studied within an asymmetrically tilting ratchet scheme (see Section 5.12). Besides these restrictions, the tight-binding model also goes beyond the approach from Section 8.2 in that the semiclassical condition (8.18) is not required and the tilting force y(t) need not be adiabatically slow (see below). In this sense, the approaches from Section 8.2 and of the present section are complementary. A non-adiabatically tilting quantum ratchet within the above single-band tight-binding approximation has been considered in [680] for a rather general class of unbiased, asymmetric random drivings y(t), including asymmetric dichotomous noise as a special case. In the absence of the heat bath HB in (8.23), the average particle current is found to vanish in all cases (for the same model, but with a periodic driving y(t), see also [166,167]). In the presence of the heat bath, the occurrence of a Fnite current is generically observed. Current inversions upon variation of di5erent model parameters are also reported. Especially, such an inversion may occur when the temperature is changed, which, for reasons detailed above, cannot be explained by the heuristic argument from Section 8.3 and thus represent a genuine feature of the non-adiabatic driving. Regarding a more detailed discussion of the e5ective di5usion coeCcient (3.6) within this model we refer to the original paper [680]. The same model, but with an asymmetric periodic driving y(t) of the harmonic mixing form (5.58) has been addressed in [681]. The emerging quantum current exhibits multiple reversals, characteristic for the non-adiabatic nature of the driving, and a stochastic resonance-like, bell-shaped behavior upon variation of the temperature. Via control of the phase and the amplitudes of the driving signal (5.58) it is furthermore possible to selectively control the magnitude of both the quantum current and di5usion, as well as the current direction. For further theoretical and experimental works along related lines see [166,167,381–383,780,781] and references therein. 135 While the rich behavior of the single-band tight-binding ratchet model can be obtained by means of sophisticated analytical approximations [680,681] which go beyond our present scope, simple intuitive explanations can usually not be given. Generalizations of the single-band tight-binding model (8.23) have been addressed in [683– 685]. The main new ingredient is an extra “potential”-term HV of the form ∞ |nn| Vn mod N [1 + f(t)] (8.25) HV = n=−∞

on the right-hand side of (8.23), reminiscent of a spatially discretized, asymmetric ratchet potential with period N ¿ 3. The case with y(t) ≡ 0 in (8.23), corresponding to a =uctuating potential ratchet, has been treated in [683,685]. The opposite case with f(t) ≡ 0 but again with an adiabatically slow, symmetric rocking force y(t) has been addressed in [684]. At Frst glance, such an extra term (8.25) in order to upgrade the single-band tight-binding model (8.23) into a veritable ratchet system with a broken spatial symmetry is indeed suggestive. However, the present author was only able to Fgure out very arti8cial actual physical situations which may be captured by such a model with a non-trivial 135

Closer inspection indicates [166,682] that the conclusions from [383] in the case of a dissipationless (collisionless) single-band model are at most valid for very special (non-generic) initial conditions.

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term (8.25). In particular, due to the nearest-neighbor hopping term in (8.23), such a description clearly cannot properly account for more than one single band. Further quantum ratchet works, some of which go beyond the adiabatic limit, and which are not based on the single-band tight-binding approximation (8.23) of the original dynamics (8.1), are [490,541,685,686]. However, the present author Fnds these studies questionable with respect to the conceptual basis and=or the technical methodology. A quantum Smoluchowski–Feynman-type model (equilibrium system) has been investigated on the basis of a widely used standard approximation in [658] and further references therein. The observed appearance of a ratchet e5ect in such an equilibrium model underlines once more the warning at the beginning of Section 8.1 that even well-established ad hoc approximations for a quantum thermal environment may easily lead to inconsistencies with fundamental principles of statistical mechanics. For a periodic driving y(t) and in the absence of the heat bath HB in (8.1) the quantum mechanical counterpart of the Hamiltonian rocking ratchet model from equation (5.34) in Section 5.8 is recovered. Within a single-band tight-binding approximation (8.23) this type of model has been solved in closed analytical form in [166,167,680]. Though the chaotic features of the classical counterpart cannot be captured in this way, a dependence of the current on the initial conditions is found [166,167] which is quite similar to the classical results from [165], while strongly non-classical features [680] arise in the presence of a Fnite static tilt F (i.e. y(t) in (8.23) is replaced by y(t) + F). A Frst step into the direction of a chaotic (Hamiltonian) quantum ratchet system has been taken in [408,409] and [687] with the main focus on the semiclassical regime and on mesoscopic electron billiard devices, respectively, see also Section 5.8 for the classical limit. 8.4. Experimental quantum ratchet systems As a Frst candidate for an experimental realization of a quantum ratchet we consider the SQUID rocking ratchet model [354] from equation (5.38). As argued at the beginning of this chapter, this stochastic dynamics (5.38) as it stands represents a classical ratchet system [94]. The question of how to properly “quantize” such a “classical” dynamics, which itself arises as an e5ective description of characteristic quantum e5ects, has been discussed extensively in the literature, see [94,664] and references therein. Leaving aside devices which contain ultra small tunnel junctions, ample theoretical [664,688– 690] as well as experimental (see references in [66]) justiFcation has been given that, after proper renaming of symbols, Eqs. (8.1) and (8.8) provide the basis for an adequate quantum mechanical extension of the classical model (5.38) when the temperature is decreased below a few times Tcmax from (8.19). Conceptionally, it is interesting to note [94] that we are dealing here with quantum e5ects which manifest themselves via the macroscopic phase-variable ’. In other words, the observation of transport properties characteristic for a quantum ratchet is not necessarily restricted to the realm of microscopic systems. So far, an actual experimental realization of a SQUID ratchet system [25,26,182,183] is only available for the two-dimensional modiFcation (5.52), (5.53) of the archetypal rocking ratchet setup (5.38). While the experiment from [182,183] works with high-Tc SQUIDS at temperatures too large to see any traces of quantum mechanical tunneling of the phase ’, an analogous experiment with conventional superconductors is presently under construction, with the intention to reveal such quantum mechanical e5ects. A second potential realization of a tilting quantum ratchet system is based on the motion of ultracold atoms in the presence of standing electromagnetic waves, creating a ratchet potential through

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the radiation-pressure forces of the counterpropagating light beams [691– 695]. For suCciently weak potentials and low temperatures, quantum e5ects will clearly play a dominant role in the atomic motion and may be roughly captured by a model like in (8.1). Especially, the tilting force may be created by exploiting the mapping of the model onto an improper traveling potential ratchet from Section 4.4.2. The corresponding accelerating optical potentials have been experimentally realized, e.g. in [695]. As detailed in Section 4.5.2, a somewhat related system has indeed been experimentally studied in [336]. Due to the remaining considerable di5erences between this system and the theoretical model (8.1), a direct comparison is, however, not possible. A third promising class of experimental tilting quantum ratchet devices are semiconductor heterostructures. The lacking periodicity of a single diode (n-p junction) can be readily remedied, in the simplest case by connecting identical diodes by normal conducting wires. Similarly as in the above discussed case of SQUID ratchets, such a simple array of diodes realizes a classical ratchet system in so far as (at the usual working temperatures) the essential transport processes across the junctions are governed by classical thermal di5usion rather than quantum mechanical tunneling (see also Sections 2.9, 5.6 and 6.1). Closely related devices are spatially periodic semiconductor superlattices. Examples with broken spatial symmetry (so-called sawtooth superlattices) have been experimentally realized since long [783,784] but have never been studied so far from the viewpoint of the ratchet e5ect. Heterostructures consisting of alternating layers of GaAs and AlGaAs in quantum mechanically dominated temperature regimes have been experimentally explored e.g. in [696 – 698]. The motion of a (quasi-) particle (dressed electron) in such a superlattice may be roughly described by an e5ective, one-dimensional model of the form (8.1), where the heat bath takes into account the e5ects of the crystal phonons [680,681,697,698]. In the simplest case of a semiconductor superlattice with only two di5erent alternating layers, a symmetric periodic potential V (x) in (8.1) arises, thus the asymmetrically tilting quantum ratchet scheme from Section 8.3 has to be employed. The quantitative estimates from [681] furthermore show that the one-band tight-binding model (8.23) may be a valid approximation for a typical experimental setting [696 – 698]. Moreover, if the driving y(t) is provided by the usual electromagnetic waves in the THz-regime, one is indeed dealing with the non-adiabatic regime from Section 8.3. On the other hand, the semiclassical theory from Section 8.2 cannot be applied to such an experimental situation not only because the driving is not adiabatically slow, but also since the semiclassical condition (8.18) is typically not satisFed. Important progress towards an adiabatically rocking quantum ratchet in one-dimensional Josephson junction arrays, consisting of three “cells” (e5ective periods) with broken spatial symmetry, has been achieved very recently in [699]: somewhat similar as in the systems from Section 5.7.3, but operating in the quantum mechanical regime, the voltage due to the dynamical response of the vortices (directed transport of quasi particles) against an applied bias current exhibits an asymmetry when the sign of this bias is inverted, cf. Fig. 2.4. In particular, the theoretically expected asymptotic temperature independence of the e5ect in the deep quantum cold is experimentally recovered. A molecular rectiFer for electrons, combining the quantum ratchet with the Coulomb blockade e1ect, has been proposed in [422]. For additional experiments which may be considered to some extent as quantum ratchet systems we also refer to the applications (especially single electron pumps) of the genuine traveling potential ratchet scheme discussed in Section 4.4.1. We close this section with the experimental realization of an adiabatically rocking quantum ratchet by Linke and colleagues on the basis of a quantum dot array with broken spatial symmetry. We skip the preliminary experiment on AC-driven electron transport through a single triangular shaped

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219

current [nA]

0.08

0.04

0

-0.04 0

1

2

3

4

temperature [K]

Fig. 8.4. Scanning electron micrograph of an array of triangular shaped quantum dots, etched from a GaAs=AlGaAs semiconductor heterostructure. The depicted top view deFnes the x–y-plane accessible to the two-dimensional conducting electron gas. The etched areas (dark regions) are insulating domains for the electrons. Shown are 4 out of the 10 triangles used in the actual experiment [705]. The period L of the triangles is about 1:2 m. Fig. 8.5. Electrical current I = −e x ˙ along the quantum dot array from Fig. 8.4 versus temperature for an unbiased rocking voltage y(t) which periodically jumps between ±1 mV at a frequency of 191 Hz [705].

quantum dot [414,700 –704] (see also [423]) and immediately turn to the exploration of an entire array of such triangular dots [415,677,678,705 –707]. The basic setup [705] is depicted in Fig. 8.4: A two-dimensional conducting electron gas is constricted by two insulating boundary-regions (dark areas in Fig. 8.4). In other words, the “conducting channel” along the x-axis is laterally conFned to a width of about 1 m. Roughly speaking, the corresponding lateral conFnement energy creates an e5ective ratchet-shaped potential V (x) for the particle dynamics along the x-axis of a qualitatively similar character as in Fig. 8.1. The two “side gates” in Fig. 8.4 allow one to externally modify this e5ective potential by putting them on di5erent electrical potentials. The actual rocking force y(t) is created by applying an AC-voltage along the x-axis, periodically switching between the two values ±F, with a typical voltage F of about 1 mV. The driving frequency of 191 Hz used in the experiment is deFnitely deep within the adiabatic regime. The “bottlenecks” which connect neighboring triangles in Fig. 8.4 are chosen such that quantum tunneling dominates at low temperatures, while for higher temperatures the conduction electrons can also substantially proceed by way of thermal activation across the corresponding e5ective potential barriers. The measured [705] current through the quantum dot array as a function of temperature is exempliFed in Fig. 8.5. The two main features theoretically predicted in Section 8.2 are clearly reproduced, namely a current inversion upon decreasing the temperature and a saturation of the current as temperature approaches absolute zero. 136 Though a model along the lines of (8.1) is obviously 136

Note that Fig. 8.2 is an Arrhenius plot (logx ˙ versus 1=T ), while Fig. 8.5 depicts the bare quantities “electrical current” versus “temperature”.

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a very crude description of the experimental situation, the basic features and thus the heuristic explanation of the current inversion from Section 8.2.1, are apparently still qualitatively correct. Note that also the direction of the current is in agreement with Fig. 8.2 by taking into account that the relevant e5ective potential for the experiment is of the same qualitative shape as in Fig. 8.1 and that the electrical current is opposite to the particle current for the negatively charged electrons. For a somewhat more realistic theoretical model, which also reproduces the main qualitative features of the experiment, we refer to [705,706]. Based on the observation from Section 8.2.1, namely that “cold” and “hot” particles move in opposite directions (as long as their individual “temperatures” (kinetic energy) change suCciently little), an interesting idea is [559,708] to apply the above quantum rocking ratchet setup in the absence of a net particle transport, i.e. operating at the current inversion point, for “cooling” purposes. 137 However, the quantitative analysis of the experiment in [705] shows that the heating due to the external rocking force exceeds the cooling e5ect due to the above separation of particles with di5erent temperatures [708]. On the other hand, assuming the existence of “ideal Flters”, which let pass in both directions only particles with one speciFc energy, it is possible to modify the original setup such that it can act either as refrigerator or as heat engine arbitrarily close to the maximal Carnot eCciency [559]. In contrast to the standard framework for considerations on the eCciency of particle transport from Section 6.9, here a zero particle current situation is addressed, and the relevant mechanical work is now associated with the external driving force. In other words, the particle motion is now considered as an internal part of the engine under consideration, and no longer as the resulting e5ect of the engine.

9. Collective e9ects At the focus of this chapter are collective e5ects that arise when several copies of “single” classical ratchet systems, as considered in extenso in the previous sections 2–6, start to interact with each other. Accordingly, the general working model (3.1) goes over into N coupled stochastic di5erential equations of the form

x˙i (t) = −V (xi (t); fi (t)) + yi (t) + F + i (t) −

9 (x1 (t); : : : ; xN (t)) 9xi

(9.1)

with i = 1; : : : ; N . The last term accounts for the interaction through an interaction potential which is assumed to be spatially homogeneous and inversion symmetric, i.e. no preferential direction is introduced through the interaction. The assumption of a thermal equilibrium environment implies that the thermal noises i (t) are mutually independent Gaussian white noises with correlation i (t)j (s) = 2 kB Tij (t − s) :

137

(9.2)

Of foremost interest in this context are “single-period setups” (e.g. a single triangular quantum dot) in contact with two electron-reservoirs at either the same or di5erent temperatures.

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The drivings fi (t) and=or yi (t) are usually assumed to be either mutually independent random processes or equal to the same periodic function for all i. In any case, these drivings as well as the interaction in (9.1) have to respect the equivalence of all the “single particles” i. Therefore, the average particle current x ˙ from (3.5) will be the same for each particle i and consequently independent of whether or not an additional average over i is performed. The case of foremost interest is usually the zero load (F = 0) situation, but also the response when a Fnite F is acting will lead to quite remarkable observations in Section 9.2. We recall that various examples with N = 2 interacting systems (9.1) have been discussed already in Sections 5:9; 6:5 and 7:6. In the present chapter, our main interest will concern collective e5ects in the case of a large number N → ∞ of interacting systems 138 (thermodynamic limit). In doing so, two basic types of questions can be addressed. First, one may consider cases for which already in the absence of the interaction in (9.1) each single system exhibits a ratchet e5ect, i.e. both, thermal equilibrium and spatial symmetry are broken. In such a case, one may study the modiFcation of the current x ˙ in magnitude and possibly even in sign when the interaction is included. A survey of such explorations will be presented in Section 9.1. A second type of questions regards genuine collective e1ects, namely spontaneous ergodicity breaking, entailing phase transitions, the coexistence of di5erent (meta-) stable phases, and hysteretic behavior in response to the variation of certain parameters. While all these collective phenomena are well known also in equilibrium systems, the second law of thermodynamics precludes a Fnite particle current in such systems even if their spatial symmetry is broken. We thus focus on interacting systems (9.1) out of equilibrium, which for the usual interactions is the case if and only if already the uncoupled systems (9.1) are out of equilibrium. Such genuine non-equilibrium collective e5ects have already been encountered in the context of the Huxley–Julicher–Prost model for cooperating molecular motors in Section 7.4.4. There, the main emphasis was put on systems with a built-in spatial asymmetry already of the single (uncoupled) systems in (9.1), which is then inherited by the coupled model. In contrast, in Section 9.2 we will address coupled non-equilibrium systems (9.1) which are fully symmetric under spatial inversion. The essential idea is then that instead of a built-in asymmetry, a perfectly symmetric system may create the asymmetry, which is necessary for the manifestation of a ratchet e5ect, by itself, namely through spontaneous symmetry breaking. While the occurrence of such a “spontaneous current” has been pointed out already for a spatially symmetric special case of the JMulicher–Prost model in [550], we will focus in Section 9.2 on a simpler model which admits a partial analytical treatment and exhibits additional, quite remarkable collective non-equilibrium features. We close with two remarks: Frst, the subject under study in this chapter is intimately related with many other topics, like for instance non-equilibrium phase transitions, reaction–di5usion systems, pattern formation, driven di5usive systems, Frenkel–Kontorova models, Josephson junction arrays, sine-Gordon equations, and coupled phase oscillators. A detailed discussion of any of these adjacent topics goes, however, beyond the scope of our present review. Second, while we feel that very 138 A deterministic collective model which does not Ft into this general framework is due to [709,710]: It works for N ¿ 3 particles with Fnite mass m in a static, not necessarily asymmetric potential V (x). A worm-like deterministic motion is generated by active changes of the interaction in a wave-like manner along the chain of particles xi (t). A similar model with non-Newtonian interaction forces (action = reaction) is due to [711]. Moreover, reaction-di5usion model for interacting Brownian motors has been discussed in Section 6.5.

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interesting and unexpected theoretical discoveries are still to come, on the experimental side the Feld is even more so at a very underdeveloped stage. 9.1. Coupled ratchets In this section we review investigations of N → ∞ coupled ratchet systems in the case that each single particle i exhibits a ratchet e5ects already in the absence of the interaction in (9.1). For related discussions of models for cooperative molecular motors we also refer to Section 7.4. Throughout this section, we restrict ourselves to the case F = 0 in (9.1) and to potentials V (x) with a broken symmetry, i.e. ratchet potentials, as exempliFed by Figs. 2.2 and 4.1. The case of interacting rocking ratchets

x˙i (t) = −V (xi (t)) + y(t) + i (t) + Ib (xi+1 (t); xi (t); xi−1 (t))

(9.3)

with a hard core repulsive interaction Ib such as to guarantee xi+1 (t)¿xi (t)+b for all i and t has been explored in [712]. Pictorially speaking, all particles are thus moving in the same one-dimensional, periodically rocked ratchet potential V (x) − xy(t) and they have a Fnite extension b which sets a lower limit for their mutual distance. The central (numerical) Fnding in [712] is a current inversion upon variation of the average density of particles along the x-axis. This inversion is robust against various modiFcations, especially of the driving y(t) in (9.3) [e.g. stochastic instead of periodic, or with small, i-dependent variations of the driving-period T] and implies according to Section 3.6 analogous inversions upon variation of practically any other parameter of the model (9.3). For adiabatically slow driving y(t) and simultaneously almost densely packed particles, an analytical treatment is possible, revealing an extremely complex dependence of the current upon the particle extension b. Somewhat similarly as in the JMulcher–Prost model [550], also in the present case the magnitude of the current x ˙ depends sensitively on whether the spatial period L is commensurate or not with the average interparticle distance xi+1 (t) − xi (t). Such e5ects may become practically relevant for separating particles at high densities e.g. according to the drift ratchet scheme from Section 6.6. A related, spatially discrete model with an adiabatically slow driving has been considered in [513], thus establishing contact with the methods and concepts of the so-called driven di5usive systems [602]. A second basic model consists of a chain of linearly coupled 9uctuating force ratchets

x˙i (t) = −V (xi (t)) + yi (t) + i (t) + A[xi+1 (t) − 2xi (t) + xi−1 (t)] ;

(9.4)

where A is the spring constant (interaction strength) and yi (t) are independent Ornstein–Uhlenbeck noise sources (cf. Eqs. (3.13) and (3.15)). In the continuum limit, one obtains a sine-Gordon-type model, which has been analyzed by means of the sophisticated analytical machinery in this Feld in [713]. The main result is the appearance of a ratchet e5ect in the form of a stationary directed transport of kinks and antikinks in opposite directions. As a rule, the kink and hence the entire particle chain move into the same direction as in the uncoupled, Ornstein–Uhlenbeck noise driven =uctuating force ratchet (cf. Section 5.5), however with a highly non-trivial modiFcation of the quantitative behavior of the current. Similar results for models of the type (9.4) have been reached also in [714 –717] and for an analogous coupled temperature ratchet model in [476]. Possible applications include the dynamics of dislocations in solids, solitonic =uxes in long Josephson junction arrays and magnetically ordered crystals, and models for friction and stick–slip motion such as the

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Frenkel–Kontorova model. For related studies in the context of coupled Josephson junction arrays see also Section 5.7.3. Next, we turn to the interacting on–o1 ratchet counterpart of (9.3), i.e.

x˙i (t) = −V (xi (t))[1 + fi (t)] + i (t) + Ib (xi+1 (t); xi (t); xi−1 (t))

(9.5)

˙ may even change with fi (t) ∈ {±1}. In this case [225], the direction of the particle current x many times as the density of particles is varied. For high particle densities and slow on–o5 cycles, an extremely complex dependence of x ˙ on the particle size b similarly as for the model (9.3) is recovered. Such e5ects clearly become relevant for the various experiments from Section 4.2.1 at high particle densities. An experiment which may be considered to some extent as related to the theoretical model (9.5) has been realized in [718,719]. In this work, the horizontal transport of granular particles in a vertically vibrated system, whose base has a ratchet-shaped proFle, has been measured. 139 The resulting material =ow exhibits current inversions and other complex collective behavior as a function of the particle density and the driving frequency, displaying a rough qualitative similarity with the theoretical model (9.5). A coupled rocking ratchet model, but in contrast to (9.3) with a global, Kuramoto-type interaction [624,721,722] with the same period L as the ratchet potential V (x), i.e. N K 2

x˙i (t) = −V (xi (t)) + y(t) + i (t) + sin (9.6) [xj (t) − xi (t)] N j=1 L has been addressed in [723]. Upon increasing the coupling strength K, the current may change direction and moreover the e5ect of the noise becomes weaker and weaker: For K → ∞ all particles in (9.6) are lumped (modulo L) into one single e5ective “superparticle” subjected to an e5ectively deterministic single-particle rocking ratchet dynamics like in Section 5.7. The existence of current inversions upon variation of other model parameters than the coupling strength immediately follows from Section 3.6. Considering that a single particle (N = 1) rocking ratchet can be realized by means of three Josephson junctions (see Eq. (5.38)), the coupled model (9.6) may well be of relevance for Josephson junction arrays 140 [724]. Universal properties of particle density =uctuations at long wavelengths and times for a large class of short-range interaction ratchet models like for instance in (9.3), (9.5) have been revealed in [725]. More precisely, the steady state density–density correlation function exhibits dynamical scaling according to the Kadar–Parisi–Zhang universality class [725]. 9.2. Genuine collective e1ects For non-interacting periodic systems, the basic result of the previous sections 2– 6 is that necessary, and generically also suCcient conditions for the occurrence of directed transport are that the system is out of thermal equilibrium and that its spatial symmetry is broken. The essential idea of this 139 A computer animation (Java applet) which graphically visualizes a somewhat related e5ect is available on the internet under [720]. 140 A nearest neighbor instead of the global coupling in (9.6) may then be a more realistic choice. Such a modiFcation is, however, not expected to change the basic qualitative features of the model (at least in d ¿ 2 dimensions), see also Figs. 9.2 and 9.3 below.

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section is to abandon the latter condition of a built-in asymmetry. Instead, the system may create an asymmetry by itself as a collective e1ect, namely by way of spontaneous symmetry breaking. As a consequence, according to Curie’s principle, a collective ratchet e5ect in the form of a “spontaneous current” is then expected. 141 It turns out that this idea can indeed be realized, and in fact even in several di5erent ways [14,524,550,551,617,726 –730]. Here, we will focus on a particularly simple example of globally coupled =uctuating potential ratchets [524,726,729]. We Fnally remark that the appearance of a “spontaneous current” has also been predicted in a rather di5erent theoretical mean Feld model for driven semiconductor superlattices in [731,732]. 9.2.1. Model As a combination of the =uctuating potential ratchet scheme from (4.11) and of our general working model for interacting systems (9.1) we take as starting point the following set of i =1; : : : ; N coupled stochastic equations x˙i (t) = −V (xi (t))[1 + fi (t)] + i (t) +

N K sin(xj (t) − xi (t)) : N j=1

(9.7)

For the sake of simplicity only, we consider a Kuramoto-type, sinusoidal global coupling [624,721,722], and we will restrict ourselves to attractive interactions K¿0. Furthermore, we have adopted dimensionless units (see Section A.4 in Appendix A) with

= kB = 1;

L = 2 :

(9.8)

[For esthetical reasons we will often continue to use the symbol L.] In particular, the potential V (x) and the interaction respect the same periodicity 142 L = 2 . However, in contrast to “conventional” =uctuating potential ratchets without interaction (see Section 4.3), we exclude any built-in spatial asymmetry of the system (9.7), which can be achieved if the potential V (x) respects the symmetry condition V (−x) = V (x) ;

(9.9)

independently of any further properties of fi (t), see the discussion below (3.23). Finally, in view of the analytic tractability in the absence of interaction (see Section 4.3.2) we specialize to potential =uctuations fi (t) which are given by independent Ornstein–Uhlenbeck processes (3.13), (3.15) of strength ∞ dt fi (t)fj (s) = 2Qij (9.10) −∞

(cf. Eq. (4.14)) and a negligibly small correlation time & in comparison to all the other relevant time scales of the system. 141

In contrast to “permanent currents”, appearing for instance in mesoscopic rings at thermal equilibrium, the “spontaneous currents” which we have here in mind can be exploited to do useful work and are moreover a purely classical phenomenon. 142 Mathematically, we avoid in this way additional complications due to incommensurability e5ects. Physically, this assumption is especially natural if the state variables xi are originally of a phase-like nature, see Section 3.4.2. Some generalizations will be addressed in Section 9.2.5 below.

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9.2.2. Spontaneous symmetry breaking In this section, we Frst present a somewhat formal analytical demonstration of the existence of spontaneous symmetry breaking for the system (9.7) in the thermodynamic limit N → ∞, followed quantitative numerical illustrations and an intuitive explanation of the basic physical mechanism at work. The main collective features of (9.7) are captured by the particle density P(x; t) :=

N 1 (x − xi (t)) : N i=1

(9.11)

In contrast to the deFnition for non-interacting systems in (2.9), the average over the noise is omitted in (9.11) and instead an average over the particles i is included. Being an intensive quantity, P(x; t) becomes independent of the speciFc realization of the noises 143 i (t) and fi (t) when N → ∞ (self-averaging), as demonstrated in detail in [733–736]. In other words, it does actually not matter whether we consider an average over the noise as included or not in the deFnition of P(x; t) in ˆ t) as usual, cf. Section 2.4. (9.11). Finally, we go over to the reduced density P(x; By rewriting the interaction term in (9.7) as K[S cos(xi (t)) − C sin(xi (t))], where 144 L=2 L=2 ˆ t) sin x; C := ˆ t) cos x ; S := d x P(x; d x P(x; (9.12) −L=2

−L=2

the dynamics of each particle (9.7) is exactly of the type which we have considered in Section 4.3.2. By summing the corresponding single-particle Fokker–Planck equations (2.17), (4.25) according to (9.11) one recovers 9 9 9 ˆ ˆ ˆ t) ; P(x; t) = V (x) + g(x) g(x) P(x; (9.13) 9t 9x 9x Vˆ (x) := V (x) + K(S sin x + C cos x) ; (9.14) g(x) := [T + QV (x)2 ]1=2 :

(9.15)

ˆ t)Note that (9.13) represents a non-linear Fokker–Planck equation due to the implicit P(x; ˆ dependence of V (x) via (9.12) and (9.14). Especially, the linear superposition principle is not respected. This feature re=ects the fact that while P(x; t) in (9.11) is self-averaging with respect to the noises fi (t) and yi (t), it describes the particle density for a system with an arbitrary but 8xed initial distribution of particles P(x; t0 ). A statistical ensemble average over di5erent initial particle distributions is no longer captured by (9.13), in clear contrast to single-particle systems described, e.g. by a linear master equation of the form (2.17), or more general, Fnite-N particle systems. As usual in the context of phase transitions, the basic reason for this structural di5erence is the thermodynamic limit N → ∞ in concert with the mean Feld coupling in (9.7), entailing the exact self-averaging property of the particle distribution (9.11) in this limit N → ∞. The non-linear character of the Fokker–Planck equation opens the possibility that di5erent initial conditions P(x; t0 ) To be precise, this means that a convolution (average) P(x; t) h(x) d x of the particle density with an arbitrary but Fxed, smooth test function h(x) that vanishes as x → ± ∞, gives the same result with probability 1 for N → ∞, independent of the realization of the noises i (t) and fi (t). 144 For later convenience, the argument t is suppressed in S and C. 143

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display a di5erent long-time behavior, again in contrast to the typical asymptotic uniqueness (ergodicity) of linear Fokker–Planck equations [82,83,100,108,109]. The reason for this possibility of ergodicity breaking with all its consequences (spontaneous symmetry breaking, phase transitions, etc.) is that the thermodynamic limit N → ∞ does not commute with the “ergodicity limit” t → ∞. In conclusion, Eqs. (9.11) – (9.15) display the typical structure of a mean 8eld theory, with S and C in (9.12) playing the role of order parameters which have to be determined self-consistently with the mean Feld equation (9.13) for the particle density. ˆ t) in (9.13) for asymptotically large coupling strengths K in Next, we discuss the behavior of P(x; (9.7). To keep things simple, we further assume that multiples of L are the only minima of V (x). As a consequence, all particles in (9.7) are forced to occupy practically the same position (t) modulo ˆ t) takes the form L and hence P(x; ˆ t) = P(x;

∞

(x − (t) + nL) :

n=−∞

Introducing (9.16) into (9.13) and operating on both sides with motion for (t) takes the form of a simple relaxation dynamics

(9.16) (t)+L=2 (t)−L=2

d x x : : : the equation of

(t) ˙ = −UW ((t)) ;

(9.17)

UW (x) := V (x) − QV (x)2 =2 :

(9.18)

For small Q, the extrema of UW (x) in (9.18) are identical to those of V (x). So, for any initial condition (t0 ) ∈ (−L=2; L=2), the center of mass (t) in (9.17) moves for t → ∞ towards the minimum ˆ t) approaches a stationary, symmetric limit Pˆ st (x) = P st (−x). However, this x = 0 of V (x), and P(x; stationary solution (t) ≡ 0 of (9.17) looses stability and two new stable Fxed points appear when Q in (9.18) exceeds the critical value Qc := 1=V (0) :

(9.19)

One thus recovers a so-called noise induced nonequilibrium phase transition [64,65,737–745] with a concomitant spontaneous symmetry breaking of P st (x). If the coupling strength K is no longer assumed to be very large, one has to solve the non-linear Fokker–Planck equation (9.18) numerically until transients have died out and for a representative sample of di5erent initial conditions. In this way, a stationary and—apart from the obvious dest generacy when the symmetry is spontaneously broken—unique long time limit Pˆ (x) is obtained. In the symmetric phase (P st (−x) = P st (x)), the order parameter S from (9.11) vanishes, while a spontaneously broken symmetry is generically monitored by a non-zero S-value, see Fig. 9.1. Moreover, for large K, the above analytical prediction is conFrmed by the numerics, for moderate K, one recovers a re-entrant behavior as a function of the potential =uctuation strength Q, and for small K, a phase with broken symmetry ceases to exist [64,738]. For an intuitive understanding of why the system-intrinsic symmetry can be spontaneously broken, we return to a one-particle dynamics of the form (4.17). By averaging over the noise, this equation takes the form x ˙ = −V (x(t))= + g(x(t))(t). On the other hand, evaluating the particle current by means of the probability current (4.25) according to (2.19), one obtains x ˙ = −V (x(t))= +

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15

.

<x> =/ 0 S =/ 0

K

10

.

5

0

<x> = 0 S = 0

0

2

4

6

8

10

12

14

Q

Fig. 9.1. Phase diagram for model (9.7) – (9.10) with V (x) = −cos x − 0:15 cos(2x) (cf. Eq. (9.23) below) and T = 2 in the thermodynamic limit N → ∞ by numerically evolving the non-linear Fokker–Planck equation (9.13) until a steady state was reached. x ˙ is the particle current, S the order parameter from (9.12), and the arrow indicates the asymptotic phase boundary (9.19) for K → ∞.

g (x(t))g(x(t))=2. Upon comparison of these two expressions one recovers that 1 d 2 [g(x(t))] : g(x(t))(t) = 4 dx

(9.20)

In other words, the white noise (t) induces a systematic drift into the direction of increasing e5ective local temperature Te5 (x) := g2 (x) (see (9.15)). To get a rough heuristic picture of how this so-called Stratonovich drift term [99] comes about, we imagine a force-free, overdamped Brownian particle starting at x(0) = 0 in the presence of a high temperature in the region x¿0 and a low temperature for x¡0. Though the particle spends on the average the same amount of time on either side of x =0, the thermal random motion within x¿0 is enhanced, leading to a net bias of the average particle position x(t) towards the right. 145 One can readily see by comparison with (9.15) that this noise-induced drift term (9.20) is indeed the origin of the second term on the right-hand side of the e5ective potential (9.18), which governs the relaxation dynamics of the particle peak (t) in (9.17). Since the intensity of the multiplicative noise fi (t) in (9.7) has a minimum at the origin (modulo L), the noise-induced drift pushes the particles away from this point x = 0 and may lead, if the noise is strong enough and the particles cluster together suCciently strongly, to a spontaneous dislocation of the peak of particles (t) towards one or the other side of the origin. If, on the other hand, the interaction is too weak in comparison to either the thermal or the potential =uctuations, then the random motions of the single particles are not suCciently coordinated and a collective spontaneous symmetry breaking is therefore not expected. These heuristic arguments are conFrmed by, and essentially explain the numerical phase diagram in Fig. 9.1. 145

Strictly speaking, the issue is rather subtle with respect to the correct order of the overdamped limit m → 0 in (2.1), the white noise limit & → 0 in (4.15), and the limit of a discontinuous temperature at x = 0. Only if the limits are taken in the latter order (m → 0 Frst, discontinuous temperature last), this explanation of the Stratonovich drift can be applied, see also (A.3) in Appendix A and the corresponding discussion in Section 4.3.2.

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9.2.3. Spontaneous ratchet e1ect We start by rewriting (9.7) in the form x(t) ˙ = −Ve5 (x(t); f(t)) + (t) ;

(9.21)

where we dropped the subscript i and where Ve5 (x; f(t)) := V (x)[1 + f(t)] + K(C cos x + S sin x) :

(9.22)

st st If there is no spontaneous symmetry breaking (Pˆ (−x)= Pˆ (x)), then (9.12) implies S =0 and hence the pulsating potential (9.22) respects the symmetry condition (3.16) with Ux =0 due to (9.9). If, on the other hand, the symmetry of the system is spontaneously broken, then—in the generic case—we have that S = 0. Hence the symmetry condition (3.16) is generically violated and the occurrence of a ratchet e5ect with x ˙ = 0 is expected according to Curie’s principle. There is, however, one prominent exception, namely a supersymmetric potential (9.22) excludes a current even if the symmetry of the system is spontaneously broken. For our present purposes it is suCcient to focus on the supersymmetry condition (3.41). Since the white noise f(t) is time-inversion invariant, we see that for instance a pure cosine-potential V (x) indeed leads to a supersymmetric e5ective potential in (9.22), whatever the values of S and C are. In order to break this supersymmetry, we can either modify the interaction in (9.7), or consider a colored noise fi (t), or, as we will do in the following, choose an augmented cosine potential of the form

V (x) = −cos x − A cos(2x)

(9.23)

with A = 0. Given that potential (9.22) respects neither symmetry nor supersymmetry, each particle (9.21) is expected to exhibit a ratchet e5ect x ˙ = 0 in the generic case [524,726,729], as conFrmed by 146 the numerical result in Fig. 9.1. The underlying mechanism is clearly of the general pulsating ratchet type, and according to (9.22) similar but not exactly identical to a 9uctuating potential ratchet scheme from Section 4.3. With the notation from (9.14), (9.15) we can rewrite (9.21) in yet another from, namely ˆ ; x(t) ˙ = −Vˆ (x(t)) + (t)

(9.24)

ˆ (s) ˆ (t) = 2g(x(t))(t − s) :

(9.25)

While g(x) from (9.15) has its minima at the integer multiples of L=2, the potential Vˆ (x) from (9.14) exhibits for S = 0 not only an asymmetric, ratchet-shaped proFle, but also its extrema are generically shifted with respect to those of g(x). From this viewpoint, the ratchet mechanism to which every single particle is subjected in the symmetry broken phase is thus of the Seebeck ratchet type from Section 6.1. 146 A computer animation of this collective phenomenon is available on the internet under [746]. It is based on simulations of (9.7) – (9.10), (9.23) with N = 1000; T = 2; Q = 4; K = 10; A = 0:15. Out of the 1000 particles, 100 are shown as green dots and one “tracer-particle” as a red dot. The position x = − is identiFed with x = (periodic boundary conditions). The initial particle distribution is symmetric about x = 0. After a spontaneous breaking of the symmetry “to the right” (S¿0) an average particle current “to the left” (x ¡0) ˙ can be observed.

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Quantitatively, once the values of the order parameters S and C in the long time limit are known, the current follows readily along the lines of Section 4.3.2 with the result e−)(x) x+L e)(y) st ˆ ; (9.26) dy P (x) = N g(x) x g(y) x ˙ = LN[1 − e)(L) ] ; x d xW Vˆ (x)=g W 2 (x) W ; )(x) := 0

(9.27) (9.28)

where the normalization N is Fxed through (2.25). Thus, the current is Fnite unless )(L) = 0, and its sign is given by that of −)(L). Specializing once again to large coupling strengths, we can exploit (9.16) to recast (9.28) into the simpliFed form )(L) = −Kˆ 1 sin ; Kˆ n :=

L=2

−L=2

d x[K cos x]n =g(x)2 ;

(9.29) (9.30)

where := (t → ∞) follows from (9.17). Here, a remarkable feature arises, entailing even more striking consequences later on. Namely, if Q¿Qc , and Kˆ 1 ¡0, which is the case whenever A¿0 in (9.23), then the sign of (t) from (9.17) will, in the long time limit, be opposite to that of x. ˙ In other words, for a symmetry broken P(x) with a peak to one side of x = 0, the =ux of particles will move just in the opposite direction. On average the particles surprisingly prefer to travel from their typical position, say (t → ∞)¡0 down to the potential minimum of V (x) at x = 0 and then over the full barrier to their right rather than to directly surmount the partial remaining barrier that they typically see to their left. 9.2.4. Negative mobility and anomalous hysteresis We now come to the response of the steady state current x ˙ when an additional external force F is added on the right-hand side of (9.7). After making the replacement V (x) → V (x) − xF ;

(9.31)

the entire analysis from Sections 9.2.1–9.2.3 can be repeated basically unchanged. For small F in combination with Q¡Qc and large K we can then infer from (9.17), (9.19), (9.27), (9.29) after some calculations that ˆ1 K (9.32) + O(F 3 ) : x ˙ = FLN Kˆ 0 + [V (0)]2 [Qc − Q] Thus, for suCciently large, negative Kˆ 1 , a negative zero-bias mobility (also called absolute negative mobility) is predicted 147 [524,726]. A numerical example for this remarkable behavior is shown in 147

Note that such a current x ˙ opposite to the applied force F is not in contradiction with any kind of “stability criteria”, cf. the discussion below (2.39).

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Fig. 9.2 (solid line). Apparently, the e5ect of pulling the particles to one side is analogous to that of a spontaneous symmetry breaking: it generates an e5ective, coupling-induced ratchet dynamics (9.21) in which the non-equilibrium =uctuations promote a current opposite to F. Upon approaching the phase boundary, the linear response of P(x) to variations of F diverges, hence the denominator Qc − Q in (9.32) and the very steep response curve in Fig. 9.2. We remark that for networks with dead-ends (see [416] and further references therein) and in the ratchet works [116,417,422,423], a negative di1erential mobility (far away from F = 0) has been reported, but not a current opposite to the applied force as in Fig. 9.2. Further, as illustrated by Fig. 2.5, in the current versus force characteristics for “standard”, non-interacting ratchet models, a current opposite to the applied force is possible as well. However, as discussed in Section 2.6.2, the ratchet e5ect is characterized by a current x ˙ which is non-zero for F = 0 and does not change its direction within an entire neighborhood of F = 0. Accordingly, it inevitably involves some kind of symmetry breaking (for F = 0), cf. Section 3.2. In contrast, according to the characteristics of negative zero-bias mobility exempliFed in Fig. 9.2 the current x ˙ is always opposite to the (not too large) force F, independently of whether F is positive or negative. Furthermore, the symmetry of the system (for F = 0) is neither externally, nor intrinsically, nor spontaneously broken. In other words, the negative zero-bias mobility and the ratchet e5ect exhibit some striking similarities but also some fundamental di5erences. We also mention that so-called absolute negative conductance has been theoretically and experimentally studied in detail in the context of semiconductor devices [698,732,747–755], photovoltaic e5ects in ruby crystals [29,756 –758], tunnel junctions between superconductors with unequal energy gaps [759 –761], and has been theoretically predicted for certain ionized gas mixtures [762–764]. While these e5ects are in fact basically identical to negative zero-bias mobility, their origin is of a genuine quantum mechanical character which does not leave room for any kind of classical counterpart. 148 For more general F- and Q-values but still large K, the qualitative dependence of x ˙ on F follows from (9.29) by observing how moves in the adiabatically changing potential UW (x) from (9.18), (9.17). In this way, not only the continuation of the zero-bias negative conductance beyond F 0 in Fig. 9.2 can be readily understood, but also its even more spectacular counterpart when Q¿Qc , namely an anomalous hysteresis-loop [524,726], see Fig. 9.3. Its striking di5erence in comparison with a “normal” hysteresis-cycle, as observed, e.g. in a ferromagnet or in the JMulicher–Prost model [550], is as follows: Given a spontaneous current in one or the other direction, we can apply a small additional force F in the same direction, with the expected result of an increased current in that direction. But upon further increasing F, the current will, all of a sudden, switch its direction and run opposite to the applied force. In short, the anomalous response curves in Figs. 9.2 and 9.3 are basically the result of a competition between the e5ect of the bias F, favoring a current in that direction, and the ratchet-e5ect, which arises as a collective property and pumps particles in the opposite direction for Kˆ 1 ¡0. The st coexistence of two solutions Pˆ (x) over a certain F-interval when Q¿Qc gives rise to the hysteresis, and the destabilization of one of them to the jumps of x ˙ in Fig. 9.3.

148

Moreover, in the last three examples spatial periodicity is either not crucial or absent and in the case of tunnel junctions the spatial symmetry is intrinsically broken.

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231

0.1 0.15 0.1 0.

0.05

.

<x>

<x>

0.05

.

0

0

-0.05

-0.05 -0.1 -0.15

-0.1 -0.8

-0.4

0

0.4

0.8

F

-1.2

-0.8

-0.4

0

0 0.4 .4

0.8 0. 8

1 1.2 .2

F

Fig. 9.2. Solid line: Steady state current x ˙ versus force F for the model (9.7) – (9.10), (9.23), (9.31) with T = 2; Q = 2; K = 8; A = 0:15 in the thermodynamic limit N → ∞ by solving the non-linear Fokker–Planck equation (9.13). Interconnected dots: Simulations of (9.7) with nearest neighbor instead of global coupling (9.33) for a 64 ∗ 64 square lattice with periodic boundary conditions and modiFed parameters Q = 6; K = 15, averaged over 10 realizations. Fig. 9.3. Same as in Fig. 9.2 but for Q = 4; K = 10 (global coupling) and Q = 10; K = 20 (nearest-neighbor coupling).

9.2.5. Perspectives In this section we brie=y discuss some generalizations and potential applications of our above considerations. A Frst natural modiFcation of the model (9.7) consist in replacing the global coupling in by a nearest-neighbor coupling in d dimensions, i.e. N

K K sin(xj (t) − xi (t)) → sin(xj (t) − xi (t)) ; N j=1 2d

(9.33)

ij

by associating the indices i with the vertices of some d-dimensional lattice with periodic boundary conditions. As Figs. 9.2 and 9.3 demonstrate, e.g. for a square lattice (d = 2), the same qualitative phenomena as for global coupling are recovered, though the quantitative details are of course di5erent. Further generalizations [726] are: (i) The bare potential, represented by the “1” in the Frst term on the right-hand side of (9.7) plays a very minor role; even without this term all results remain qualitatively unchanged. Similarly, the thermal noise strength T is arbitrary, except that it must not vanish in the present model, but may even vanish in a somewhat modiFed setup [727]. (ii) A strictly periodic interaction K(x) is not necessary. For instance, one may add on top of the periodic an (not too strong) attractive interaction such as to keep the “cloud” of particles xi in (9.7) always well clustered.

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Closely related studies on nonequilibrium phase transitions [765,766] suggest that also periodic instead of stochastic drivings fi (t) in (9.7) will lead to qualitatively similar results, see also [767]. Furthermore, a =uctuating force or rocking ratchet scheme instead of a =uctuating potential model, amounting in (9.7) to a substitution − V (xi (t)) [1 + fi (t)] → −V (xi (t)) + yi (t) ;

(9.34)

can apparently be employed as well [551]. Especially, the characteristic time scale of the driving fi (t) in (9.7) [728] and of yi (t) in (9.34) [551] may become asymptotically large. It might appear [64,738,768] that taking the overdamped limit m → 0 in (2.1) before the white noise limit & → 0 of the Ornstein–Uhlenbeck noise fi (t) in (9.7), (9.10) (see also (4.15) and (4.16)) is an indispensable prerequisite for spontaneous symmetry breaking and spontaneous current, since only in this way [99,291] a white noise fi (t) in the sense of Stratonovich and a concomitant noise induced drift term can arise. Our detailed analysis, however, reveals that the same phenomena can in fact still be encountered even if the white noise limit & → 0 is performed prior to m → 0, see also [769]. In other words, 8nite inertia terms are also admissible on the left hand side of (9.7). While a spontaneous ergodicity-breaking with all its above-discussed consequences is clearly possible only in the thermodynamic limit N → ∞, the same a priori restriction does not hold for the phenomenon of negative zero-bias mobility. Indeed, a stylized, spatially discretized descendant of the above-discussed working model (9.7) with negative mobility for N ¿ 4 has been presented in [730]. A di5erent, experimentally realistic single particle system (N = 1) in two dimensions with negative mobility has been introduced in [770], while a game theoretic counterpart of the e5ect (cf. Section 6.7) is due to [771]. In conclusion, the above-revealed main phenomena seem to be rather robust against modiFcations and extensions of the considered model (9.7). Much like in equilibrium phase transitions, such an extremely simple model is thus expected to be of interest for a variety of di5erent systems, corresponding to a “normal form” description that subsists after the irrelevant terms have been eliminated. Models of this type may be of relevance not only in the context of molecular motors (see Section 7.4), but also for coupled phase oscillators [624,721,722], active rotator systems [772], charge density waves [773], and many other physical, chemical, and biological systems [622,774 – 776]. For instance, one may also look at (9.7) as a planar XY-spin-model [777] exposed to a strong (but incoherent) electromagnetic irradiation [13,28,33,284 –286,340,779], with the various e5ects of the photon-impacts (scattering, excitations of the host-crystal ions, etc.) roughly described by the non-equilibrium =uctuations fi (t). An experimental realization in a granular gas system is presently under construction. 10. Conclusions The central theme of our review are transport phenomena in spatially periodic “Brownian motors” or “ratchet systems” induced by unbiased perturbations of the thermal equilibrium. Letting aside variations and extensions like di5usive transport, quenched spatial disorder, or questions of eCciency, our extensive discussions may be summarized under three main categories: (i) Understanding and predicting the “ratchet e5ect” per se, i.e. the occurrence (or not) of a directed average long-time

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current x. ˙ (ii) Exploring qualitative features of the current as a function of various parameters, for example the sign of the current and the possible appearance of current inversions, monotonic versus non-monotonic “resonance-like” behavior with some type of “optimum”, or the asymptotic behavior for fast, slow, and weak perturbations, etc. (iii) On the one hand, identifying particularly simple or counterintuitive “minimal models” and very general “normal forms” exhibiting a ratchet e5ect and=or current inversions. On the other hand, elaborating realistic models and their quantitative features with some speciFc experimental situation in mind. For several of these questions, symmetry considerations play an important role. This is so basically due to Curie’s principle, stating that in the absence of prohibiting “systematic” symmetries, the appearance of a certain phenomenon (here: the ratchet e5ect) will be the rule, while its absence will be the exception. In our case, there are three such “systematic” symmetry conditions, each of which is suCcient to rule out the appearance of a ratchet e5ect: (1) Detailed balance symmetry, implying that we are dealing with an equilibrium system and that a thermal equilibrium state will thus be approached in the long time limit. (2) (Spatial) symmetry as detailed in Section 3.2. (3) Supersymmetry as detailed in Section 3.5 in the overdamped limit and its counterpart (5.35) in the underdamped (deterministic Hamiltonian) limit. Closely related to these symmetry conditions, there are in addition a couple of “systematic” no go theorems for certain classes of ratchet systems, see at the beginning of Section 4.3, at the end of Section 4.4.2, and in Section 6.4.1. If all three above systematic symmetry conditions are violated, then a vanishing current is the exception, which may be termed an “accidental symmetry”, and which is usually connected with a current inversion. A very general method of tailoring such current inversions has been elaborated in Section 3.6 together with a very simple and in fact obvious necessary and suCcient condition for their existence. Our ratchet classiFcation scheme from Section 3.3 is mainly based on the speciFc manner in which the second of the above systematic symmetries is broken. Depending on whether current inversions exist or not, we may speak of a “non-trivial” and an “obvious” ratchet e5ect, exempliFed by =uctuating potential and tilting ratchets and by (proper) traveling potential ratchets, respectively. In the Frst case, the direction of the current is obvious in some simple cases, but not at all in general, while in the second it is always rather clear. We remark that for both, “systematic” and “accidental” symmetries, the result x ˙ = 0 is unstable against completely general, generic variations of the model, while the property x ˙ = 0 is robust against such variations, i.e. “a Fnite current is the rule”. The only di5erence is that for “systematic” symmetries, the hyperspace of parameters with x=0 ˙ (and thus the deFnition of the symmetry itself) can be easily expressed in terms of “natural” model parameters, while for “accidental” symmetries such a hyperspace exists as well but is very diCcult to characterize. In this sense, there are actually no “accidental” symmetries, they are only very diCcult to deFne and therefore “overlooked” within any “natural” invariance-considerations of the problem. We note that the above symmetries (1)–(3) refer strictly speaking to the (asymptotic) state and not to the system itself. Since the thermodynamic limit of inFnitely many interacting subsystems may not commute with the long-time limit, and so an asymmetry of the initial condition may never disappear, some symmetry property of the system dynamics alone does not yet imply the corresponding asymptotic symmetry of the state (solution) in the case of extended systems. While this implication is still correct (leaving aside glass-like systems) for the Frst of the above-mentioned symmetries (an equilibrium system implies an asymptotic equilibrium state), it may be incorrect in the second case of (spatial) symmetry: Even in a perfectly symmetric system, a spontaneous

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symmetry breaking of the asymptotic state may occur, leading to a spontaneous ratchet e5ect, see Section 9. Regarding future perspectives of the Feld, the fact that many of the above symmetry considerations became clear only very recently suggest that further new theoretical results on a very basic conceptual level may still be discovered. If a speciFc direction has to be named then the still rather fresh topic of coupled Brownian motors appears to be a particularly promising candidate, both theoretically (Section 9) and with respect to biological applications (Section 7.4). Further, there is a remarkably large and rapidly increasing number of exciting experimental studies, some of them with promising perspectives regarding technological applications. Whether Brownian motors o5er just a new view or an entirely new paradigm with respect to the modeling of molecular motors (Section 7) remains to be seen as well. Acknowledgements The present review would not have been possible without the continuous support by P. HManggi and his group and the very fruitful collaborations with C. Van den Broeck, R. Kawai, R. Bartussek, R. HMaussler, M. Schreier, E. Pollak, C. Kettner, F. MMuller, T.C. Elston, B. Lindner, L. SchimanskyGeier, I. Bena, M. Nagaoka, M. Grifoni, G.J. Schmid, and J. Lehmann. In preparing this work, I have furthermore proFted a lot from the scientiFc interaction with I. Goychuk, J. Luczka, P. Talkner, R.D. Astumian, M. Bier, J. Prost, F. JMulicher, C.R. Doering, P. Jung, G.-L. Ingold, H. Linke, D. Koelle, S. Weiss, I.M. Sokolov, T. Dittrich, R. Ketzmerick, H. Schanz, S. Flach, O. Yevtushenko, A. Lorke, C. Mennerat-Robilliard, C.M. Arizmendi, F. Sols, I. Zapata, T. T]el, M. Rubi, A. PerezMadrid, M. Thorwart, R. Eichhorn, and J.M.R. Parrondo. Special thanks is due to P. HManggi and I. Goychuk, for providing Refs. [1] and [670], respectively, to J.M.R. Parrondo and H. Linke for providing Figs. 2.1, 6.1 and 8.4, respectively, and to N.G. van Kampen for providing a copy of his thesis [668]. Financial support by the DFG-Sachbeihilfe HA1517=13-2,13-4 and the Graduiertenkolleg GRK283 is gratefully acknowledged. Appendix A. Supplementary material regarding Section 2.1.1 We have modeled the two e5ects of the environment on the right-hand side of Eq. (2.1) phenomenologically, and we will discuss in the next three subsections the rather far reaching implications of this speciFc phenomenological ansatz. Especially, we will argue that the assumptions of the environment being at thermal equilibrium and of a dissipation mechanism of the form − x(t) ˙ completely Fx the statistical properties of the additive =uctuations (t) in (2.1). While our line of reasoning will be conducted on a heuristic physical level, it still captures the essential ideas of mathematically more sophisticated and rigorous approaches [66,77–97], see also Sections 3:4:1, 6:4:3, and 8:1. A.1. Gaussian white noise The fact that the friction force on the right-hand side of (2.1) is linear in x(t), ˙ i.e. no spatial direction is preferred, suggests that—due to their common origin—also the thermal =uctuations are

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unbiased, that is (cf. (2.4)) (t) = 0

(A.1)

for all times t, where · indicates the average over independent realizations of the random process (t). Similarly, the fact that the friction force only depends on the present state of the system and not on what happened in the past has its counterpart in the assumption that the random =uctuations are uncorrelated in time, i.e. (t)(s) = 0

if t = s :

(A.2)

Furthermore, the fact that the friction involves no explicit time dependence has its correspondence in the time-translation invariance of all statistical properties of the =uctuations, i.e. the noise (t) is a stationary random process. Finally, the fact that the friction force acts permanently in time indicates that the same will be the case for the =uctuations. In other words, a noise (t) exhibiting rare but relatively strong “kicks”, caused e.g. by impacts of single molecules in a diluted gas, is excluded. Technically speaking, one says that (t) cannot contain a shot noise component [178,323–326]. During a small time interval, the e5ect of the environment thus consists of a large number of small and, according to (A.2) practically independent, contributions. Due to the central limit theorem 149 the net e5ect of all these contributions on the particle x(t) will thus be Gaussian distributed. Such a Gaussian random process which is unbiased (A.1) and uncorrelated in time (A.2) is called Gaussian white noise. A.2. Fluctuation–dissipation relation A crucial implicit assumption in (2.1) is the independence of the friction force, and hence also of the =uctuation force, from the system x(t), i.e. 150 (t)x(s) = 0 for all times 151 t ¿ s. It re=ects the assumption that the environment is given by a bath so that its properties are practically not in=uenced by the behavior of the “small” Especially, the statistical properties of the =uctuations will not depend on the choice of V (x) and we may set V (x) ≡ 0 in the following. One readily veriFes that in this case of motion (2.1) is solved by 1 t −( =m)(t −t ) −( =m)(t −t0 ) + dt e (t ) : x(t) ˙ = x(t ˙ 0 )e m t0

(A.3) “huge” heat system x(t). the potential the equation (A.4)

149 In its simplest version—suCcient for our present purposes—the central limit theorem [100] states that if r1 ; : : : ; rN are independent, identically distributed random variables with zero mean and unit variance then the sum N −1=2 [r1 + · · · + rN ] converges for N → ∞ towards a Gaussian random variable of zero mean and unit variance. 150 We remark that m¿0 (cf. (2.1)) is understood in (A.3). Properties (A.2) and (A.3) lead for m → 0 to a Gaussian white noise (t) in the so-called Ito-sense [99,101]. 151 The case t¡s is somewhat subtle and not needed in the following.

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Choosing as initial time t0 = −∞ it follows that t t 1 2 x˙ (t) = 2 dt dt e−( =m)(2t −t −t ) (t )(t ) : m −∞ −∞

(A.5)

In view of (A.2), the integrand only contributes if t = t and the upper limit t in the second integral can be furthermore extended to +∞, i.e. t ∞ 1 x˙2 (t) = 2 dt e−(2 =m)(t −t ) dt (t )(t ) : (A.6) m −∞ −∞ Since the statistical properties of the =uctuations (t) are time-translation invariant, the second integral has the same value for all times t and we can conclude that ∞ ds(t)(s) = 2 mx˙2 (t) (A.7) −∞

for all times t. The left hand side of this equation is called the intensity of the noise (t) or the noise strength. At this point, we make use of the fact that the environment is a heat bath at thermal equilibrium with temperature T . Since we have chosen as initial time t0 = −∞, all transients have died out and the particle is in thermal equilibrium with the bath, satisfying the equipartition principle (for a one-dimensional dynamics) m 2 1 x˙ (t) = kB T ; 2 2

(A.8)

where kB is Boltzmann’s constant. Collecting (A.2), (A.7), (A.8) we obtain the so-called 9uctuation– dissipation relation [79 –81] (cf. (2.5)) (t)(s) = 2 kB T(t − s) ;

(A.9)

where (t) is Dirac’s delta function. In other words, (t) is a Gaussian white noise of intensity 2 kB T . Note that since (t) is a Gaussian random process, all its statistical properties are completely determined [99 –101] already by the mean value (A.1) and the correlation (A.9). A.3. Einstein relation In the absence of the potential V (x) in (2.1), we know that the particle exhibits a free thermal di1usion in one dimension with a di5usion constant D, i.e. for asymptotically large times t we have that 152 x2 (t) = 2Dt :

152

(A.10)

Corrections of order o(t) are omitted in (A.10) and we will tacitly assume that their time derivative approaches zero for t → ∞ [67]. Furthermore, we note that this asymptotic result (A.10) is independent of the initial condition x(0).

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On the other hand, upon multiplying Eq. (2.1) by x(t), averaging, and exploiting (A.3), we obtain mx(t)x(t) M = − x(t)x(t) ˙ :

(A.11)

The left-hand side of this equation can be rewritten as mx(t)x(t) M =m

d x(t)x(t) ˙ − mx˙2 (t) : dt

(A.12)

By di5erentiating (A.10) we have (for large t) that x(t)x(t) ˙ = D and hence dx(t)x(t)=dt ˙ = 0. Observing (A.8) we Fnally obtain from (A.11), (A.12) the so-called Einstein relation [77] (cf. (2.10)) D = kB T= :

(A.13)

Its most remarkable feature is that the di5usion in (A.10) does not depend on the mass m of the particle x(t) for asymptotically large times t. A.4. Dimensionless units and overdamped dynamics The objective of this section is to recast the stochastic dynamics (2.1), (A.9) into a dimensionless form, useful for qualitative theoretical considerations and indispensable for a numerical implementation. We start with deFning the barrier height UV := max{V (x)} − min{V (x)} ; x

x

(A.14)

between adjacent local minima of the periodic potential V (x). Next, we introduce for the threeˆ and UVˆ , which for the dimensionful quantities , L, and UV dimensionless counterparts , ˆ L, moment can still be freely chosen. With the deFnitions of the dimensionless quantities tˆ := 0t;

2 UV ˆLˆ ; 0 := 2

L UVˆ

(A.15)

Lˆ x( ˆ tˆ) := x(tˆ=0) ; L

(A.16)

UVˆ ˆ ; Vˆ (x) ˆ := V (xL= ˆ L) UV

(A.17)

we can rewrite (2.1) in the dimensionless form mˆ

d 2 x( ˆ tˆ) d Vˆ (x( ˆ tˆ)) d x( ˆ tˆ) ˆ tˆ) ; + ( =− + ˆ 2 d xˆ d tˆ d tˆ

(A.18)

ˆ tˆ) is a dimensionless Gaussian white noise with correlation where ( ˆ tˆ)( ˆ s) ˆ : ( ˆ = 2 ˆkˆB Tˆ (tˆ − s)

(A.19)

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Furthermore, the dimensionless mass in (A.18) is deFned as 2

mˆ := m

UV Lˆ ˆ2

2 L2 UVˆ

(A.20)

and the dimensionless temperature in (A.19) as kB T UVˆ ; Tˆ := UV kˆB

(A.21)

where kˆB may be chosen arbitrarily, e.g. kˆB = 1. ˆ UVˆ , and kˆB all equal to unity. For the typically very small systems one Next, we choose , ˆ L, has in mind, and for which thermal =uctuations play any notable role at all, the rescaled mass (A.20) then often turns out to be smaller than unity by many orders of magnitude, see e.g. in [288], while the dimensionless temperature (A.21) is of order unity or smaller. 153 On the other hand, the period Lˆ and the barrier height UVˆ of the potential Vˆ (x) ˆ are both unity, so the derivative of this potential is typically of order unity as well. It is therefore quite plausible that in (A.18) the inertia 2 term mˆ d 2 x( ˆ tˆ)=d tˆ can be dropped in very good approximation. Admittedly, from a mathematical viewpoint, dropping the highest order derivative in a di5erential equation, especially in the presence ˆ tˆ), may rise some concerns. A more careful of such an elusive object as the Gaussian white noise ( treatment of this problem has been worked out e.g. in [291,463,465,466,469,482,483] with the same conclusion as along our simple heuristic argument. We Fnally note that letting m → 0 a5ects neither the =uctuation–dissipation relation (A.9) nor Einstein’s relation (A.13). Finally, we turn to the typical case that mˆ is known to be a small quantity and we thus can set formally m=0 in (2.1). We thus recover the “minimal” Smoluchowski–Feynman ratchet model from (2.6). Introducing dimensionless units like before, one arrives again at (A.18) but now with mˆ = 0 ˆ and UVˆ may still be chosen arbitrarily. However, in right from the beginning. In principle, , ˆ L, most concrete cases it is convenient to assume that Lˆ and UVˆ are of order unity, but not necessary equal to 1 (e.g. Lˆ = 2 or Vˆ 0 = 1 in (2.3) may sometimes be a more convenient choice), while ˆ may still be a variable “control parameter” of the model. The implication of a dimensionless solution x( ˆ tˆ) for the original, dimensionful system x(t) is obvious. Especially, varying one parameter (e.g. ˆ or Tˆ ) and keeping the others Fxed, corresponds to exactly the same parameter-variation in the dimensionful system. We Fnally remark that in the end one usually drops again the “hat” symbols of the dimensionless quantities. Depending on the context, equation (2.6) may thus represent either the dimensionful or the dimensionless version of the model.

153

In the opposite case, i.e. if UV=kB T is a small quantity (especially if V (x) = const:) one has to replace UV by kB T in the deFnition (A.15) of 0, and similarly in (A.17), (A.20). On condition that m kB T= 2 L2 is small, one can then drop the inertia term. The condition for arbitrary UV=kB T is thus that the dimensionless quantity m −2 L−2 max{UV; kB T } has to be a small quantity.

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Appendix B. Alternative derivation of the Fokker–Planck equation In this appendix we give a derivation of the Fokker–Planck equation (2.14) by considering the corresponding overdamped stochastic dynamics (2.6) as limiting case of the discretized dynamics (2.7) when Ut → 0. To simplify notation, we use dimensionless units (see below Eq. (2.6)) with kB = = 1. Next we recall that n in (2.7) are independent, Gaussian distributed random variables with n = 0 and 2n = 2T=Ut (see (2.8)). It follows that for a particle (2.7), the conditional probability P(x|y) to start out at time t = tn = nUt from the point x n = y and to arrive one time step Ut later at the point x n+1 = x is Gaussian distributed about x = y − UtV (y) with variance (Utn )2 = 2T Ut, i.e. [x − y + UtV (y)]2 −1=2 : (B.1) exp − P(x|y) = (4 T Ut) 4T Ut Furthermore, the probability distribution P(x; t + Ut) at time t + Ut is obviously related to that at time t through the so-called Chapman–Kolmogorov equation [100] ∞ P(x; t + Ut) = dy P(x|y)P(y; t) : (B.2) −∞

√ After a change of the integration variable according to z = (x − y)= Ut we obtain √ √ ∞ √ dz [z + Ut V (x − Ut z)]2 P(x − Ut z; t) : exp − P(x; t + Ut) = 1=2 4T −∞ (4 T )

(B.3)

Under the assumption that P(x; √ t) behaves suCciently well as Ut → 0, we can expand the right-hand side of (B.3) in powers of Ut and perform the remaining Gaussian integrals, with the result 9 92 {V (x)P(x; t)} + UtT 2 P(x; t) + o(Ut) : (B.4) 9x 9x √ In particular, there is no contribution proportional to Ut. In the limit Ut → 0, the Fokker–Planck equation (2.14) now readily follows. P(x; t + Ut) = P(x; t) + Ut

Appendix C. Perturbation analysis In this appendix we solve the Fokker–Planck equation (2.52) perturbatively for small time-periods T in (2.48) and zero load F = 0. We recall that for evaluating the particle current (2.53) we can ˆ t) which are L-periodic in space and T-periodic in time and that focus on probability densities P(x; ˆ the function T (h) from (2.56) is assumed to be T-independent. The latter assumption suggests to introduce ˆ hT) ; WT (x; h) := P(x;

(C.1)

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so that the Fokker–Planck equation (2.52) takes the form 9 V (x) kB Tˆ (h) 92 9 WT (x; h) = T WT (x; h) + WT (x; h) : 9h 9x

9x2

(C.2)

The small quantity T on the right-hand side of this equation furthermore suggest a power series ansatz WT (x; h) =

∞

Tn Wn (x; h)

(C.3)

n=0

ˆ t) one readily with T-independent functions Wn (x; h). From the periodicity and normalization of P(x; Fnds that Wn (x + L; h) = Wn (x; h + 1) = Wn (x; h) ; 0

L

d x Wn (x; h) = n; 0

(C.4) (C.5)

for n ¿ 0, where i; j is the Kronecker delta. Next the usual perturbation analysis argument is invoked: Introducing the ansatz (C.3) into the Fokker–Planck equation (C.2) and observing that this equation is supposed to hold for arbitrary T it follows that the coeCcients of each power of T must be equal to zero separately. In the lowest order T0 it follows that 9 W0 (x; h) = 0 ; (C.6) 9h i.e. W0 (x; h) is equal to a h-independent but otherwise still unknown function W0 (x). By introducing this function into (C.3), equating order T1 -terms, and averaging over one time period T one obtains kB TW 92 9 V (x) W0 (x) + W0 (x) ; (C.7) 0= 9x

9x2 where the time-averaged temperature TW is deFned in (2.57). This ordinary second order equation for W0 (x) can now be readily solved, with the two emerging integration constants being determined by the periodicity and normalization conditions (C.4), (C.5). The result is W

W0 (x) = Z −1 e−V (x)=kB T ; Z :=

0

L

W

d x e−V (x)=kB T

(C.8) (C.9)

and the corresponding contribution of order T0 to the particle current (2.53) is found to vanish. In other words, we have recovered in the limit T → 0 the same results as for a constant, time-averaged temperature TW in Section 2.4, in accordance with what one may have expected. Proceeding in exactly the same way up to the next order T1 still gives a zero contribution to the particle current. It is only in the second order T2 that the Frst non-trivial contribution (2.58) is encountered.

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Physics Reports 361 (2002) 267–417

Singular or non-Fermi liquids C.M. Varmaa; b;1 , Z. Nussinovb , Wim van Saarloosb; ∗ b

a Bell Laboratories, Lucent Technologies, Murray Hill, NJ 07974, USA Universiteit Leiden, Instituut-Lorentz, Postbus 9506, 2300 RA Leiden, Netherlands

Received June 2001; editor: C:W:J: Beenakker

Contents 1. Introduction 1.1. Aim and scope of this paper 1.2. Outline of the paper 2. Landau’s Fermi liquid 2.1. Essentials of Landau Fermi liquids 2.2. Landau Fermi liquid and the wavefunction renormalization Z 2.3. Understanding microscopically why Fermi-liquid theory works 2.4. Principles of the microscopic derivation of Landau theory 2.5. Modern derivations 2.6. Routes to breakdown of Landau theory 3. Local Fermi liquids and local singular Fermi liquids 3.1. The Kondo problem 3.2. Fermi-liquid phenomenology for the Kondo problem 3.3. Ferromagnetic Kondo problem and the anisotropic Kondo problem 3.4. Orthogonality catastrophe 3.5. X-ray edge singularities

269 269 272 273 273 275 280 286 290 291 295 296 299 300 300 301

3.6. A spinless model with Anite range interactions 3.7. A model for mixed-valence impurities 3.8. Multichannel Kondo problem 3.9. The two-Kondo-impurities problem 4. SFL behavior for interacting fermions in one dimension 4.1. The one-dimensional electron gas 4.2. The Tomonaga–Luttinger model 4.3. Thermodynamics 4.4. One-particle spectral functions 4.5. Correlation functions 4.6. The Luther–Emery model 4.7. Spin–charge separation 4.8. Spin–charge separation in more than one dimension? 4.9. Recoil and the orthogonality catastrophe in one dimension and higher 4.10. Coupled one-dimensional chains 4.11. Experimental observations of onedimensional Luttinger liquid behavior 5. Singular Fermi-liquid behavior due to gauge Aelds

∗

Corresponding author. Tel.: +31-71-5275501; fax: +31-71-5275511. E-mail address: [email protected] (W. van Saarloos). 1 Present and permanent address: Bell Laboratories, Lucent Technologies, Murray Hill, NJ 07974, USA.

c 2002 Elsevier Science B.V. All rights reserved. 0370-1573/02/$ - see front matter PII: S 0 3 7 0 - 1 5 7 3 ( 0 1 ) 0 0 0 6 0 - 6

302 305 306 310 314 315 320 321 322 324 327 328 330 332 335 335 338

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5.1. SFL behavior due to coupling to the electromagnetic Aeld 5.2. Generalized gauge theories 6. Quantum critical points in fermionic systems 6.1. Quantum critical points in ferromagnets, antiferromagnets, and charge density waves 6.2. Quantum critical scaling 6.3. Experimental examples of SFL due to quantum criticality: open theoretical problems 6.4. Special complications in heavy fermion physics 6.5. EKects of impurities on quantum critical points 7. The high-Tc problem in the copper-oxide-based compounds 7.1. Some basic features of the high-Tc materials 7.2. Marginal Fermi liquid behavior of the normal state

338 341 343 343 344 353 359 360 361 361

7.3. General requirements in a microscopic theory 7.4. Microscopic theory 8. The metallic state in two dimensions 8.1. The two-dimensional electron gas 8.2 Non-interacting disordered electrons: scaling theory of localization 8.3. Interactions in disordered electrons 8.4. Finkelstein theory 8.5. Compressibility, screening length and a mechanism for metal–insulator transition 8.6. Experiments 8.7. Discussion of the experiments in light of the theory of interacting disordered electrons 8.8. Phase diagram and concluding remarks Acknowledgements References

372 373 379 380 381 385 389 391 392 400 404 406 406

364

Abstract An introductory survey of the theoretical ideas and calculations and the experimental results which depart from Landau Fermi liquids is presented. The common themes and possible routes to the singularities leading to the breakdown of Landau Fermi liquids are categorized following an elementary discussion of the theory. Soluble examples of singular or non-Fermi liquids include models of impurities in metals with special symmetries and one-dimensional interacting fermions. A review of these is followed by a discussion of singular Fermi liquids in a wide variety of experimental situations and theoretical models. These include the eKects of low-energy collective Muctuations, gauge Aelds due either to symmetries in the Hamiltonian or possible dynamically generated symmetries, Muctuations around quantum critical points, the normal state of high-temperature superconductors and the two-dimensional metallic state. For the last three systems, the principal experimental results are summarized and the outstanding theoretical c 2002 Elsevier Science B.V. All rights reserved. issues are highlighted. PACS: 7.10.Ay; 71.10.Hf; 71.10.Pm; 71.27.+a

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1. Introduction 1.1. Aim and scope of this paper In the last two decades a variety of metals have been discovered which display thermodynamic and transport properties at low temperatures which are fundamentally diKerent from those of the usual metallic systems which are well described by the Landau Fermi-liquid theory. They have often been referred to as Non-Fermi liquids. A fundamental characteristic of such systems is that the low-energy properties in a wide range of their phase diagram are dominated by singularities as a function of energy and temperature. Since these problems still relate to a liquid state of fermions and since it is not a good practice to name things after what they are not, we prefer to call them singular Fermi liquids (SFL). The basic notions of Fermi-liquid theory have actually been with us at an intuitive level since the time of Sommerfeld: He showed that the linear low-temperature speciAc heat behavior of metals as well as their asymptotic low-temperature resisitivity and optical conductivity could be understood by assuming that the electrons in a metal could be thought of as a gas of non-interacting fermions, i.e., in terms of quantum mechanical particles which do not have any direct interaction but which do obey Fermi statistics. Meanwhile, Pauli calculated that the paramagnetic susceptibility of non-interacting electrons is independent of temperature, also in accord with experiments in metals. At the same time it was understood, at least since the work of Bloch and Wigner, that the interaction energies of the electrons in the metallic range of densities are not small compared to the kinetic energy. The rationalization for the qualitative success of the non-interacting model was provided in a masterly pair of papers by Landau [152,153] who initially was concerned with the properties of liquid 3 He. This work epitomized a new way of thinking about the properties of interacting systems which is a cornerstone of our understanding of condensed matter physics. The notion of quasiparticles and elementary excitations and the methodology of asking useful questions about the low-energy excitations of the system based on concepts of symmetry, without worrying about the myriad unnecessary details, is epitomized in Landau’s phenomenological theory of Fermi liquids. The microscopic derivation of the theory was also soon developed. Our perspective on Fermi liquids has changed signiAcantly in the last two decades or so. This is due both to changes in our theoretical perspective, and due to the experimental developments: on the experimental side, new materials have been found which exhibit Fermi-liquid behavior in the temperature dependence of their low-temperature properties with the coePcients often a factor of order 103 diKerent from the non-interacting electron values. These observations dramatically illustrate the power and range of validity of the Fermi-liquid ideas. On the other hand, new materials have been discovered whose properties are qualitatively different from the predictions of Fermi-liquid theory (FLT). The most prominently discussed of these materials are the normal phase of high-temperature superconducting materials for a range of compositions near their highest Tc . Almost every idea discussed in this review has been used to understand the high-Tc problem, but there is no consensus yet on the solution. It has of course been known for a long time that FLT breaks down in the Muctuation regime of classical phase transitions. This breakdown occurs in a more substantial region of the phase

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Fig. 1. Schematic phase diagram near a quantum critical point. The parameter along the x-axis can be quite general, like the pressure or a ratio of coupling constants. Whenever the critical temperature vanishes, a QCP, indicated with a dot in the Agure, is encountered. In the vicinity of such a point quantum mechanical, zero-point Muctuations become very important. However, when Tc is Anite, critical slowing down implies that the relevant frequency scale goes as ! ∼ |Tc − T |z , dwarAng quantum eKects; the standard classical critical methodology then applies. An example of a phase diagram of this type for MnSi is shown in Fig. 34 below.

diagram around the quantum critical point (QCP) where the transition temperature tends to zero as a function of some parameter, see Fig. 1. This phenomenon has been extensively investigated for a wide variety of magnetic transitions in metals where the transition temperature can be tuned through the application of pressure or by varying the electronic density through alloying. Heavy fermions, with their close competition between states of magnetic order with localized moments and itinerant states due to Kondo eKects, appear particularly prone to such QCPs. Equally interesting are questions having to do with the change in properties due to impurities in systems which are near a QCP in the pure limit. The density–density correlations of itinerant disordered electrons at long wavelengths and low energies must have a diKusive form. In two dimensions this leads to logarithmic singularities in the eKective interactions when the interactions are treated perturbatively. The problem of Anding the ground state and low-lying excitations in this situation is unsolved. On the experimental side, the discovery of the metal–insulator transition in two dimensions and the unusual properties observed in the metallic state make this an important problem to resolve. The one-dimensional electron gas reveals logarithmic singularities in the eKective interactions even in a second-order perturbation calculation. A variety of mathematical techniques have been used to solve a whole class of interacting one-dimensional problems and one now knows the essentials of the correlation functions even in the most general case. An important issue is whether and how this knowledge can be used in higher dimensions.

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The solution of the Kondo problem and the realization that its low-temperature properties may be discussed in the language of FLT has led in turn to the formulation and solution of impurity models with singular low-energy properties. Such models have a QCP for a particular relation between the coupling constants; in some examples they exhibit a quantum critical line. The thermodynamic and transport properties around such critical points or lines are those of local singular Fermi liquids. Although the direct experimental relevance of such models (as of one-dimensional models) to experiments is often questionable, these models, being soluble, can be quite instructive in helping to understand the conditions necessary for the breakdown of FLT and associated quasiparticle concepts. The knowledge from zero-dimensional and one-dimensional problems must nevertheless be extrapolated with care. A problem which we do not discuss but which belongs in the study of SFLs is the quantum Hall eKect problem. The massive degeneracy of two-dimensional electrons in a magnetic Aeld leads to spectacular new properties and involves new fractional quantum numbers. The essentials of this problem were solved following Laughlin’s inspired variational calculation. The principal reason for the omission is Arstly that excellent papers reviewing the developments are available [213,72,111] and secondly that the methodology used in this problem is in general distinct from those for discussing the other SFLs which have a certain unity. We will however have occasions to refer to aspects of the quantum Hall eKect problem often. Especially interesting from our point of view is the weakly singular Fermi liquid behavior predicted in the = 12 quantum Hall eKect [118]. With less justiAcation, we do not discuss the problem of superconductor to insulator and=or to metal transitions in two-dimensional disordered systems in the limit of zero temperature with and without an applied magnetic Aeld. Interesting new developments in this problem with references to substantial earlier work may be found in [176,177,247]. The problem of transitions in Josephson arrays [247] is a variant of such problems. One of the principal aspects that we want to bring to the foreground in this review is the fact that SFLs all have in common some fundamental features which can be stated usefully in several diKerent ways. (i) They have degenerate ground states to within an energy of order kB T . This degeneracy is not due to static external potentials or constraints as in, for example the spin-glass problem, but degeneracies which are dynamically generated. (ii) Such degeneracies inevitably lead to a breakdown of perturbative calculations because they generate infra-red singularities in the correlation functions. (iii) If a bare particle or hole is added to the system, it is attended by a divergent number of low-energy particle–hole pairs, so that the one-to-one correspondence between the one-particle excitation of the interacting problem and those of the non-interacting problem, which is the basis for FLT, breaks down. (iv) Since SFLs are concerned with dynamically generated degeneracies within energies of order of the measuring temperature, the observed properties are determined by quantum-mechanical to classical crossovers and in particular by dissipation in such a crossover. On the theoretical side, one may now view Fermi-liquid theory as a forerunner of the renormalization group ideas. The renormalization group has led to a sophisticated understanding of singularities in the collective behavior of many-particle systems. It is likely that these methods have an important role to play in understanding the breakdown of FLT. The aim of this paper is to provide a pedagogical introduction to SFLs, focused on the essential conceptual ideas and on issues which are settled and which can be expected to

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survive future developments. Therefore, we will not attempt to give an exhaustive review of the literature on this problem or of all the experimental systems which show hints of SFL behavior. The experimental examples we discuss have been selected to illustrate both what is essentially understood and what is not understood even in principle. On the theoretical side, we will shy away from presenting in depth the sophisticated methods necessary for a detailed evaluation of correlation functions near QCP—for this we refer to the book by Sachdev [225]—or for an exact solution of local impurity models (see, e.g. [124,227,259]). Likewise, for a discussion of the application of quantum critical scaling ideas to Josephson arrays or quantum Hall eKects, we refer to the nice introduction by Sondhi et al. [247]. 1.2. Outline of the paper The outline of this paper is as follows. We start by summarizing in Section 2 some of the key features of Landau’s FLT—in doing so, we will not attempt to retrace all of the ingredients which can be found in many of the classic textbooks [208,37]; instead our discussion will be focused on those elements of the theory and the relation with its microscopic derivation that allow us to understand the possible routes by which the FLT can break down. This is followed in Section 3 by the Fermi-liquid formulation of the Kondo problem and of the SFL variants of the Kondo problem and of two interacting Kondo impurities. The intention here is to reinforce the concepts of FLT in a diKerent context as well as to provide examples of SFL behavior which oKer important insights because they are both simple and solvable. We then discuss the problem of one spatial dimension (d = 1), presenting the principal features of the solutions obtained. We discuss why d = 1 is special, and the problems encountered in extending the methods and the physics to d ¿ 1. We then move from the comforts of solvable models to the reality of the discussion of possible mechanisms for SFL behavior in higher dimensions. First we analyze in Section 5 the paradigmatic case of long-range interactions. Coulomb interactions will not do in this regard, since they are always screened in a metal, but transverse electromagnetic Aelds do give rise to long-range interactions. The fact that as a result no metal is a Fermi liquid for suPciently low temperatures was already realized long ago [127]—from a practical point of view, this mechanism is not very relevant, since the temperatures where these eKects become important are of order 10−16 K; nevertheless, conceptually this is important since it is a simple example of a gauge theory giving rise to SFL behavior. Gauge theories on lattices have been introduced to discuss problems of fermions moving with the constraint of only zero or single occupation per site. We then discuss in Section 6 the properties near a quantum critical point, taking Arst an example in which the ferromagnetic transition temperature goes to zero as a function of some externally chosen suitable parameter. We refer in this section to several experiments in heavy fermion compounds which are only partially understood or not understood even in principle. We then turn to a discussion of the marginal Fermi liquid phenomenology for the SFL state of copper-oxide high-Tc materials and discuss the requirements on a microscopic theory that the phenemenology imposes. A sketch of a microscopic derivation of the phenemenology is also given. We close the paper in Section 8 with a discussion of the metallic state in d = 2 and the state of the theory treating the diKusive singularities in d = 2 and its relation to the metal–insulator transition.

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2. Landau’s Fermi liquid 2.1. Essentials of Landau Fermi liquids The basic idea underlying Landau’s Fermi-liquid theory [152,153,208,37] is that of analyticity, i.e., that states with the same symmetry can be adiabatically connected. Simply put, this means that whether or not we can actually carry out the calculation we know that the eigenstates of the full Hamiltonian of the same symmetry can be obtained perturbatively from those of a simpler Hamiltonian. At the same time states of diKerent symmetry cannot be obtained by “continuation” from the same state. This suggests that given a tough problem which is impossible to solve, we may guess a right simple problem. The low energy and long wavelength excitations, as well as the correlation and the response functions of the impossible problem bear a one-to-one correspondence with the simpler problem in their analytic properties. This leaves Axing only numerical values. These are to be determined by parameters, the minimum number of which is Axed by the symmetries. Experiments often provide an intuition as to what the right simple problem may be: for the interacting electrons, in the metallic range of densities, it is the problem of kinetic energy of particles with Fermi statistics. (If one had started with the opposite limit, just the potential energy alone, the starting ground state is the Wigner crystal—a bad place to start thinking about a metal!) If we start with non-interacting fermions, and then turn on the interactions, the qualitative behavior of the system does not change as long as the system does not go through (or is close to) a phase transition. Owing to the analyticity, we can even consider strongly interacting systems—the low-energy excitations in these have strongly renormalized values of their parameters compared to the non-interacting problem, but their qualitative behavior is the same as that of the simpler problem. The heavy fermion problem provides an extreme example of the domain of validity of the Landau approach. This is illustrated in Fig. 2, which shows the speciAc heat of the heavy fermion compound CeAl3 . As in the Sommerfeld model, the speciAc heat is linear in the temperature at low T , but if we write Cv ≈ T at low temperatures, the value of is about a thousand times as large as one would estimate from the density of states of a typical metal, using the free electron mass. For a Fermi gas, the density of states N (0) at the Fermi energy is proportional to an eKective mass m∗ : N (0) =

m∗ kF ; 2 ˝2

(1)

with kF the Fermi wavenumber. Then the fact that the density of states at the chemical potential is a thousand times larger than for normal metals can be expressed by the statement that the eKective mass m∗ of the quasiparticles is a thousand times larger than the free electron mass m. Likewise, as Fig. 3 shows, the resistivity of CeAl3 at low temperatures increases as T 2 . This also is a characteristic sign of a Fermi liquid, in which the quasiparticle lifetime at the Fermi surface, determined by electron–electron interactions, behaves as ∼ 1=T 2 . 2 However, just as 2

In heavy fermions, at least in the observed range of temperatures, the transport lifetime determining the temperature dependence of resistivity is proportional to the single-particle lifetime.

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Fig. 2. SpeciAc heat of CeAl3 at low temperatures from Andres et al. [28]. The slope of the linear speciAc heat is about 3000 times that of the linear speciAc heat of, say, Cu. However, the high-temperature cut-oK of this linear term is smaller than that of Cu by a similar amount. The rise of the speciAc heat in a magnetic Aeld at low temperatures is the nuclear contribution, irrelevant to our discussion.

Fig. 3. Electrical resistivity of CeAl3 below 100 mK, plotted against T 2 . From Andres et al. [28].

the prefactor of the speciAc heat is a factor thousand times larger than usual, the prefactor of the T 2 term in the resistivity is a factor 106 larger—while scales linearly with the eKective mass ratio m∗ =m, the prefactor of the T 2 term in the resistivity increases for this class of Fermi liquids as (m∗ =m)2 .

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It should be remarked that the right simple problem is not always easy to guess. The right simple problem for liquid 4 He is not the non-interacting Bose gas but the weakly interacting Bose gas (i.e., the Bogoliubov problem [45,154]). The right simple problem for the Kondo problem (a low-temperature local Fermi liquid) was guessed [197] only after the numerical renormalization group solution was obtained by Wilson [289]. The right simple problem for two-dimensional interacting disordered electrons in the “metallic” range of densities (Section 8 in this paper) is at present unknown. For SFLs, the problem is diKerent: usually one is in a regime of parameters where no simple problem is a starting point—in some cases the Muctuations between solutions to diKerent simple problems determines the physical properties, while in others even this dubious anchor is lacking. 2.2. Landau Fermi liquid and the wavefunction renormalization Z Landau theory is the forerunner of our modern way of thinking about low-energy eKective Hamiltonians in complicated problems and of the renormalization group. The formal statements of Landau theory in their original form are often somewhat cryptic and mysterious—this reMects both Landau’s style and his ingenuity. We shall take a more pedestrian approach. Let us consider the essential diKerence between non-interacting fermions and an interacting Fermi liquid from a simple microscopic perspective. For free fermions, the momentum states |k are also eigenstates of the Hamiltonian with eigenvalue k =

˝2 k 2

2m

:

(2)

Moreover, the thermal distribution of particles n0k , is given by the Fermi–Dirac function where denotes the spin label. At T = 0, the distribution jumps from 1 (all states occupied within the Fermi sphere) to zero (no states occupied within the Fermi sphere) at |k| = kF and energy equal to the chemical potential . This is illustrated in Fig. 4. A good way of probing a system is to investigate the spectral function; the spectral function A(k; !) gives the distribution of energies ! in the system when a particle with momentum k is added or removed from it (remember that removing a particle excitation below the Fermi energy means that we add a hole excitation). As sketched in Fig. 5(a), for the non-interacting system, A0 (k; !) is simply a -function peak at the energy k , because all momentum states are also energy eigenstates A0 (k; !) = (! − (k − ))

for ! ¿ ;

1 1 1 = − Im = − Im G 0 (k; !) : ! − (k − ) + i

(3) (4)

Here, is small and positive; it reMects that particles or holes are introduced adiabatically, and it is taken to zero at the end of the calculation for the pure non-interacting problem. The Arst step of the second line is just a simple mathematical rewriting of the delta function. In the second line the Green’s function G 0 for non-interacting electrons is introduced. More generally

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Fig. 4. Bare-particle distribution at T = 0 for a given spin direction in a translationally invariant Fermi system with interactions (full line) and without interactions (dashed line). Note that the position of the discontinuity, i.e., the Fermi wavenumber kF , is not renormalized by interactions.

Fig. 5. (a). The non-interacting spectral function A(k; !) at Axed k as a function of !; (b) the spectral function of single-electron excitations in a Fermi liquid at Axed k as a function of !. If (1=)A(k; !) is normalized to 1, signifying one bare particle, the weight under the Lorentzian, i.e., the quasiparticle part, is Z. As explained in the text, at the same time Z is the discontinuity in Fig. 4.

the single-particle Green’s function G(k; !) is deAned in terms of the correlation function of particle creation and annihilation operators in standard textbooks [195,4,222,168]. For our present purpose, it is suPcient to note that it is related to the spectral function A(k; !), which has a clear physical meaning and which can be deduced through-angle resolved photoemission experiments ∞ A(k; x) G(k; !) = dx : (5) ! − − x + i sgn(! − ) −∞ A(k; !) thus is the spectral representation of the complex function G(k; !). Here we have deAned the so-called retarded Green’s function which is especially useful since its real and imaginary

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Fig. 6. Schematic illustration of the perturbative expansion (8) of the change of wavefunction as a result of the addition of an electron to the Fermi sea due to interactions with the particles in the Fermi sea.

parts obey the Kramers–Kronig relations. In the problem with interactions G(k; !) will differ from G 0 (k; !). This diKerence can be quite generally deAned through the single-particle self-energy function (k; !): (G(k; !))−1 = (G 0 (k; !))−1 − (k; !) :

(6)

Eq. (5) ensures the relation between G(k; !) and A(k; !) 1 A(k; !) = − Im G(k; !) :

(7)

With these preliminaries out of the way, let us consider the form of A(k; !) when we add a particle to an interacting system of fermions. Due to the interaction (assumed repulsive) between the added particle and those already in the Fermi sea, the added particle will kick particles from below the Fermi surface to above. The possible terms in a perturbative description of this process are constrained by the conservation laws of charge, particle number, momentum and spin. Those which are allowed by these conservation laws are indicated pictorially in Fig. 6, and lead to an expression of the type |

1=2 † N +1 N k = Zk ck |

+

1 V 3=2

k1 ;k2 ;k3 1 ;2 ;3

× k; k1 −k2 +k3 (; 1 ; 2 ; 3 )|

N

k1 1 k2 2 k3 3 ck† 3 ck2 ck† 1

+ ::: :

(8)

Here the ck† ’s and ck ’s are the bare particle creation and annihilation operators, and the dots indicate higher-order terms, for which two or more particle–hole pairs are created and (; 1 ; 2 ; 3 ) expresses conservation of spin under vector addition. The multiple-particle–hole pairs for a Axed total momentum can be created with a continuum of momentums of the individual bare particles and holes. Therefore, an added particle with Axed total momentum has a wide distribution of energies. However, if Zk deAned by Eq. (8) is Anite, there is a well-deAned feature in this distribution at some energy which is in general diKerent from the non-interacting value ˝2 k 2 =(2m). The spectral function in such a case will then be as illustrated in Fig. 5(b). It is useful to separate the well-deAned feature from the broad continuum by writing the spectral function

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as the sum of two terms, A(k; !) = Acoh (k; !) + Aincoh (k; !). The single-particle Green’s function can similarly be expressed as a sum of two corresponding terms, G(k; !) = Gcoh (k; !) + Gincoh (k; !). Then Zk Gcoh (k; !) = ; (9) ! − ˜k + i=k which for large lifetimes k gives a Lorentzian peak in the spectral density at the quasiparticle energy ˜k ≡ k −. The incoherent Green’s function is smooth and hence for large k corresponds to the smooth background in the spectral density. The condition for the occurrence of the well-deAned feature can be expressed as the condition that the self-energy (k; !) has an analytic expansion about ! = 0 and k = kF and that its real part is much larger than its imaginary part. One can easily see that were it not so, then expression (9) for Gcoh could not be obtained. These conditions are necessary for a Landau Fermi liquid. Upon expanding (k; !) in (12) for small ! and small deviations of k from kF and writing it in the form (9), we make the identiAcations 1 ˜k = k Zk Zˆ k ; = − Zk Im (kF ; ! = 0) ; (10) k where 1 9 9 −1 ; Zˆ = 1 + : (11) Zk = 1 − 9! !=0; k=kF vF 9k !=0; k=kF From Eq. (8), we have a more physical deAnition of Zk : Zk is the projection amplitude of N +1 onto the state with one bare particle added to the ground state, since all other terms k in the expansion vanish in the thermodynamic limit in the perturbative expression embodied by (8):

|

Zk1=2 =

N +1 † N |ck | k

:

(12)

In other words, Zk is the overlap of the ground state wavefunction of a system of interacting N ± 1 fermions of total momentum k with the wavefunction of N interacting particles and a bare particle of momentum k. Zk is called the quasiparticle amplitude. The Landau theory tacitly assumes that Zk is Anite. Furthermore, it asserts that for small ! and k close to kF , the physical properties can be calculated from quasiparticles which carry the same quantum numbers as the particles, i.e., charge, spin and momentum and which may be deAned simply by the creation operator †k; : |

N +1 = †k; | N k

:

(13)

Close to kF , and for T small compared to the Fermi energy, the distribution of the quasiparticles is assumed to be the Fermi–Dirac distribution in terms of the renormalized quasiparticle energies. The bare particle distribution is quite diKerent. As is illustrated in Fig. 4, it is depleted below kF and augmented above kF , with a discontinuity at T = 0 whose value is shown in microscopic theory to be Zk . A central result of Fermi liquid theory is that close to the Fermi energy at zero temperature, the width 1=k of the coherent quasiparticle peak is proportional to (˜k − )2 so that near the Fermi energy the lifetime is long and quasiparticles are well-deAned.

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Likewise, at the Fermi energy 1=k varies with temperature as T 2 . From the microscopic derivation of this result, it follows that the weight in this peak, Zk , becomes equal to the jump Z in nk when we approach the Fermi surface: Zk → Z for k → kF . For heavy fermions, as we already mentioned, Z can be of the order of 10−3 . However, as long as Z is non-zero, one has Fermi liquid properties for temperatures lower than about ZEF . Degeneracy is eKectively lost for temperatures much higher than ZEF and classical statistical mechanics prevails. 3 An additional result from microscopic theory is the so-called Luttinger theorem, which states that the volume enclosed by the Fermi surface does not change due to interactions [195,4]. The mathematics behind this theorem is that with the assumptions of FLT, the number of poles in the interacting Green’s function below the chemical potential is the same as that for the non-interacting Green’s function. Recall that the latter is just the number of particles in the system. Landau actually started his discussion of the Fermi liquid by writing the equation for the deviation of the (Gibbs) free energy from its ground state value as a functional of the deviation of the quasiparticle distribution function n(k; ) from the equilibrium distribution function n0 (k; ) n(k; ) = n(k; ) − n0 (k; )

(14)

as follows: G = G0 +

1 1 (˜k − )nk + f nk nk + · · · V 2V 2 kk ; k;

(15)

kk ;

Note that (˜k − ) is itself a function of n; so the Arst term contains at least a contribution of order (n)2 which makes the second term quite necessary. In principle, the unknown function fkk ; depends on spin and momenta. However, spin rotation invariance allows one to write the spin part in terms of two quantities, the symmetric and antisymmetric parts fs and fa . Moreover, for low energy and long-wavelength phenomena only momenta with k ≈ kF play a role; if we consider the simple case of 3 He where the Fermi surface is spherical, rotation invariance implies that for momenta near the Fermi momentum f can only depend on the relative angle between k and k ; this allows one to expand in Legendre polynomials Pl (x) by writing k≈k ≈kF s; a N (0)fkk → ;

∞ l=0

Fls; a Pl (kˆ · kˆ ) :

(16)

From expression (15) one can then relate the lowest order so-called Landau coePcients F0 and F1s and the eKective mass m∗ to thermodynamic quantities like the speciAc heat Cv , the compressibility %, and the susceptibility &: m∗ Cv = ; Cv0 m 3

m∗ % = (1 + F0s ) ; %0 m

m∗ & = (1 + F0a ) : &0 m

It is an unfortunate common mistake to think of the properties in this regime as SFL behavior.

(17)

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Here subscripts 0 refer to the quantities of the non-interacting reference system, and m is the mass of the fermions. For a Galilean invariant system (like 3 He), there is a simple relation between the mass enhancement and the Landau parameter F1s , and there is no renormalization of the particle current j; however, there is a renormalization of the velocity: one has Fs m∗ j = k=m; v = k=m∗ ; (18) = 1+ 1 : m 3 The transport properties are calculated by deAning a distribution function n(k; r; t) which is slowly varying in space and time and writing a Boltzmann equation for it [208,37]. It is a delightful conceit of the Landau theory that the expressions of the low-energy properties in terms of the quasiparticles in no place involve the quasiparticle amplitude Zk . In fact in a translationally invariant problem such as liquid 3 He; Zk cannot be measured by any thermodynamic or transport measurements. A masterly use of conservation laws ensures that Z’s cancel out in all physical properties (one can extract Z from measurement of the momentum distribution. By neutron scattering measurements, it is found that Z ≈ 1=4 [112] for He3 near the melting line). This is no longer true on a lattice, in the electron–phonon interaction problem [212] or in heavy fermions [265] or even more generally in any situation where the interacting problem contains more than one type of particle with diKerent characteristic frequency scales. 2.3. Understanding microscopically why Fermi-liquid theory works Let us try to understand from a more microscopic approach why the Landau theory works so well. We present a qualitative discussion in this subsection and outline the principal features of the formal derivation in the next subsection. As we already remarked, a crucial element in the approach is to choose the proper noninteracting reference system. That this is possible at all is due to the fact that the number of states to which an added particle can scatter due to interactions is severely limited due to the Pauli principle. As a result, non-interacting fermions are a good stable system to perturb about; they have a Anite compressibility and susceptibility in the ground state, and so collective modes and thermodynamic quantities change smoothly when the interactions are turned on. This is not true for non-interacting bosons which do not support collective modes like sound waves. So one cannot perturb about the non-interacting bosons as a reference system. Landau also laid the foundations for the formal justiAcation of Fermi liquid theory in two and three dimensions. The Murry of activity in this Aeld following the discovery of high-Tc phenomena has led to new ways of justifying Fermi-liquid theory (and understanding why the one-dimensional problem is diKerent). However, the principal physical reason, which we now discuss, remains the phase-space restrictions due to kinematical constraints. We learned in Section 2.2 that in order to deAne quasiparticles, it was necessary to have a Anite ZkF , which in turn needed a self-energy function (kF ; !) which is smooth near the chemical potential, i.e., at ! = 0. Let us Arst see why a Fermi gas has such properties when interactions are introduced perturbatively.

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Fig. 7. The three second-order processes in a perturbative calculation of the correction to the bare interaction in a Fermi liquid.

We will explicitly consider only short-range interactions in this section, so that they can be characterized at all momentum transfers by a single parameter. Nevertheless, the essential results of Landau theory remain valid in the presence of Coulomb interactions because screening makes the interactions essentially short-ranged. The coupling constant g below may then be considered to parametrize the screened interaction. In Fig. 7, we show the three possible processes that arise in second-order perturbation theory for the scattering of two particles with Axed initial energy ! and momentum q. Note that in two of the diagrams, Fig. 7(a) and (b) the intermediate state has a particle and a hole while the intermediate state in diagram 7(c) has a pair of particles. We will And that, for our present purpose, the contribution of diagram 7(a) is more important than the other two. It gives a contribution g2

k

fk+q − fk : ! − (Ek+q − Ek ) + i

(19)

Here, g is a measure of the strength of the scattering potential (the vertex in the diagram) in the limit of small q. The denominator ensures that the largest contribution to the scattering comes from small scattering momenta q: for these the energy diKerence is linear in q; Ek+q − Ek ≈ q · vk , where vk is a vector of length vF in the direction of k. Moreover, the term in the numerator is non-zero only in the area contained between two circles (for d = 2) or spheres (for d = 3) with their centers displaced by q—here the phase-space restriction is due to the Pauli principle. This area is also proportional to q · vk , and so in the small q approximation from diagram 7(a) we get a term proportional to g2

q · vk df : ! − q · vk + i dk

(20)

Now we see why diagram 7(a) is special. There is a singularity at ! = q · vk and its value for small ! and q depends on which of the two is smaller. This singularity is responsible for the low-energy long-wavelength collective modes of the Fermi liquid in Landau theory. At low temperatures, df=dk = − (k − ), so the summation is restricted to the Fermi surface. The real part of (19) therefore vanishes in the limit qvF =! → 0, while it approaches a =nite limit

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Fig. 8. (a) Restriction on allowed particle–hole excitations in a Fermi sea due to kinematics. The plasmon mode has been drawn for the case d = 3; (b) the absorptive part of the particle–hole susceptibility (in the charge, current and spin channels) for ! ¡ qvF in the Fermi gas.

for ! → 0. The imaginary part in this limit is proportional to 4 !: ! Im &(q; !) = g2 N (0) for ! ¡ qvF ; qvF

(21)

while Im &(q; !) = 0 for ! ¿ vF q. This behavior is sketched in Fig. 8(b). An explicit evaluation for the real part yields ! − qvF ! 2 ; Re &(q; !) = g N (0) 1 + ln (22) qvF ! + qvF which gives a constant (leading to a Anite compressibility and spin susceptibility) at ! small compared to qvF . For diagram 7(b), we get a term ! − (Ep1 −p2 +k+q − Ek ) in the denominator. This term is always Anite for general momenta p1 and p2 , and hence the contribution from this diagram can always be neglected relative to the one from 7(a). Along similar lines, one Ands that diagram 7(c), which describes scattering in the particle–particle channel, is irrelevant except when p1 = − p2 , when it diverges as ln !. Of course, this scattering process is the one which gives superconductivity. Landau noticed this singularity but ignored its implication. 5 Indeed, as long as the eKective interactions do not favor superconductivity or as long as we are at temperatures much higher than the superconducting transition temperature, it is not important for Fermi-liquid theory. Let us now look further at the absorptive spectrum of particle–hole excitations in two and three dimensions, i.e., we examine the imaginary part of Eq. (19). When the total energy ! of the pair is small, both the particle and the hole have to live close to the Fermi surface. In 4

This behavior implies that this scattering contribution is a marginal term in the renormalization group sense, which means that it aKects the numerical factors, but not the qualitative behavior. 5 Attractive interactions in any angular momentum channel (leading to superconductivity) are therefore marginally relevant operators.

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Fig. 9. The single-particle self-energy diagram in second order.

this limit, we can make any excitation with momentum q 6 2kF . For Axed but small values of q, the maximum excitation energy is ! ≈ qvF ; this occurs when q is in the same direction as the main momentum k of each quasiparticle. For q near 2kF , the maximum possible energy is ! = vF |q − 2kF |. Combining these results, we obtain the sketch in Fig. 8(a), in which the shaded area in the !–q space is the region of allowed particle–hole excitations. 6 From this spectrum, one can calculate the polarizability, or the magnetic susceptibility. The behavior sketched above is valid generally in two and three dimensions (but as we will see in Section 4, not in one dimension). The important point to remember is that the density of particle–hole excitations decreases linearly with ! for ! small compared to qvF . We shall see later that one way of undoing Fermi-liquid theory is to have ! ∼ k 2 in two dimensions or ! ∼ k 3 in three dimensions. We can now use Im &(q; !) to calculate the single-particle self-energy to second order in the interactions. This is shown in Fig. 9 where the wiggly line denotes &(q; ) which in the present approximation is just given by the diagram of Fig. 7(a). For the perturbative evaluation of this process, the intermediate particle with energy– momentum (! + ); (k + q) is a free particle. Second-order perturbation theory then yields an imaginary part, or a decay rate, 2 1 ! 2 Im (k; !) = (23) = g N (0) (k; !) EF in three dimensions for k ≈ kF . In two dimensions, the same process yields Im (kF ; !) ∼ !2 ln(EF =!). The !2 decay rate is intimately related to the analytic result (22) for Im &(q; !) exhibited in Fig. (8). As may be found in textbooks, the same calculation for electron–phonon interactions or for interaction with spin waves in an antiferromagnetic metal gives Im (kF ; !) ∼ (!=!c )3 , where !c is the phonon Debye frequency in the former and the characteristic zone-boundary spin-wave frequency in the latter. The real part of the self-energy may be obtained directly or by Kramers–Kronig transformation of (23). It is proportional to !. Therefore, if the quasiparticle amplitude ZkF is evaluated

6

In the presence of long-range Coulomb interactions, in addition to the particle–hole excitation spectrum associated with the screened (and hence eKectively short-ranged) interactions one gets a collective mode with a =nite plasma √ frequency as q → 0 in d = 3 and a ! ∼ q behavior in d = 2. The plasma mode is a high-frequency mode in which the motion of the light electrons cannot be followed by the heavy ions: screening is absent in this regime and the long-range Coulomb interactions then give rise to a Anite plasma frequency in d = 3.

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Fig. 10. Single-particle energy k in one dimension, in the approximation that the dispersion relation is linearized about kF . Note that the Fermi surface consists of just two points. The spectrum of particle–hole excitations is given by !(q) = (k + q) − (k) = kF q=m. Low-energy particle–hole excitations are only possible for q small or for q near 2kF . Fig. 11. Phase space for particle–hole excitation spectrum in one dimension compared with the same in higher dimensions, Fig. 8. For linearized single-particle kinetic energy k = ± vF (k − kF ), particle–hole excitations are only possible on lines going through k = 0 and k = 2kF .

perturbatively 7 ZkF ≈ 1 − 2g2 N (0)=EF :

(24)

Thus in a perturbative calculation of the eKect of interactions the basic analytic structure of the Green’s function is left the same as for non-interacting fermions. The general proof of the validity of Landau theory consists in showing that what we have obtained to second order in g remains valid to all orders in g. The original proofs [4] are self-consistency arguments—we will consider them brieMy in Section 2.4. They assume a Anite Z in the exact single-particle Green’s functions and eKectively show that to any order in perturbation theory, the polarizability functions retain the analytic structure of the non-interacting theory, which in turn ensures a Anite Z. In one dimension, phase-space restrictions on the possible excitations are crucially diKerent. 8 Here the Fermi surface consists of just two points in the one-dimensional space of momenta— see Fig. 10. As a result, whereas in d = 2 and 3 a continuum of low-energy excitations with Anite q is possible, in one dimension at low-energy only excitations with small k or k ≈ 2kF are possible. The subsequent equivalent of Fig. 8 for the one-dimensional case is the one shown in Fig. 11. Upon integrating over the momentum k with a cut-oK of O(kF ) the contribution from 7 This quantity has been precisely evaluated by Galitski [104] for the model of a dilute Fermi gas characterized by a scattering length. 8 It might appear surprising that they are not diKerent in any essential way between higher dimensions.

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Fig. 12. (a) The nested Fermi surface obtained in a tight binding model on a square lattice with nearest-neighbor hopping; (b) a partially nested Fermi surface which leads to charge-density wave or antiferromagnetic instabilities.

this particle–hole scattering channel to Re &(q; !) is kF 1 dk ∼ ln[(! + qvF )=EF ] : ! + (2k + q)vF 0

(25)

(Note that (25) is true for both qkF and |q − 2kF |kF .) This in turn leads to a singleparticle self-energy calculated by the process in Fig. 9 to be Re (kF ; !) ∼ ! ln ! and so Z ∼ ln ! giving a hint of trouble. The Cooper (particle–particle) channel has the same phasespace restrictions, and gives a contribution to Re (kF ; !) proportional to ! ln ! too. The fact that these singular contributions are of the same order, leads to a competition between charge=spin Muctuations and Cooper pairing Muctuations, and in the exact calculation to powerlaw singularities. The fact that instead of the continuum of low-energy excitations present in higher dimensions, the width of the band of allowed particle–hole excitations vanishes as ! → 0, is the reason that the properties of one-dimensional interacting metals can be understood in terms of bosonic modes. We will present a brief summary of the results for the single-particle Green’s function and correlation functions in Section 4.9. In special cases of nesting in two or three dimensions, one can have situations that resemble the one-dimensional case. When the non-interacting Fermi surface in a tight binding model has the square shape sketched in Fig. 12(a) (which occurs for a tight-binding model with the nearest neighbor hopping on a square lattice at half-Alling) a continuous range of momenta on opposite sides of the Fermi surface can be transformed into each other by one and the same wavenumber. This so-called nesting leads to log and log2 singularities for a continuous range of k in the perturbation theory for the self-energy (k; !). Likewise, the partially nested Fermi surface of Fig. 12(b) leads to charge density wave and antiferromagnetic instabilities. We will come back to these issues in Sections 2.6 and 6.

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2.4. Principles of the microscopic derivation of Landau theory In this section, we will sketch how the conclusions in the previous section based on secondorder perturbation calculation are generalized to all orders in perturbation theory. This section is slightly more technical than the rest; the reader may choose to skip to Section 2.6. We follow the microscopic approach whose foundations were laid by Landau himself and which is discussed in detail in excellent textbooks [197,208,37,4]. For more recent methods with the same conclusions, see [237,128]. Our emphasis will be on highlighting the assumptions in the theory so that in the next section we can summarize the routes by which the Fermi-liquid theory may break down. These assumptions are usually not stated explicitly. The basic idea is that due to kinematic constraints, any perturbative process with n particle– hole pairs in the intermediate state provides contributions to the polarizability proportional to (!=EF )n . Therefore, the low-energy properties can be calculated with processes with the same “skeletal” structure as those in Fig. 7, which have only one particle–hole pair in the intermediate state. So one may concentrate on the modiAcation of the four-legged vertices and the single-particle propagators due to interactions to all orders. Accordingly, the theory is formulated in terms of the single-particle Green’s function G(p) and the two-body scattering vertex -(p1 ; p2 ; p1 + k; p2 − k) = -(p1 ; p2 ; k) :

(26)

Here and below we use, for the sake of brevity, p, etc. to denote the energy–momentum four vector (p; !) and we suppress the spin labels. The equation for - is expanded in one of the two particle–hole channels as 9 d4 q (1) -(p1 ; p2 ; k) = - (p1 ; p2 ; k) − i -(1) (p1 ; q; k)G(q)G(q + k)-(q; p2 ; k) ; (27) (2)4 where -(1) is the irreducible part in the particle–hole channel in which Eq. (27) is expressed. In other words, -(1) cannot be split into two parts by cutting two Green’s function lines with total momentum k. So -(1) includes the complete vertex in the other (often called cross-) particle–hole channel. The diagrammatic representation of Eq. (27) is shown in Fig. 13. In the simplest approximation -(1) is just the bare two-body interaction. Landau theory assumes that -(1) has no singularities. 10 An assumption is now further made that G(p) does have a coherent quasiparticle part at |p| pF and ! 0: G(p) =

9

Z + Ginc ; ! − ˜p + i sgn(p )

(28)

To second order in the interactions the correction to the vertex in the two possible particle–hole channels has been exhibited in the Arst two parts of Fig. 7. 10 The theory has been generalized for Coulomb interactions [208,197,4].The general results remain unchanged because a screened short-range interaction takes the place of -(1) .This is unlikely to be true in the critical region of a metal–insulator transition, because on the insulating side, the Coulomb interaction is unscreened.

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Fig. 13. Diagrammatic representation of Eq. (27).

where ˜p is to be identiAed as the excitation energy of the quasiparticle, Z its weight, and Ginc the incoherent non-singular part of G. (The latter provides the smooth background part of the spectral function in Fig. 5(b) and the former the sharp peak, which is proportional to the function for ˜p = p − .) It follows [195,4] from (28) that G(q)G(q + k) =

2iz 2 vq · k ()(|q| − pF ) + .(q) vF ! − vq · k

(29)

for small k and !, and where and ( + !) are frequencies of the two Green’s functions. Note the crucial role of kinematics in the form of the Arst term which comes from the product of the quasiparticle parts of G; .(q) comes from the scattering of the incoherent part with itself and with the coherent part and is assumed smooth and featureless (as it is indeed, given that Ginc is smooth and featureless and the scattering does not produce an infrared singularity at least perturbatively in the interaction). The vertex - in regions close to k ≈ kF and ! ≈ 0 is therefore dominated by the Arst term. The derivation of Fermi-liquid theory consists in proving that Eqs. (27) for the vertex and (28) for the Green’s function are mutually consistent. The proof proceeds by deAning a quantity -! (p1 ; p2 ; k) through d4 q ! (1) - (p1 ; p2 ; k) = - (p1 ; p2 ; k) − i -(1) (p1 ; q; k).(q)-! (q; p2 ; k) : (30) (2)4 -! contains repeated scattering of the incoherent part of the particle–hole pairs among itself and with the coherent part, but no scattering of the coherent part with itself. Then, provided the irreducible part of -(1) is smooth and not too large, -! is smooth in k because .(q) is by construction quite smooth. Using the fact that the Arst part of (29) vanishes for vF |k |=! → 0, and comparing (27) and (30) one can write the forward scattering amplitude lim lim -(p1 ; p2 ; k) = -! (p1 ; p2 ) : (31) !→0 k→0

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Fig. 14. Diagram for the exact single-particle self-energy in terms of the exact vertex - and the exact single-particle Green’s function.

This is now used to write the equation for the complete vertex - in terms of -! : vq · k Z 2 pF2 -(p1 ; p2 ; k) = -! (p1 ; p2 ) + -! (p1 ; q)-(q; p2 ; k) d/q ; ! − vq · k (2)3 vF

(32)

where in the above |q| = pF and one integrates only over the solid angle /q . Given a non-singular -! , a non-singular - is produced (unless the denominator in Eq. (32) produces singularities after the indicated integration—the Landau–Pomeranchuk singularities discussed below). The one-particle Green’s function G can be expressed exactly in terms of -—see Fig. 14. This leads to Eq. (28) proving the self-consistency of the ansatz with a Anite quasiparticle weight Z. The quantity Z 2 -! is then a smooth function and goes into the determination of the Landau parameters. The Landau parameters can be written in terms of the forward scattering amplitude. In effect, they parametrize the momentum and frequency independent scattering of the incoherent parts among themselves and with the coherent parts so that the end result of the theory is that the physical properties can be expressed purely in terms of the quasiparticle part of the single-particle Green’s function and the Landau parameters. No reference to the incoherent parts needs to be made for low-energy properties. For single-component translational invariant fermions (like liquid 3 He) even the quasiparticle amplitude Z disappears from all physical properties. This last part is not true for renormalization due to electron–phonon interactions and in multicomponent systems such as heavy fermions. Special simpliAcations of the Landau theory occur in such problems and in other problems where the single-particle self-energy is nearly momentum independent [180,280,105,265,187]. As we also mentioned, the single-particle self-energy can be written exactly in terms of the vertex -: the relation between the two is represented diagrammatically in Fig. 14. The relations between and - are due to conservation laws which Landau theory, of course, obeys. However, the conservation laws are more general than Landau theory. It is often more convenient to express these conservation laws as relations between the self-energy and the three-point vertices, 0 (p; q!) which couple external perturbations to either the density (the fourth component,

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Fig. 15. Vertex for coupling to external perturbations at energy-momentum (!; q); 1 is the bare vertex.

= 4) or the current density in the = (1; 2; 3) direction. The diagrammatic representation of the equation for 0 is shown in Fig. 15. The following relations (Ward identities) have been proven for translationally invariant problems: p ! 9 lim → 0; q → 0 0 (p; q!) = − (p; ) ( = 1; 2; 3) ; (33) q m 9p

lim lim lim

! 9(p; ) → 0; q → 0 04 (p; q!) = 1 + ; q 9

q

p d → 0; ! → 0 0 (p; q!) = − (p; )

q

d(p; ) → 0; ! → 0 04 (p; q!) = 1 + :

! !

m

dp d

(34) ( = 1; 2; 3) ;

(35) (36)

A relation analogous to (34) is derived for Aelds coupling to spin for the case that interactions conserve spin. The total derivative in (35) and (36) [rather than the partial derivative in (33) and (34)] is represented such that d=d is the variation in when is changed to + d together with to + d, and d=dp represents the variation when the momentum p as well as the Fermi surface is translated by dp . Eq. (36) is an expression of energy conservation, and Eq. (34) of particle number conservation. Eqs. (33) and (34) together signify the continuity equation. Eq. (35) represents current conservation. 11

11 The Ward identity Eq. (35) does not hold for an impure system where the Fermi surface cannot be deAned in momentum space. Since energy is conserved, a Fermi surface can still be deAned in energy space, and hence the other Ward identities continue to hold. This point is further discussed in Section 7.

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In Landau theory, the right-hand sides in Eqs. (33) – (36) are expressible in terms of the Landau parameters. These relations are necessary for the derivation of the renormalization of the various thermodynamic quantities quoted in Eqs. (17) and (18) as well as the Landau transport equation. Needless to say, any theory of SFL must also be consistent with the Ward identities. 2.5. Modern derivations The modern derivations of Fermi-liquid theory similarly start by assuming the existence of a Fermi surface. Kinematics then inevitably leads to similar considerations as above. Instead of the division into coherent and incoherent parts made in Eq. (28), the renormalization group procedures are used to systematically generate successively lower energy and small momentum Hamiltonians with excitations of particle ever closer to the Fermi surface. The calculations are carried out either in terms of fermions [237] or newly developed bosonization methods in arbitrary dimensions [128]. The end result is equivalent to Eqs. (28), (30) and (32). These methods may well turn out to be very important in Anding the structure of SFLs and in systematizing them. These derivations carry out the calculation in an arbitrary dimension d and conclude that the forward scattering amplitude is d−1 k ! - (p1 ; p2 ; k) ∼ f(p1 ; p2 ; k) ; (37) kF where f is a smooth function of all of its arguments. In one-dimension, the forward scattering amplitude has a logarithmic singularity, as we noted earlier. We can rephrase the conceptual framework of Landau Fermi-liquid theory in the modern language of renormalization group theory [237]. As we discussed, in Fermi-liquid theory one treats a complicated strongly interacting fermion problem by writing the Hamiltonian H as H = Hsimple + Hrest :

(38)

In our discussion, Hsimple was the non-interacting Hamiltonian. The non-interacting Hamiltonian is actually a member of a “line” of Axed-point Hamiltonians H∗ all of which have the same symmetries but diKer in their Landau parameters Fls; a , etc. The Fl ’s, obtained from the forward scattering in Landau theory are associated with marginal operators and distinguish the properties of the various systems associated with the line of Axed points. Landau Fermi-liquid theory is primarily a statement regarding the domain of attraction of this line of Axed points. The theory also establishes the universal low-temperature properties due to the “irrelevant” operators generated by Hrest due to scattering in channels other than the forward channel. Landau theory does not establish (at least completely) the domain of attraction of the “critical surface” bounding the domain of attraction of the Fermi liquid Axed line from those of other Axed points or lines. If Hrest generates a “relevant” operator (i.e., eKective interactions which diverge at low energies and temperatures) the scheme breaks down. For example, attractive interactions between fermions generate relevant operators—they presage a transition to superconductivity,

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a state of diKerent symmetry. However, if we stay suPciently above Tc , we can usually continue using Landau theory. 12 2.6. Routes to breakdown of Landau theory From Landau’s phenomenological theory, one can only say that the theory breaks down when the physical properties—speciAc heat divided by temperature, 13 compressibility, or the magnetic susceptibility—diverge or when the collective modes representing oscillations of the Fermi surface in any harmonic and singlet or triplet spin combinations become unstable. The latter, called the Landau–Pomeranchuk singularities, are indeed one route to the breakdown of Landau theory and occur when the Landau parameters Fls; a reach the critical value −(2l + 1). A phase transition to a state of lower symmetry is then indicated. The new phase can again be described in Landau theory by deAning distribution functions consistent with the symmetry of the new ground state. The discussion following Eq. (8) in Section 2.2 allows us to make a more general statement. Landau theory breaks down when the quasiparticle amplitude Zk becomes zero; i.e., when the states ck† | N and | kN +1 are orthogonal. This can happen if the series expansion in Eq. (8) in terms of the number of particle–hole pairs is divergent. In other words, the addition of a particle or a hole to the system creates a divergent number of particle–hole pairs in the system so that the leading term does not have a Anite weight in the thermodynamic limit. From Eq. (11), which links the Z’s to ’s, this requires that the single-particle self-energy be singular as a function of ! at k kF . This in turn means that the Green’s functions of SFLs contain branch cuts rather than the poles unlike Landau Fermi liquids. The weakest singularity of this kind is encountered in the borderline “marginal Fermi liquids” where 14

!c (kF ; !) 1 ! ln (39) + i|!| : ! If a divergent number of low-energy particle–hole pairs is created upon the addition of a bare particle, it means that the low-energy response functions (which all involve creating particle– hole pairs) of SFLs are also divergent. Actually, the single-particle self-energy can be written in terms of integrals over the complete particle–hole interaction vertex as in Fig. 14. The implication is that the interaction vertices are actually more divergent than the single-particle self-energy. 12 We note that in a renormalization group terminology, all Landau parameters fkk ; originating from forward scattering (i.e., zero momentum transfer), are “marginal operators” [151,237]. All other operators that determine Anite temperature observable properties are “irrelevant”. Thus, in a “universal” sense, condensed matter physics may be deemed to be an “irrelevant” Aeld. So much for technical terminology! 13 The speciAc heat of a system of fermions can be written in terms of integrals over the phase angle of the exact single-particle Green’s function [4]. Given any singularity in the self-energy, Cv =T is never more singular than ln T . This accounts for the numerous experimental examples of such behavior that we will come across. 14 To see why this is the borderline case, note that a requisite for the deAnition of a quasiparticle is that the 1 quasiparticle peak width − k = 2 should vanish faster than linear in !, the quasiparticle energy. Thus, ∼ ! is the Arst power for which this is not true. The ! ln(!c =!) term in Eq. (39) is then dictated by the Kramers–Kroning relation.

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Yet another route to SFLs is the case in which the interactions generate new quantum numbers which are not descriptive of the non-interacting problem. This happens in particular in the Quantum Hall problems and in one-dimensional problems (Section 4) as well as in problems of impurity scattering with special symmetries (Section 3). In such cases, the new quantum numbers characterize new low-energy topological excitations. New quantum numbers of course imply Z = 0, but does Z = 0 imply new quantum numbers? One might wish to conjecture that this is so. However, no general arguments on this point are available. 15 Ultimately, all breakdowns of Landau theory are due to degeneracies leading to singular low-energy Muctuations. If the characteristic energy of the Muctuations is lower than the temperature, a quasiclassical statistical mechanical problem results. On the basis of our qualitative discussion in Section 2.3 and the sketch of the microscopic derivation in Section 2.4, we may divide the various routes to breakdown of Landau theory into the following (not necessarily orthogonal) classes: (i) Landau–Pomeranchuk singularities: Landau theory points to the possibility of its breakdown through the instability of the collective modes of the Fermi-surface which arise from the solution of the homogeneous part of Eq. (32). These collective modes can be characterized by the angular momentum ‘ of oscillation of the Fermi surface and whether the oscillation is symmetric “s” or antisymmetric “a” in spin. The condition for the instability derived from the condition of zero frequency of the collective modes are [208,37] F‘s 6 − (2‘ + 1);

F‘a 6 − (2‘ + 1) :

(40)

The ‘ = 0 conditions refer to the divergence in the compressibility and the (uniform) spin susceptibility. The former would in general occur via a Arst-order transition, so is uninteresting to us. The latter describes the ferromagnetic instability. No other Landau–Pomeranchuk instabilities have been experimentally identiAed. However, such new and exotic possibilities should be kept in mind. Thus, for example, an F1s -instability corresponds to the Fermi velocity → 0, a F2s instability to a “d-wave-like” instability of the particle–hole excitations on the Fermi surface etc. Presumably, these instabilities are resolved by reconstruction of the Fermi surface. The microscopic interactions necessary for the Landau–Pomeranchuk instabilities and the critical properties near such instabilities have not been well investigated, especially for fermions with a lattice potential. 16 It is also worth noting that some of the instabilities are disallowed in the limit of translational invariance. Thus, for example, time-reversal breaking states, such as the “anyon-state” [156,60] cannot be realized because in a translationally invariant problem the current operator cannot be renormalized by the interactions, as we have learnt from Eqs. (18), (33). 15

It would indeed be a signiAcant step forward if such a conjecture could be proven to be true or if the conditions in which it is true were known. One might think that this should be obvious in any given case. However, it is not so. In the SFL problems of magnetic impurities (Section 3) and in the SFL behavior of one-dimensional fermions (Section 4), a complete description of the solution was given without introducing new quantum numbers. That these problems may be discussed in such terms was only realized later, lending to the solutions additional elegance besides insight. 16 See, however, two recent papers [116,200]. In [273] the pseudogap in the underdoped cuprates arises as a consequence of a Landau–Pomeranchuk instability.

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(ii) Critical regions of large Q-singularities: Landau theory concerns itself only with longwavelength response and correlations. A Fermi liquid may have instabilities at a non-zero wavevector, for example a charge-density wave (CDW) or spin-density wave (SDW) instability. Only a microscopic calculation can provide the conditions for such instabilities and therefore such conditions can only be approximately derived. An important point to note is that they arise perturbatively from repeated scattering between the quasiparticle parts of G while the scattering vertices (irreducible interactions) are regular. The superconductive instability for any angular momentum is also an instability of this kind. In general, such instabilities are easily seen in RPA and=or t-matrix calculations. Singular Fermi liquid behavior is generally expected to occur only in the critical regime of such instabilities [117,164]. If the transition temperature Tc is Anite then there is usually a stable low-temperature phase in which unstable modes are condensed to an order parameter, translational symmetry is broken, and gaps arise in part or all of the Fermi surface. For excitations on the surviving part of the Fermi surface, Fermi-liquid theory is usually again valid. The Muctuations in the critical regime are classical, i.e., with characteristics frequency !f‘ kB Tc . If the transition is tuned by some external parameter so that it occurs at zero temperature, one obtains, as illustrated already in Fig. 1, a quantum critical point (QCP). If the transition is approached at T = 0 as a function of the external parameter, the Muctuations are quantum-mechanical, while if it is approached as a function of temperature for the external parameter at its critical value, the Muctuations have a characteristic energy proportional to the temperature. A large region of the phase diagram near QCPs often carries SFL properties. We shall discuss such phenomena in detail in Section 6. (iii) Special symmetries: The Cooper instability at q = 0, Fig. 7(c), is due to the “nesting” of the Fermi surface in the particle–particle channel. Usually, indications of Anite-q CDW or SDW singularities are evident pertubatively from Fig. 7(a) or (b) for special Fermi surfaces, nested in some q-direction in particle–hole channels. One-dimensional fermions are perfectly nested in both particle–hole channels and particle–particle channels (Fig. 7(a) – (c)) and hence they are both logarithmically singular. Pure one-dimensional fermions also have the extra conservation law that right going and left going momenta are separately conserved. These introduce special features to the SFL of one-dimensional fermions such as the introduction of extra quantum numbers. These issues are discussed in Section 4.9. Several soluble impurity problems with special symmetries have SFL properties. Their study can be illuminating and we discuss them in Section 3. (iv) Long-range interactions: The breakdown of Landau Fermi liquid may come about through long-range interactions, either in the bare Hamiltonian through the irreducible interaction or through a generated eKective interaction. The latter, of course, occurs in the critical regime of phase transitions such as discussed above. Coulomb interactions will not do for the former because of screening of charge Muctuations. The fancy way of saying this is that the longitudinal electromagnetic mode acquires mass in a metal. The latter is not true for current Muctuations or transverse electromagnetic modes which must remain massless due to gauge invariance. This is discussed in Section 5.1, where it is shown that no metal at low enough temperature is a Fermi liquid. However, the cross-over temperature is too low to be of experimental interest.

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An oK-shoot of an SFL through current Muctuations is the search for extra (induced) conservation laws for some quantities to keep their Muctuations massless. This line of investigation may be referred to generically as gauge theories. Extra conservation laws imply extra quantum numbers and associated orthogonality. We discuss these in Section 5.2. The one-dimensional interacting electron problem and the Quantum Hall eKect problems may be usefully thought of in these terms. (v) Singularities in the irreducible interactions: In all the possibilities discussed in (i) – (iii) above, the irreducible interactions -(1) deAned after Eq. (27) are regular and not too large. As noted after Eq. (30), this is necessary to obtain a regular -! . When these conditions are satisAed the conceivable singularities arise only from the repeated scattering of low-energy particle–hole (or particle–particle pairs) as in Eq. (32) or its equivalent for large momentum transfers. A singularity in the irreducible interaction of course invalidates the basis of Landau theory. Such singularities imply that the parts of the problem considered harmless perturbatively because they involve the incoherent and high-energy parts of the single-particle spectral weight as in Eq. (30) are, in fact, not so. This is also true if -(1) is large enough such that the solution of Eq. (30) is singular. Very few investigations of such processes exist. How can an irreducible interaction be singular when the bare interaction is perfectly regular? We know of two examples: In disordered metals, the density correlations are diKusive with the characteristic frequency ! scaling with q2 . The irreducible interactions made from the diKusive Muctuations and interactions are singular in d = 2. This gives rise to a new class of SFLs which are discussed in Section 8. One Ands that in this case, the singularity in the cross-particle–hole channel (the channel diKerent from the one through which the irreducibility of -(1) is deAned) feeds back into a singularity in -(1) . This is very special because the cross-channel is integrated over and the singularity in it must be very strong for this to be possible. The second case concerns the particle singularities in the irreducible interactions because of excitonic singularities. Usually, the excitonic singularities due to particle–hole between diKerent bands occur at a Anite energy and do not introduce low-energy singularities. However, if the interactions are strong enough, these singularities occur near zero frequency. In eKect, eliminating high-energy degrees of freedom generates low-energy irreducible singular vertices. This only requires that the appropriate bare interactions are large compared to characteristic inter-band energies. Consider, for example, the band structure of a solid with more than one atom per unit cell with (degenerate) valence band maxima and minima at the same points in the Brillouin zone, as in Fig. 16. Let the conduction band be partially Alled, and the energy diKerence between and the valence band below E0 in Fig. 16 be much smaller than the attractive particle–hole interactions V between states in the valence (v) band and the conduction band (c). For any Anite V , excitonic resonances form from scattering between v and c states, as in the X-ray edge problem to be discussed in Section 3.5. For large enough V , such resonances occur at asymptotically low energy so that the Fermi liquid description of states near the chemical potential in terms of irreducible interaction among the c states is invalid. The eKective irreducible interactions among the low-energy states integrate over the excitonic resonance and will in general be singular if the resonance is near zero energy.

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Fig. 16. A model band structure for a solid with more than one atom per unit cell. Actually in CaB6 where excitonic singularities have been invoked to produce a ferromagnetic state [295] there are three equivalent points in the Brillouin zone where the conduction band minima and the valence band maxima occur.

Such singularities require interactions above a critical magnitude and are physically and mathematically of an unfamiliar nature. In a two-band one-dimensional model, exact numerical calculations have established the importance of such singularities [252,249]. Recently, it has been found that CaB6 or SrB6 with low densities of trivalent Eu or quadrivalent Ce ions substituting for (Ca; Sr) are ferromagnets [295]. The most plausible explanation [303,34,36] is that this is a realization of the excitonic ferromagnetism predicted by Volkov et al. [279]. The instability to such a state occurs because the energy to create a hole in the valence band and a particle in the conduction band above the Fermi energy goes to zero if the attractive particle–hole (interband) interactions are large enough. This problem has been investigated only in the mean Aeld approximation. Fluctuations in the critical regime of such a transition are well worth studying. Excitonically induced singularities in the irreducible interactions are also responsible for the Marginal Fermi-liquid state of Cu–O metals in a theory to be discussed in Section 7. 3. Local Fermi liquids and local singular Fermi liquids In this section, we discuss a particular simple form of Fermi liquid formed by electrons interacting with a dilute concentration of magnetic impurity. Many of the concepts of Fermi-liquid theory are revisited in this problem. Variants of the problem provide an interesting array of soluble problems of SFL behavior and illustrate some of the principal themes of this article.

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3.1. The Kondo problem The Kondo problem is at the same time one of the simplest and one of the most subtle examples of the eKects of strong correlation eKects in electronic systems. The experiments concern metals with a dilute concentration of magnetic impurities. In the Kondo model, one considers only a single impurity; the Hamiltonian then is † H=t ci cj + J S · c0† c0 ; (41) ij ; † ) denote the annihilation and creation operators of a conduction electron at site where (ci ; ci i with projection in the z-direction of spin . The second term is the exchange interaction between a single magnetic impurity at the origin (with spin S = 12 ) and a conduction electron spin. When the exchange constant J ¿ 0, the system is a Fermi liquid. Although not often discussed, the ferromagnetic (J ¡ 0) variant of this problem is one of the simplest examples of a singular Fermi liquid. There are two simple starting points for the problem: (i) J = 0: This turns out to describe the unstable high-temperature Axed point. 17 The term proportional to J is a marginal operator about the high-temperature Axed point because, as discovered by Kondo [146], in a third-order perturbation calculation the eKective interaction acquires a singularity ∼ J 3 =t 2 ln(t=!). (ii) t = 0: The perturbative expansion about this point is well behaved. This turns out to describe the low-temperature Fermi liquid Axed point. One might be surprised by this, considering that typically the bare t=J is of order 10+3 . But such is the power of singular renormalizations. 18 17 For the reader unfamiliar with reading a renormalization group diagram like that of Fig. 17(b) or Fig. 18, the following explanation might be helpful. The Mow in a renormalization group diagram signiAes the following. The original problem, with bare parameters, corresponds to the starting point in the parameter space in which we plot the Mow. Then we imagine “integrating out” the high-energy scales (e.g. virtual excitations to high-energy states); eKectively, this means that we consider the system at lower-energy (and temperature) scales by generating eKective Hamiltonians with new parameters so that the low-energy properties remain invariant. The “length” along the Mow direction is essentially a measure of how many energy scales have been integrated out—typically, as in the Kondo problem, this decrease is logarithmic along the trajectory. Thus, the regions towards which the Mow points signify the eKective parameters of the model at lower and lower temperatures. Fixed points towards which all trajectories Mow in a neighborhood describe the universal low-temperature asymptotic behavior of the class of models to which the model under consideration belongs. When a Axed point of the Mow is unstable, it means that a model whose bare parameters initially lie close to it Mows away from this point towards a stable Axed point; hence it has a low-temperature behavior which does not correspond to the model described by the unstable Axed point. A Axed line usually corresponds to a class of models which have some asymptotic behavior, e.g. an exponent, which varies continuously. 18 A particularly lucid discussion of the renormalization procedure may be found in [151]. BrieMy, the procedure consists in generating a sequence of Hamiltonians with successively lower-energy cut-oKs that reproduce the low-energy spectrum. All terms allowed by symmetry besides those in the bare Hamiltonian are retained. The coePcients of these terms scale with the cut-oK. Those that decrease proportionately to the cut-oK or change only logarithmically, are coePcients of marginal operators, those that grow=decrease (algebraically) are coePcients of relevant=irrelevant operators. Marginal operators are marginally relevant or marginally irrelevant. Upon renormalization, the Mow is towards the strong coupling J = ∞ Axed point, see Fig. 17. The terms generated from t = 0 serve as irrelevant operators at this Axed point; this means that they do not aKect the ground state but determine the measurable low-energy properties.

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Fig. 17. (a) Hartree–Fock excitation spectrum of the Anderson model in the two limits of zero hybridization, V = 0 and zero interaction, U = 0; (b) renormalization group Mow of the Kondo problem.

The interaction between conduction electrons and the localized electronic level is not a direct spin interaction. It originates from quantum-mechanical charge Muctuations that (through the Pauli principle) depend on the relative spin orientation. To see this explicitly, it is more instructive to consider the Anderson model [19] in which † † H=t ci cj + d c0 c0 + Uc0;† ↑ c0; ↑ c0;† ↓ c0; ↓ + (Vk ck;† c0; + h:c:) : (42) ij

k;

The last term in this Hamiltonian is the hybridization between the localized impurity state and the conduction electrons, in which spin is conserved. In the particle–hole symmetric case, d = − U=2 is the one-hole state on the impurity site in the Hartree–Fock approximation and the one-particle state has the energy U=2. Following a perturbative treatment in the limit t=V; U=V 1 the Anderson model reduces to the Kondo Hamiltonian with an eKective exchange constant JeK ∼ (V 2 =t)2 =U . The Anderson model has two simple limits, which are illustrated in Fig. 17: (i) V = 0: This describes a local moment with Curie susceptibility & ∼ B2 =T . This limit is the correct point of departure for an investigation of the high-temperature regime. As noted one soon encounters the Kondo divergences.

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Fig. 18. Renormalization group Mows for the Kondo problem, displaying the line of critical points for the “ferro∗ = Jz∗ → ∞ for the “antiferromagnetic” problem. magnetic” problem and the Mow towards the Axed point J±

(ii) U = 0: In this limit, the impurity forms a resonance of width - ∼ V 2 =t at the chemical potential which in the particle–hole symmetric case is half-occupied. The ground state is a spin singlet. This limit is the correct starting point for an examination of the low-temperature properties (T TK ). A temperature independent contribution to the susceptibility and a linear contribution to the speciAc heat (∼ N (0)T=-) are contributed by the resonant state. Hence the name local Fermi liquid. The conceptually tough part of the problem was to realize that (ii) is the correct stable low-temperature Axed point and the technically tough problem is to derive the passage from the high-temperature regime to the low-temperature regime. This was Arst done correctly by Wilson [289] through the invention of the numerical renormalization group (and almost correctly by Anderson and Yuval [22,23] by analytic methods). The analysis showed that under renormalization group scaling transformations, the ratio (J=t) increases monotonically as illustrated in Fig. 17(b)—continuous RG Mows are observed from the high-temperature extreme (i) to the low-temperature extreme (ii) and a smooth crossover between the two regimes occurs at the Kondo temperature TK ∼ t exp(−t=2J ) :

(43)

Since all Mow is towards the strong-coupling Axed point, universal forms for the thermodynamic functions are found. For example, the speciAc heat Cv and the susceptibility & scale as Cv = Tfc (T=TK );

& = B2 f& (T=TK ) ;

(44)

where the f’s are universal scaling functions. An important theoretical result is that compared to a non-interacting resonant level at the chemical potential, the ratio of the magnetic susceptibility enhancement to the speciAc heat enhancement RW =

&=& Cv =Cv

(45)

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for spin 1=2 impurities at T TK is precisely 2 [289,197]. In a non-interacting model, this ratio, nowadays called the Wilson ratio, is equal to 1, since both & and Cv are proportional to the density of states N (0). Thus, the Wilson ratio is a measure of the importance of correlation eKects. It is in fact the analogue of the Landau parameter F0a of Eq. (18). 3.2. Fermi-liquid phenomenology for the Kondo problem Following Wilson’s solution [289], NoziZeres [197] showed that the low-temperature properties of the Kondo problem can be understood simply through a (local) Fermi-liquid framework. This is a beautiful example of the application of the concept of analyticity and of symmetry principles about a Axed point. We present the key arguments below. For the application of this line of approach to the calculation of a variety of properties, we refer the reader to papers by NoziZeres and Blandin [197,198]. The properties of a local impurity can be characterized by the energy-dependent s-wave phase shift (), which in general also depends on the spin of the conduction electron being scattered. In the spirit of Fermi-liquid theory the phase shift may be written in terms of the deviation of the distribution function n() of conduction electrons from the equilibrium distribution () = 0 () + (; )n ( ) + · · · : (46)

About a stable Axed point, the energy dependence is analytic near the chemical potential ( = 0), so that we may expand 0 () = 0 + + · · · ;

(; ) = . + · · · :

(47)

Just as the Landau parameters are expressed in terms of symmetric and antisymmetric parts, we can write .↑↑ = .↓↓ = .s + .a ;

.↓↑ = .↑↓ = .s − .a :

(48)

Taken together, this leaves three parameters , .s and .a to determine the low-energy properties. NoziZeres [197] showed that in fact there is only one independent parameter (say which is of O(1=TK ), with a prefactor which can be obtained by comparing with Wilson’s detailed numerical solution). To show this, note that by the Pauli principle same spin states do not interact, therefore [197] .↑↑ = .s + .a = 0 :

(49)

Secondly, a shift of the chemical potential by and a simultaneous increase in n by N (0) should have no eKect on the phase shift, since the Kondo eKect is tied to the chemical potential. Therefore according to (46) and (47) [ + N (0).s ] = 0 ;

⇒ .s = − =N (0) :

(50)

Thirdly, one may borrow from Wilson’s solution that the Axed point has 0 = =2. This expresses that the tightly bound spin singlet state formed of the impurity spin and conduction electron spin

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completely blocks the impurity site to other conduction electrons; this in turn implies maximal scattering and therefore phase shift of =2 for the eKective scattering potential [197]. In other words, it is a strong-coupling Axed-point where one conduction electron state is pushed below the chemical potential in the vicinity of the impurity to form a singlet resonance with the impurity spin. One may now calculate all physical properties in terms of . In particular, one Ands Cv =Cv = 2=(VN (0)) and a similar expression for the enhancement of &, such that the Wilson ratio is 2. 3.3. Ferromagnetic Kondo problem and the anisotropic Kondo problem The ferromagnetic Kondo problem provides us with the simplest example of SFL behavior. We will discuss this below after relating the problem to a general X-ray edge problem in which the connection to the so-called orthogonality catastrophe is clearer. As discussed in Section 2, orthogonality generally plays an important role in SFLs. We start with the anisotropic generalization of the Kondo Hamiltonian, which is the proper starting model for a perturbative scaling analysis [21,100], † † † H =t ci cj + [J± (S + ck↓ ck ↑ + S − ck↑ ck ↓ ) + Jz S z (ck;† ↑ ck ; ↑ − ck;† ↓ ck ; ↓ )] : (51) ij ;

k; k

Long before the solution of the Kondo problem, perturbative renormalization group for the effective vertex coupling constants J± and Jz as a function of temperature were obtained [21,100]. The scaling relation between them is found to be exact to all orders in the J s: (Jz2 − J±2 ) = const

(J+ = J− ) :

(52)

In the Mow diagram of Fig. 18, we show the scaling trajectories for the anisotropic problem. In the antiferromagnetic regime, the Mows continuously veer towards larger and larger (J± ; Jz ) values; at the attracting Axed point (J ∗± ; J ∗z ) = (∞; ∞) singlets form between the local moment and the conduction electrons. The “ferromagnetic” regime spans the region satisfying the inequalities Jz ¡ 0 and |Jz | ¿ J± . Upon reducing the bandwidth the coupling parameters Mow towards negative Jz values. Observe the line of =xed points on the negative Jz axis. Such a continuous line is also seen in the Kosterlitz–Thouless transition [147] of the two-dimensional XY model. Moreover, in both problems continuously varying exponents in physical properties are obtained along these lines (in fact, the renormalization group Mow equations of the Kondo model for small coupling are mathematically identical to those for the XY model). This is an instance of a zero temperature quantum critical line. The physics of the quantum critical line has to do with an “orthogonality catastrophe” which we describe next. Such orthogonalities are generally an important part of the physics of SFLs. 3.4. Orthogonality catastrophe As we saw in Section 2, a Fermi-liquid description is appropriate so long as the spectrum retains a coherent single-particle piece of Anite weight Z ¿ 0. So if by some miracle the

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evaluation of Z reduces to an overlap integral between two orthogonal wavefunctions then the system is an SFL. In the thermodynamic (N → ∞) limit, such a miracle is more generic than might appear at Arst sight. In fact, such an orthogonality catastrophe arises if the injection of an in=nitely massive particle produces an eKective Anite range scattering potential for the remaining N electrons [20] (see Section 4.9). Such an orthogonality is exact only in the thermodynamic limit: The single-particle wavefunctions are not orthogonal. It is only the overlap between the ground state formed by their Slater determinants 19 which vanishes as N tends to inAnity. More quantitatively, if the injection of the additional particle produces an s-wave phase shift 0 for the single-particle wavefunctions (all N of them) .(kr) =

sin(kr + 0 ) sin kr → kr kr

(53)

then an explicit computation of the Slater determinants reveals that their overlap diminishes as

N| N

2

2

∼ N −0 = :

(54)

Here, | N is the determinant Fermi sea wavefunction for N particles and | N is the wavefunction of the system after undergoing a phase shift by the local perturbation produced by the injected electron. 20 Quite generally such an orthogonality (Z = 0) arises also if two N -particle states of a system possess diKerent quantum numbers and almost the same energy. These new quantum numbers might be associated with novel topological excitations. This is indeed the case in the quantum Hall liquid where new quantum numbers are associated with fractional charge excitations. The SFL properties of the interacting one-dimensional fermions (Section 4) may also be looked at as being due to orthogonality. Often orthogonality has the eKect of making a quantum many-body problem approach the behavior of a classical problem. This will be one of the leitmotifs in this review. We turn Arst to a problem where this orthogonality is well understood to lead to experimental consequences, although not at low energies. 3.5. X-ray edge singularities The term X-ray edge singularity is used for the line shape for absorption in metals by creating a hole in an atomic core level and a particle in the conduction band above the chemical potential. In the non-interacting particle description of this process, the absorption starts at the threshold frequency !D , as sketched in Fig. 19. In this case, a Fermi edge reMecting the density of unoccupied states in the conduction band is expected to be visible in the spectrum.

19

The results also hold true for interacting fermions, at least when a Fermi-liquid description is valid for both of the states. 20 Through the Friedel sum rule, 0 = has a physical meaning; it is the charge that needs to be transported from inAnity to the vicinity of the impurity in order to screen the local potential [80].

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Fig. 19. Absorption line shape for transitions between a lower dispersionless level and a conduction band, (a) for zero interaction between the conduction electrons and local level, and (b) for small interaction.

However, when a hole is generated in the lower level, the potential that the conduction electrons see is diKerent. The relevant Hamiltonian is now 1 1 1 1 † † † † H = d d d − + k ck ck + V (k; k ) ck ck − d d− ; (55) 2 L 2 2 k

k; k

where spin indices have been suppressed. The operators (d; d† ) annihilate or create holes in the core level, which is taken to be dispersionless. The Arst two terms in the Hamiltonian represent the unperturbed energies of the core hole and the free electrons. The last term depicts the screened Coulomb interaction between the conduction electrons and the hole in the core level. As a consequence of the interactions, the line shape is quite diKerent. This is actually an exactly solvable problem [196]. There are two kinds of eKects, (a) excitonic—the particle and the hole attract, leading to a shift of the edge and a sharpening of the edge singularity—and (b) an orthogonality eKect of the type just discussed above, which smoothens the edge irrespective of the sign of the interaction. This changes the absorption spectrum to that of Fig. 19(b) in the presence of interactions. The form of the singularity is [167,196] 2

2

A(!) ∼ (! − !˜ D )−20 =+0 = :

(56)

The exponent 20 =2 is a consequence of the orthogonality catastrophe overlap integral; the exponent (−20 =) is due to the excitonic particle–hole interactions. If the hole has Anite mass, we have a problem with recoil which is not exactly solvable. Notwithstanding this, we do know the essential features of the problem. As we will discuss later in Section 4.9, recoil removes the singularity in two and three dimensions and the absorption edge acquires a characteristic width of the order of the dispersion of the hole band. If the hole moves only in one dimension, the singularity is not removed. 3.6. A spinless model with =nite range interactions A model which is a generalization of the ferromagnetic Kondo problem and in which the low-energy physics is dominated by the orthogonality catastrophe, is given by the following

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Hamiltonian: † 1 1 t † 1 † † H= k k; l k; l + √ (k; 0 d + h:c:) + V − d d− : L l k; l k ; l 2 2 L k k;l k; k

303

(57)

The operators (; † ) are the annihilation and creation operators of spinless conduction electrons with kinetic energy k , while as before d+ and d create or annihilate electrons in the localized level. The local chemical potential has been set to zero (d = 0) and the Hamiltonian is particle– hole symmetric. The new index l is an orbital angular momentum index (or a channel index). Hybridization conserves point-group symmetry, so the localized orbital hybridizes with only one channel (l = 0). By contrast, the impurity couples to all channels via the interaction Vl . As we are summing over all moments (k; k ) this interaction is local. This problem may be mapped onto the anisotropic Kondo model [107]. Indeed, the transformation d† → S † ; J ;0 ; t → √⊥ 2a

1 → Sz ; 2 √ √ 2Vl → 2Jz; l − 2vF ( 2 − 1)l; 0

d† d −

(58)

produces H=

k;;l

k ck;† ; l ck; ; l +

1 Jz; l Sz sz; l : J⊥; 0 (S † sl− + h:c:) + 2

(59)

l

Here, a is short distance cutoK. In the resulting (anisotropic multichannel) Kondo Hamiltonian, the spin operators S and sl portray charge excitations of the local orbital and conduction band. The spin index in the resulting Kondo Hamiltonian should now be regarded as a charge label. Physically, this mapping is quite natural. The impurity may or may not have an electron, this is akin to having spin up or spin down. Similarly, the kinetic hybridization term transforms into a spin Mip interaction term of the form (S † d− + h:c:). As Vl couples to the occupancy √of the impurity site, we might anticipate Jz to scale with Vl . The additional correction (−2vF ( 2 − 1)l; 0 ) originates from the subtle transformation taking the original fermionic system into an eKective spin model. This problem has been solved by renormalization group methods (see Fig. 20). But simple arguments based on the X-ray edge singularity, orthogonality and recoil give the correct qualitative physics. When t = 0, the problem is that of the X-ray edge Hamiltonian (with d = 0). When t is Anite, the charge at the impurity orbital Muctuates (the impurity site alternately empties and Alls). This generates, in turn, a Muctuating potential. The X-ray absorption spectrum is the Fourier transform of the particle–hole pair correlator @(!) ∼ † (t)d(t)d† (0)(0)! :

(60)

This quantity, which is none other than the hybridization correlation function, should display the X-ray edge characteristics for large frequencies (! ¿ @eK ) where the eKect of recoil is

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Fig. 20. The renormalization Mow diagram for the model with Anite range interactions according to [107]. The initial values are Jx; 0 = 0; J⊥; 0 = 0:5, and Jx; l varies from 0 to 1 in increments of 0.05. When Jz; l becomes large enough (i.e., when V is large enough), the Mow veers from the usual Kondo Axed point to a zero-hybridization (J⊥; 0 = 0) singular Fermi liquid.

unimportant @(!) ≈ @0 (!=W ) ;

= −

20 2l : + 2

(61)

l

The threshold frequency @eK is determined by the recoil energy. W is the bandwidth. The bare hybridization width @0 ∼ t 2 =W . The exponent in the singularity contains an excitonic shift (−20 =) as well as an orthogonality contribution ( l 2l =2 ). The recoil is cut oK by @eK . For frequencies ! ¡ @eK the electron gas becomes insensitive to the change in the potential. As the X-ray edge singularity is cut oK at ! = O(@eK ), self-consistency implies that @eK = @(! = @eK ) :

(62)

This leads to the identiAcation @eK = W (@0 =W )B ;

B = 1=(1 − )

(63)

so that for ¡1 ;

@eK → 0

as W → ∞ :

(64)

For ¡ 1, a singular Fermi liquid emerges in which the hybridization of the localized d-orbital with the electron gas scales to zero at zero frequency. The actual value of determines the singular properties at low energy or temperature. In the single channel problem, such a scenario

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occurs if the potential V0 is suPciently attractive. On mapping to the spin problem we And that this region corresponds to the singular Fermi liquid ferromagnetic Kondo problem. The scaling of the hybridization to zero corresponds, in the spin model, to J± → 0. In a renormalization group language, the Mows impinge on the line of Axed points (J⊥ = 0; Jz ¡ 0). In this regime, we recover, once again, a continuous set of exponents. If the number of channels is large enough, the orthogonality catastrophe associated with the change in the number of particles on the impurity site is suPciently strong to drive the hybridization to zero even for the case of repulsive interactions V or antiferromagnetic Jz; l . In the singular regime various correlation functions may be evaluated [107]. For instance, the Green’s function of the localized impurity Gd () = − T d()d† (0) → e−

→0

l

(Vl =vF )2 ln(||+a)

:

(65)

The orthogonality induced by the Muctuation in the occupation number of the impurity site leads to the decay of the correlation function with a non-universal exponent. Owing to this orthogonality catastrophe the system behaves as a singular Fermi liquid. In the vicinity of the t = 0 Axed point, the self-energy due to hybridization ∼ !1− :

(66)

A line of critical points for ¡ 1 is found. This bears a resemblance to the Kosterlitz–Thouless phenomenon [147]. The analogue to the emergence of vortices in the Kosterlitz–Thouless transition are instantons—topological excitations which are built of a succession of spin Mips in time on the impurity site. 3.7. A model for mixed-valence impurities We next consider a slightly more realistic model [204,240,245] † H= kl ckl ckl + d nd; ↑ nd; ↓ + t (d† ck0 + h:c:) k;;l

+

k; k ;l

Vkk l

1 nd − 2

† ckl ck l

−1

:

(67)

In this model, both spin and charge may be altered on the impurity site. This enhanced number of degrees of freedom implies that the states need to be speciAed by more quantum numbers. This also allows, a priori, for higher degeneracy. In the following, N screening channels, all of equal strength V , will be assumed. In the U → ∞ limit, the spectrum of the impurity site may be diagonalized. The two lowest states are √ V N d † C |0 = |0; 1 with energy EC = − − ; 4 2 1=2  √ 2 V N D† |0 = |; 0 with energy ED = −  d − + t2 ; (68) 2 4

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where in the brackets, the Arst entry is the charge and spin of the impurity spin and the second one is√ the compensating charge in the screening channels. Other states are elevated by energies O(V= N ). The states satisfy the Friedel screening sum rule by having a small phase shift =2N in each of the N channels. The parameter V can be tuned to produce a degeneracy between the two states (EC = ED )—the mixed valence condition. The enhanced degeneracy produces singular Fermi liquid like behavior. 21 A perturbative calculation for small ! yields a self-energy ∼ [! ln ! + i! sgn(!)] + O[(! ln !)2 ] :

(69)

As a speculative note, we remark that this physics might be of relevance to quantum dot problems. Quantum dots are usually described in terms of the Anderson model. However, there are certainly other angular momentum channels whereby the local charge on the dot and the external environment can interact. As the external potential in the leads is varied, one is forced to pass through a potential in which this mixed valence condition must be satisAed. At this potential, the aforementioned singular behavior should be observed. 3.8. Multichannel Kondo problem Blandin and NoziZeres [198] invented the multichannel Kondo problem and gave convincing arguments for its local singular Fermi liquid behavior. Since then, it has been solved by a multitude of sophisticated methods. For an overview of these and of applications of the multichannel Kondo problem, we refer to [227]. The multichannel Kondo problem is the generalization of the Kondo problem to the case in which the impurity spin has arbitrary spin S and is coupled to n “channels” of conduction electrons. The Hamiltonian is HmcK = t

n ‘=1 i¡j;

(‘)† (‘) ci cj + h:c: + J S · c0(‘)† c0(‘) :

(70)

Here ‘ is the channel index. Degeneracy, the key to SFL behavior, is enforced through having an equal antiferromagnetic coupling J ¿ 0 in all the channels. When the couplings to the various channels are not all the same, at low enough temperatures a crossover to local Fermi-liquid behavior in the channel with the largest J‘ always occurs [7]. This crossover temperature is in general quite large compared to TK because channel asymmetry is a relevant perturbation about the symmetric Axed point. Therefore, in comparing this theory with SFL behavior in experiments, one should ensure that one is above the crossover temperature. The simple Kondo problem is the case 2S = 1 = n. In this case, at low temperatures a singlet state of the impurity state and the conduction electron electrons in the appropriate channel is formed. In the general multichannel case in which 2S = n, the physics is essentially the same, since there are exactly the right number of conduction electron channels to compensate the impurity spin at low temperatures. Thus, at low temperatures an eKective spin 0 state is formed 21

There is a singularity only in the local charge response at the impurity, not in the magnetic response. In this respect, the results of [244] are not completely correct.

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Fig. 21. In a multichannel problem, a (Hund’s rule coupled) spin S is compensated by total spin s1 (= n=2) in n diKerent conduction electron channels leaving a net uncompensated spin as shown. Alternatively exact compensation or over-compensation with special properties may arise, as discussed in the text.

again, and the properties of the compensated Kondo problem 2S = n at Anite temperature is that of crossover from a weakly interacting problem above the Kondo temperature TK to a strongly interacting problem below TK . The physics of the underscreened Kondo problem 2S ¿ n is diKerent. 22 In this case, there are not enough conduction electron spins to compensate the impurity spin. As a result, when the temperature is lowered and the eKective coupling increases, only a partial compensation of the impurity spin occurs by conduction electrons with spin opposite to it. As depicted in Fig. 21, a net spin in the same direction as the impurity spin then remains at the impurity site. Since the conduction electrons with their spin in the same directions as the impurity spin can then still make virtual excitations by hopping on that site while the site is completely blocked for conduction electrons with opposite spin, a net ferromagnetic coupling remains between the remaining eKective spin and the conduction electrons. As a result, the low-temperature physics of the underscreened Kondo problem is that of the ordinary ferromagnetic Kondo problem. To be more precise, the approach to the Axed point is analogous to that in the ferromagnetic Kondo problem along the boundary Jz = − J± , because the impurity must decouple (become pseudo-classical) at the Axed point. In the overscreened Kondo problem 2S ¡ n, there are more channels than necessary to compensate the impurity spin. At low temperatures, all n channels tend to compensate the impurity spin due to the Kondo eKect. Channel democracy now causes an interesting problem. As the eKective interaction J scales to stronger values, a local eKective spin with direction opposite to the impurity spin results. This eKective spin must have an eKective antiferromagnetic interaction with the conduction electrons, since now the virtual excitations of conduction electrons with spins opposite to the eCective local spin lower their energy. This then gives a new Kondo problem with a new eKective interaction, and so on. Of course, in reality one does not get a succession of antiferromagnetic Kondo problems—the net eKect is that a new stable Anite-J Axed point appears. As sketched in Fig. 22, the renormalized eKective interaction Mows to this 22 Since it is hard to imagine that the angular momentum states of the impurity are larger than that of the conduction electron states about the impurity, such models may be regarded to be of purely theoretical interest. See, however, Section 6.5.

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Fig. 22. Flow diagram for the degenerate over screened Kondo problem exhibiting a critical point.

Fig. 23. EKective shells in energy space in Wilson’s method. The Arst shell integrates over a fraction 1 of the top of the band, the next shell 1 of the rest, and so on. In the two-channel S = 1=2 problem, an S = 1=2 eKective impurity is left at every stage of interpretation.

Axed point both from the strong-coupling as well as from the weak-coupling side. One can understand this intuitively from the above picture: if one starts with a large initial value of J , then in the next order of perturbation theory about it, the interaction is smaller, since in perturbation theory the eKective interaction due to virtual excitations decreases with increasing J (see also Fig. 23). This means that J scales to smaller values. Likewise, if we start from small J , then initially J increases due to the Kondo scaling, but once J becomes suPciently large, the Arst eKect which tends to decrease J becomes more and more important. The Anite-J Axed point leads to non-trivial exponents for the low-temperature behavior of quantities like the speciAc heat (n−2)=(n+2) Cv T ˙ : (71) T TK For n = 2, the power-law behavior on the right-hand side is replaced by a ln T term.

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Fig. 24. Flow in temperature-anisotropy plane for the two-channel Kondo problem with coupling constant J1 and J2 .

Another way of thinking about the problem is in the spirit of Wilson’s renormalization group: Consider the problem of two channels interacting with a S = 12 impurity. The conduction states can be expressed as linear combinations of concentric orbitals of conduction electrons around the impurity. These successive orbitals peak further and further away from the impurity. Consider Arst the exchange coupling of the orbitals in each of the two channels peaking at the impurity site. Each of them has S = 12 . Only one linear combination of the two channels, call it red, can couple, while the other (blue) does not. So, after the singlet with the impurity is formed, we are left with a S = 12 , color blue problem. We must now consider the interaction of this eKective impurity with the next orbital and so on. It is obvious that to any order, we will be left with a spin 1=2 problem in a color. Conformal Aeld theory methods Arst showed that the ground state is left with 1=2 ln 2 impurity. A nice application of the bosonization method [85] identiAes the red and blue above as linear combinations of the fermions in the two channels so that one is purely real, the other purely imaginary. The emergence of new types of particles—the Majorana fermions in this case—often occurs at singular Fermi liquids. As a detailed calculation conArms [7], the J1 = J2 Axed point is unstable, and the Mow is like that sketched in Fig. 24. This means that the J1 = J2 Axed point is a quantum critical point: in the T − J1 =J2 phase diagram, there is a critical point at T = 0; J1 =J2 = 1. Moreover, it conArms that asymmetry in the couplings is a relevant perturbation, so that the SFL behavior is unstable to any introduction of diKerences between the couplings to the diKerent channels. The crossover temperature T× below which two-channel behavior is replaced by the approach to the Kondo Axed point is [7] T× = O(TK (J[ )[(J1 − J2 )= J[ ]2 ) ; where J[ = (J1 + J2 )=2.

(72)

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The overscreened Kondo problem is again an example where that the SFL behavior is associated with the occurrence of degeneracy; the critical point requires degeneracy of the two orthogonal channels. An interesting application of the two-channel Kondo problem is obtained by considering the spin label to play the role of the channel index, while the Kondo coupling is in the orbital angular momentum or crystal Aeld states for impurities at symmetry sites in crystals [143,69,70,227]. Another possibility that has been considered is that of scattering of conduction electrons of two-level tunneling centers with diKerent angular momentum states [302]. In this case, the tunneling model translates into a model with x and z coupling only, but this model Mows towards a Kondo-type model with equal x and y spin coupling. For both types of proposed applications, one has to worry about the breaking of the symmetry, and about the question of how dilute the system has to be for a Kondo-type model to be realistic. Interesting results in the tunneling conductance of two metals through a narrow constriction, shown in Fig. 25 appear to bear resemblance to the properties expected of a degenerate two-channel Kondo eKect [219,71], but this interpretation is not undisputed [290,220]. Applications to impurities in heavy fermions will be brieMy discussed in Section 6.4. 3.9. The two-Kondo-impurities problem In a metal with a Anite concentration of magnetic impurities, an important question is what the (weak) interaction between the magnetic impurities does to the Kondo physics—that the eKect might be substantial is already clear from the fact that the Kondo eKect is seen in logarithmic corrections about the high-temperature local moment Axed point while the RKKY interaction between the moments mediated by the conduction electrons occurs as a power law correction. 23 Stated simply, TK ∼ 1=N (0) exp(−1=N (0)J ) while the RKKY interaction I ∼ J 2 N (0). The existence of mixed-valence and heavy fermion metals makes this much more than an academic question. The question of the competition between these two eKects, and in particular whether long-range magnetic order can arise, was Arst posed by Varma [263] and by Doniach [78], who gave the obvious answer that RKKY interactions will be ineKective only when the Kondo temperature below which the local spin at each impurity is zero is much larger than the RKKY interaction I. Considering that JN (0) is usually 1, this is unlikely for S = 1=2 problems, but for large S, as encountered typically in rare earths and actinides, it is possible in some cases. However, the vast majority of rare earths and actinide compounds show magnetic order and no heavy fermion magnetic behavior. We will, however, only consider the S = 1=2 problem and work with unrealistic JN (0) so that the competition between the Kondo eKect and RKKY is possible. Heavy-fermion phenomena typically occur in solids with partially Alled inner shells (usually the f-shells of rare earth and actinide compounds) which hybridize very weakly with conduction electron bands formed of the outer orbitals (s, p and d) of the atoms [101,102]. They are usually 23 Stated technically, the Kondo Hamiltonian is a marginal operator while the RKKY operator is a relevant operator about the local moment Axed point: In a perturbation calculation, the interaction produces corrections of O(1=T ) compared to a ln(T ) correction of the Kondo eKect.

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Fig. 25. (a) DiKerential conductance G(V; T ) as a function of voltage V in measurements on metal point contacts by √ Ralph et al. [219], for various temperatures ranging from 0:4 K (bottom curve) to 5:6 K (upper curve). Note the V type behavior developing as the temperature decreases; (b) the zero√bias (V = 0) conductance as a function of temperature for three diKerent samples shows a G(0; T ) − G(0; 0) ∼ T behavior. The scaling behavior as a function of voltage and temperature is consistent with two-channel Kondo behavior [219].

modelled by a periodic array of magnetic moments interacting locally with AFM exchange interaction with conduction electrons. The two-Kondo-impurity problem therefore serves as a Arst step to understanding some of the physics of heavy fermions that is not primarily associated with the occurrence of a collective phase. In this subsection, we will summarize the results [136,137,8,9,245] for the two-Kondoimpurity problem: like the models we discussed above, this system also exhibits a quantum critical point at which SFL behavior is found. However, as for other impurity problems, an unrealistic symmetry must be assumed for a QCP and attendant SFL behavior. The two-Kondo-impurity Hamiltonian is deAned as † H=t ck ck + J [S1 · (r1 ) + S2 · (r2 )] : (73) k;

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In this form, the problem has a symmetry with respect to the midpoint between the two impurity sites r1 and r2 , and hence one can deAne even (e) and odd (o) parity states relative to this point. In the approximation that the k-dependence of the couplings is neglected [136], the two-Kondo-impurity Hamiltonian can then be transformed to H = Hnon-interacting + (S1 + S2 ) · [Je ck† e cke + Jo ck† o cko ]

+

kk

kk

Jm (S1 − S2 ) · [ck† e cko + ck† o cke ] :

(74)

The coupling constants Je ; Jo ; Jm are proportional to J with diKerent numerical prefactors. The coupling between the spin and orbital channels generates an eKective RKKY interaction HRKKY = I0 (Je ; Jo ; Jm )S1 · S2

(75) I0 = 2 ln 2(Je2 +Jo2 − 2Jm2 )

between the two impurity spins, with for t = 1. A very important point to note is that neglecting the k-dependences of the coupling introduces particle–hole symmetry in the problem, which is generically absent. The main results of a numerical Wilson-type renormalization group treatment of this model are the following: (i) For ferromagnetic coupling I0 ¿ 0 or for a small antiferromagnetic coupling I0 ¿ − 2:24TK , where TK is the Kondo temperature of the single-impurity problem, one Ands that there is a Kondo eKect with S1 · S2 = 0;

(76)

unless I0 is very small, |I0 =TK |1. Since for uncorrelated impurity spins S1 · S2 = 0, (76) expresses that although in the RG language the RKKY interaction is an irrelevant perturbation, it is quite important in calculating physical properties due to large “corrections to scaling”. Another feature of the solution in this regime is the fact that the phase shift is =2 in both channels. This means that at the Axed point, the even-parity channel and the odd-parity channel have independent Kondo eKects, each one having one electron pushed below the chemical potential in the Kondo resonance. As discussed below, this is due to particle–hole symmetry assumed in the model—without it, only the sum of the phase shifts in the two-channels is Axed. (ii) There is no Kondo eKect for I0 ¡ − 2:24TK . In this case, the coupling between the impurities is so strong that the impurities form a singlet among themselves and decouple from the conduction electrons. There is no phase shift at the Axed point. Also, in this case, the total spin Stot = 0, but the impurity spins become only singlet like, S1 · S2 ≈ −3=4, for very strong coupling, I0 − 2:24TK . So again there are important “corrections to scaling”. (iii) The point I0 = − 2:24TK is a true critical point, at which the staggered susceptibility (S1 − S2 )2 =T diverges. Moreover, at this point the speciAc heat has a logarithmic correction to the linear T dependence, Cv ∼ T ln T , while the impurity spin correlation function S1 · S2 becomes equal to −1=4 at this value. Although the approximate Hamiltonian (74) has a true quantum critical point with associated SFL behavior, we stress that the analysis shows that this critical point is destroyed by any k-dependent ck† e cko coupling. A coupling of this type is not particle–hole symmetric. As the approximate Hamiltonian (74) is particle–hole symmetric, these terms are not generated under

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the renormalization group Mow for (74). The physical two-Kondo-impurity problem (73), on the other hand, is not particle–hole symmetric. Therefore, the physical two-Kondo-impurity problem does not have a true quantum critical point—in other words, when the two-Kondo-impurity Hamiltonian (73) is approximated by (74) by ignoring k-independent interactions, relevant terms which destroy the quantum critical point of the latter Hamiltonian are also dropped. This has been veriAed by an explicit analysis retaining the symmetry breaking terms [244]. An illuminating way to understand the result for the two-impurity-Kondo problem, is to note that the Hamiltonian can be written in the following form:     H=  

Heven S=0

0

0

Hodd S=0 Hmix

0 Hmix

0

0

Hmix

Hmix

0



Heven S=1

0

0

Hodd S=1

    ;  

(77)

where S = S1 + S2 is the total impurity spin. In this representation, the Hamiltonian Hmix couples the S = 0 and 1 state, and the following interpretation naturally emerges: for large antiferromagnetic values of I0 , HS=0 is lower in energy than HS=1 , while for large ferromagnetic coupling I0 , the converse is true. The two-Kondo-impurity coupling can thus be viewed as one in which by changing I0 , we can tune the relative importance of the upper left block and the lower right block of the Hamiltonian. In general, the two types of states are mixed by Hmix , but at the Axed point H∗mix → 0. This implies that there is a critical value of I0 =TK where the S = 0 and 1 states are degenerate, and where SFL behavior occurs. At this critical value, the impurity spin is a linear combination of a singlet and triplet state with S1 · S2 = − 1=4 (i.e., a value in between the singlet value −3=4 and the triplet value 1=4) and the singular low-energy Muctuations give rise to the anomalous speciAc heat behavior. Within this scenario, the fact that the susceptibility & is divergent at the critical point signals that a term H · (S1 − S2 ) lifts the spin degeneracy. Moreover, the leading irrelevant operators about the Axed point are all divergent at the critical point—of course, this just reMects the breakdown of the Fermi-liquid description. The reason for Hmix → 0 is as follows. Hmix can only be generated from the last term in (74) which is particle–hole symmetric because under even–odd interchange, both the spin term and the fermion terms change sign. At the Kondo-Axed point, the leading operators must all be † biquadratic in fermions. An Hmix in that case would be of the form cke ck o and such a term by itself would break particle–hole symmetry, not consistent with the last term in (74). In the two-Kondo-impurity problem, one again encounters the essentials of degeneracy for quantum critical points and the need for (unphysical) constraints to preserve the singularity. Once again, new types of quantum numbers can be invoked in the excitations about the QCP. From the point of view of understanding actual phenomena for problems with a moderate concentration of impurities or in reference to heavy fermion compounds, the importance of the solution to the two-Kondo-impurity problem is the large correction to scaling found in the Wilson-type solution away from the special symmetries required to have a QCP. These survive

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quite generally and must be taken into account in constructing low-energy eKective Hamiltonians in physical situations. 4. SFL behavior for interacting fermions in one dimension We have already noted the unique phase-space restrictions for scattering of fermions in one dimension. For low-energy processes, one may conAne attention to one-particle states in the vicinity of the two Fermi-points, ±kF . These two points are always nested for particle–hole scattering (both in the singlet and the triplet spin-channels) and lead to singularities in the appropriate correlation functions. The upper cut-oK singularities is the bandwidth, just as the Cooper singularity in the particle–particle channel. The competition between these singularities lead to rather unique low-energy correlation functions in one dimension. The simplicity due to the sharp restriction in phase space allows a thorough analysis of the one-dimensional problem. A variety of elegant mathematical techniques, including exact solutions in certain non-trivial limits, have been employed to analyze the problem of interacting electrons in one dimension. There are also various diKerent ways of thinking about the one-dimensional problem, each of which provides insight into the general problem of singular Fermi liquids. We shall touch on these diKerent aspects without going into the details of the technical steps leading to their derivation. To get a Mavor for the mathematical nature of the results and their diKerences from Fermi liquids, we will present the derivation of the diagonalizable form of the Hamiltonian as well as exhibit the principal thermodynamic properties and correlation functions. Detailed reviews of the technical steps in the various solutions as well as numerical calculations may be found in [113,246]. We also discuss the special aspects of the one-dimensional problem and the methods and whether the results can be extended to higher dimensions. We Arst present the Hamiltonian and the T = 0 “phase diagram” (obtainable by perturbative RG) which identiAes the principal singularities for various coupling constants in the problem. This will be followed by a presentation of some of the results of the exact solution of the simpliAed model known as the Tomonaga–Luttinger model as well as the more general model for special values of the coupling constants (along the so-called “Luther–Emery” line [163]). Since in one dimension, hard core bosons and spinless fermions cannot go around each other, special features in their statistics may be intuitively expected. A special feature of one-dimensional physics is that the low-energy excitations can be described by either fermions or bosons. The bosonic description of the Tomonaga–Luttinger model is especially attractive and will be presented below. A related distinctive feature of one-dimensional physics is that single-particle as well as multiple-particle correlation functions are expressible in terms of independent charge and spin excitations, which, in general, propagate with diKerent velocities. 24 This feature has been shown 24

Even a Fermi liquid displays distinct energy scales for charge and spin (particle–hole) Muctuations because of the diKerence in the Landau parameters in the spin-symmetric and spin-antisymmetric channels. The phrase spin–charge separation should therefore be reserved for situations, as in one dimension, where the single-particle excitations separate into objects which carry charge alone and which carry spin alone.

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to arise due to extra conservation laws in one dimensions [51,179]. As we shall discuss, an extension of charge–spin separation to higher dimensions is unlikely because there are no such conservation laws. The one-dimensional singularities may also be seen as a manifestation of the orthogonality catastrophe [20] that we discussed in Section 3.4. We shall see that this feature disappears in higher dimension due to the eKects of recoil. Some cases where the one-dimensional models solved are experimentally realized include the edge states of quantum Hall liquids and quasi-one-dimensional organic and inorganic compounds [83,84]. In the latter case, the asymptotic low-energy properties are, however, unlikely to be those of the one-dimensional models because of the inevitable coupling to the other dimensions which proves to be a relevant perturbation. Nonetheless, data on carbon nanotubes [74] discussed in Section 4.11 show clear evidence of one-dimensional interacting electron physics. Several one-dimensional spin chains problems can also be transformed into problems of one-dimensional fermions [225]. 4.1. The one-dimensional electron gas In this section, we shall outline the special features of the one-dimensional problem which make it soluble and show its singular properties. As it often happens, solubility implies Anding the right set of variables in terms of which the Hamiltonian is expressed as a set of harmonic oscillators. In one dimension, the “Fermi surface” is reduced to the two Fermi points k = ± kF . At low energies, particles may move only to the right or to the left with momenta of almost Axed magnitude k rkF (belonging, to either the right (r = 1) or to the left (r = − 1) moving branch). Due to the constrained character of one-dimensional motion, the phase space for collisions between particles is severly limited compared to higher dimensions. Let us start oK by an inspection of the possible collisions. By energy and lattice momentum conservation alone, all the low-energy scattering processes may be classiAed into four interactions. These interactions are schematically illustrated in Fig. 26.

Fig. 26. Pictorial representation of the low-energy interaction terms in the one-dimensional problem. After [181]. The “+” and “−” points are a shorthand for the two Fermi points k = kF and (−kF ), respectively.

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By observing this small set of allowed processes we note that the general Hamiltonian describing the low-energy dynamics may be split into four parts H = H0 + Hforward + Hbackward + HUmklapp :

(78)

In Eq. (78) H0 is the free electron Hamiltonian † H0 = k ck ck :

(79)

k

The other three terms describe the interactions depicted in Fig. 26. The Umklapp process (HUmklapp ), denoted conventionally by H3 as in Fig. 26, describes interactions in which the lattice momentum is conserved yet plain momentum is not: i.e., k1in + k2in = k1out + k2out + G with G = 0 a non-vanishing reciprocal lattice “vector” (a scalar in this one-dimensional case). From Fig. 26, we note that such a process can occur only in the special case when 4kF is very close to a reciprocal lattice “vector”. It follows that except for halfAlling of the one-dimensional band, such a process cannot occur. So, in general, one needs to study only forward (H2 or H4 ) processes or backwardscattering (H1 ) processes. Note that Hforward describes interactions with small momentum transfer and Hbackward momentum transfer close to 2kF . We will now introduce a few simplifying assumptions which form the backbone of all Luttinger liquid treatments of the one-dimensional problem: (1) In Eq. (79), k may be expanded about the two Fermi points to produce the linear dispersion r (k) = vF (rk − kF ) + EF

(80)

with vF and EF denoting the Fermi velocity and the Fermi energy, respectively. As before, r = ± 1 is the right=left branch index. (2) The band cutoK is taken to be inAnite. These assumptions lead us to focus attention on a simpliAed system in which there are two independent Mavors of particles (right and left movers) each of which has a linear dispersion relation with unbounded momentum and energy. The simpliAed energy spectrum is shown in Fig. 10: an inAnite “sea” of unphysical (negative energy) states below the usual Fermi sea. The added inAnity of unphysical states with k ¡ 0 have a negligible physical eKect (as they are far removed from the chemical potential, they enable the problem to be tractable mathematically). First we explain how the one-dimensional Hamiltonian can be expressed equivalently in terms of Bosonic variables. DeAne the charge-density operators Hr and the spin-density operators Sr for the two branches, r = ±, by Hr = =±1

† r; r; ;

Srz = 12 ;

† z r; ; r;

;

where z is a Pauli matrix. The Fourier transform of the particle-density operators is † Hr; (q) = cr; ; k+q cr; ; k = H†r; (−q) : k

(81)

(82)

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A pivotal point is that, within the stated assumptions, these Fermi bilinears may be explicitly shown to obey Bose commutation relations. 25 We will soon see that these density operators are not merely bosonic but that, in the linear-band approximation, they may also be viewed as raising and lowering operators which reduce the Hamiltonian Eq. (78) to a simple quadratic (oscillator) form. With our Arst assumption, we may linearize k about the two Fermi points for low-energy processes; the energy of a particle–hole pair created by Hr; (q): r; k+q − r; k = rvF q

(83)

is independent of k in a Luttinger liquid. (This step cannot be implemented in higher dimensions.) In other words, states created by Hr; (q) are linear combinations of individual electron– hole excitations all of which have the same energy and are therefore eigenstates of H0 . It follows that for q ¿ 0, the bosonic Hr=+; (q) [Hr=−; (q)] is a raising [lowering] operator. The kinetic energy H0 may be expressed in terms of the density operators H0 =

2vF Hr; (rq)Hr; (−rq) : L r=±

(84)

q¿0

Upon separating the densities on a given branch into charge and spin pieces Hr (x) = 12 [Hr (x) + Srz (x)] ;

(85)

the free Hamiltonian may be expressed as a sum in the spin and charge degrees of freedom H0 = H0 [Hr ] + H0 [Srz ] : (86) r

r

It follows that in the non-interacting problem, spin and charge have identical dynamics and propagate in unison. Once interactions are introduced, the electron will “fractionalize” and spin and charge dynamics will, in general, diKer. We now turn to a closer examination of the various interaction terms in the Hamiltonian. The part describing the forward scattering (small momentum transfer) events → (kF ; ; kF ; ) may be further subdivided (as shown) into the processes (kF ; ; −kF ; ) → (kF ; ; −kF ; ) and (kF ; ; kF ; ), respectively Hforward = H2 + H4

(87)

25 q); Hr=+1; (q)] = Explicitly, for the right movers the only non-vanishing commutation relations read [Hr=+1; (− (n − n ). By invoking the last of the stated Luttinger liquid assumptions, we And that k − q k k (nk −q − nk ) = k k¿k0 (nk −q − nk ) = k0 −q6k 6k0 nk = Lq=2. Here, k0 is a high momentum cut-oK which is taken to inAnity at the end of the calculation. Similar relations are found for the left movers (r = − 1). Taken together, we And the

operators Hr; (q) to be unnormalized Bose operators: [Hr; (q); Hr ; ] = ; r; r q+q ; 0 (rqL=2):

318

with

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1 g2 H+; (q)H−; (−q) ; L q 1 H4 = g4 [H+; (q)H+; (−q) + H−; (q)H−; (−q)] : 2L q H2 =

(88) (89)

The operators Hr; involve a creation and annihilation operator on the same branch. Let us further deAne also operators Hr formed from bilinears of fermions on opposite branches † cr; ; k c−r; ; k+q : (90) Hr = k

In terms of these, the Hamiltonian describing backscattering interactions (the scattering event (+kF ; ; −kF ; ) → (−kF ; ; kF ; ) and its reverse) becomes − Hbackwards = H1 = g1 g1 H+ (91) (q)H (−q) q

and the Umklapp term reads 1 + − − Humklapp = g3 [H (q)H+ (−q) + H (q)H (−q)] : 2L q

(92)

The behavior of gi; (q) in momentum space translates into corresponding real-space couplings. If the couplings {gi (q)} are momentum independent constants, then the corresponding real-space interactions are local and describe contact collisions. Unless otherwise stated, this is the case that we shall consider. In all these expressions, the coupling constants may be spin-dependent

gi = gi + gi⊥ − :

(93)

As the terms H1 and H2 describe the same process, we may set g1 = 0 with no loss of generality. As already mentioned, Umklapp processes are important only when 4kF is a reciprocal lattice vector so that all scattering particles may be near the Fermi points. The condition for spin rotation invariance [H; ˜S] = 0 reduces the number of independent coupling constants further g2⊥ − g1⊥ = g2 − g1 :

(94)

On examining all four possible interactions, we note that the forward scattering H2 and H4 break no symmetries but that H1 and H3 may; The latter leads to qualitatively new properties. Backscattering breaks the SU (2)L ⊗ SU (2)R symmetry of spin currents for each of the individual left–right moving fronts. Gaps (or condensates) are usually associated with broken symmetries, and this case is no exception: A spin gap @s ¿ 0 is dynamically generated when these interactions (i.e., an attractive backscattering (H1 ) process (+kF ; ; −kF ; ) → (−kF ; ; kF ; )). Similarly, the Umklapp process (H3 ) breaks the conservation of individual charge currents; a charge gap @c ¿ 0 is associated with this broken Galilean invariance. The gaps open up only if the interactions are attractive. On a formal level, Umklapp breaks independent right and

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Fig. 27. Phase diagram for one-dimensional interacting fermions in the (Kc ; g1 ) plane with g1 = 0 in Eq. (93) and Kc given by Eq. (103). SDW and (CDW ) in the upper left quadrant indicates that both the spin density wave susceptibility and the charge-density wave susceptibility diverge as T → 0, but that the spin-density wave susceptibility diverges a factor ln2 T faster than the charge-density wave susceptibility—see Eq. (112). Other sectors are labeled accordingly. From [233].

left (U (1)L ⊗ U (1)R ) charge conservation symmetry leaving the system with only a single U (1) symmetry. We shall later show in Section 4.7 how unbroken U (1)L ⊗ U (1)R and SU (2)L ⊗ SU (2)R symmetries (in models without Umklapp or backscattering) allow independent left–right conservation laws with interesting consequences. Before discussing the exact solutions to sub-classes of the above general model, it is good to obtain physical insight through a “phase diagram” obtained by the perturbative renormalization group Mow equations [246,233]. Due to the limitation of phase space, the one-dimensional problem is subject to all manners of competing singularities. In one dimension, there are no truly ordered phases of course, but at T = 0 correlation functions diverge and one may say that there is algebraic long-range order. One may thus determine a “phase diagram” according to which susceptibilities diverge as T → 0: the one associated with singlet superconductivity (SS), triplet superconductivity (TS), a charge-density wave (CDW ) at 2kF , and a spin-density wave (SDW ) at 2kF . The expressions for these susceptibilities are given in Eq. (112) below, and the resulting “phase diagram” is shown in Fig. 27. 26 26 It is a useful exercise (left to the reader) to see how Fig. 27 corresponds to the intuitive notions of what kind of interaction, short- or long-range, in singlet or triplet channel, favors which instability. These notions are transferrable to higher dimensions.

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4.2. The Tomonaga–Luttinger model With forward scattering alone in (78) and after linearizing the kinetic energy, we obtain the Tomonaga–Luttinger (T–L) model. In terms of fermion Aeld operators r; (x) and r;† (x), and density and spin operators Hr (x) and Srz (x) in real space, the T–L model is

1 HT–L = d x −vF r r;† i9x r; + g2; c Hr (x)H−r (x) 2 r=± r;=± z + g4; c Hr (x)Hr (x) + 2 g2; s Srz (x)S−r (x) + g4; s Srz (x)Srz (x) ; (95) r=±

r=±

where gic =

gi + gi⊥ ; 2

gis =

gi − gi⊥ : 2

(96)

Note that the g2; s term is the only term which breaks SU (2) spin symmetry. The T–L model is exactly solvable. After all, as previously noted, the (Dirac-like) kinetic energy Hamiltonian H0 is also quadratic in the density operators. So the Hamiltonian is readily diagonalized by a Bogoliubov transformation whereupon the Hamiltonian becomes a sum of two independent (harmonic) parts describing non-interacting charge- and spin-density waves: the charge- and spin-density waves are the collective eigenmodes of the system. The simplest way to solve the TL model and to explicitly track down these collective modes is via the bosonization of the electronic degrees of freedom. 27 The bosonic representation of the fermionic Aelds proceeds by writing [113,193,225] r; (x) = lim a→0

exp[ir(kF x + Kr (x))] √ Fr ; 2a

(97)

where a is a short distance regulator. Kr (x) satisAes [Kr (x); Kr† ; (x )] = − ir; r ; sign(x − x ) :

(98)

The so-called Klein factors 28 Fr are chosen such that the proper fermionic anticommutation relations are reproduced. The exponential envelope exp[iKr (x)] represents the slow bosonic collective degrees of freedom which dress the rapidly oscillating part Fr exp[ikF x] describing the energetic particle excitations near the Fermi points.

27

The reader should be warned that many diKerent conventions abound in the literature. The Klein factors connect states diKering by one electron. When the thermodynamic limit is taken in a gapless system, there is, for all practical purposes (the computation of correlation functions), no diKerences between states containing N and N ± 1 particles. However, when gaps open up, giving rise to Anite correlation lengths, caution must be exercised when dealing with these operators. The literature contains several examples of calculations which were later discovered to be incorrect, precisely due to this subtlety. 28

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The slowly varying Aelds K may be written in terms of the bosonic Aelds .c; s and their conjugate momenta 9x Lc; s Kr; = =2[(Lc − r.c ) + (Ls − r.s )] : (99) In terms of the new variables, the familiar charge and spin densities are 2 1 z z Hr (x) = 9x .c ; S (x) = Sr (x) = 9x .s : H(x) = 2 r r

(100)

In the (Lc; s ; .c; s ) representation, the Tomonaga–Luttinger Hamiltonian becomes a sum of two decoupled sets of oscillators describing the gapless charge and spin density wave eigenmodes v (9x . )2 T–L 2 ≡ HsT–L + HcT–L : H = dx K (9x L ) + (101) 2 K =c; s The velocities of the collective charge and spin modes are easily read oK by analogy to a harmonic string c; s 2 g4c; s 2 g2 vc; s = vF + − : (102) Likewise, the moduli determining the power-law decay of the correlations are vF + g4c; s − g2c; s Kc; s = : vF + g4c; s + g2c; s

(103)

In Section 4.7, we shall show how the above expressions for the spin and charge density wave velocities simply follow from the conservation of left and right moving particles in the T–L model. As previously noted, the charge and spin velocities are degenerate in the non-interacting model. When interactions are introduced, the charge and spin velocities (vc and vs ) as well as the energy to create spin and charge excitations (vs =Ks and vc =Kc , respectively) become diKerent. The charge constant Kc is less than 1 for repulsive interactions, which elevates the energy of the charge excitations, while Kc is greater than 1 for attractive interactions. 4.3. Thermodynamics As evident from (101) the contributions of the independent charge and spin modes must appear independently in most physical quantities. The speciAc heat coePcient is found to be vF 1 1 =0 = + ; (104) 2 vc vs where = 0 for the non-interacting system.

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¡ Fig. 28. The zero-temperature spectral function A(k; !) = Im{Gr=+1 (k; !)} as a function of ! for the case (g2 = 0; g4 = 0)—the “one-branch Luttinger liquid” in which the spin and charge velocities diKer but for which the correlation exponents retain their canonical value Kc = 1 according to (103). In the Agure vc ¿ vs and k ¿ 0 are assumed. Note the inverse square root singularities. This is a consequence of Kc = 1 which makes c = 0. After Voit [278].

The spin susceptibility and the compressibility are also readily computed from (101) &0 = vF =vs ;

%=%0 = vF Kc =vc ;

(105)

where &0 and %0 are the susceptibility and compressibility of the non-interacting gas. The Wilson ratio, already encountered in our discussion of the Kondo problem in Section 3.2, RW =

&=&0 2vc = =0 vc + vs

(106)

deviates from its Fermi-liquid value of unity by an amount dependent on the relative separation between the spin and charge velocities. 4.4. One-particle spectral functions We display the calculated zero temperature spectral functions of the T–L model [278] in order to point out the diKerences from the Landau Fermi-liquid discussed in Section 2 A(k; !) ≈ (! − vc (k − kF ))2c −1=2 |! − vc (k − kF )|c −1=2

(vc ¿ vs ) ;

A(k; !) ≈ (! − vs (k − kF ))c −1=2 |! − vs (k − kF )|2c −1=2

(vc ¡ vs ) :

(107)

These spectral functions are sketched in Fig. 28 for the case g2 = 0 and in Fig. 29 for the general case. Note that unlike the single quasiparticle pole in A(k; !) in a Landau Fermi liquid, A(k; !)

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Fig. 29. The generic (g2 = 0; g4 = 0) zero-temperature spectral function. Note the broader range of non-trivial singularities near vs k and ±vc k. Here both the eKect of spin–charge velocity diKerence and the emergence of non-trivial exponents is visible. After Voit [277,278].

in a Lutinger liquid is smeared with a branch cut extending from the spin mode excitation energy to the charge mode excitation energy. These branch cuts split into two in an applied magnetic Aeld, see Fig. 30. Note also the important diKerence for the case shown in Fig. 28, that g2 = 0, when the left and right branches are orthogonal, from the general case shown in Fig. 29. These results are exact for small ! and small |k − kF |. Another manifestation of the SFL behavior is the behavior of the momentum distribution function derived from A(k; !) by integrating over !: nk ∼ nkF − const × sign(k − kF )|k − kF |2c ;

(108)

c; s = 18 (Kc; s + Kc;−1s − 2) :

(109)

where

In contrast to a Fermi liquid, the expression for nk does not exhibit a step-like discontinuity at the Fermi points. The exponent 2c is non-universal (as usual, an outcome of a line of critical points). The single-particle density of states obtained from A(k; !) by integrating over k: N (!) ≈ |!|2c vanishes at the Fermi surface.

(110)

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Fig. 30. The energy distribution curve (the spectrum A(k; !) at Axed k) as a function of ! in the presence of a magnetic Aeld. The dashed line is the zero-Aeld result of Fig. 28. The magnitude of the Zeeman splitting is enhanced with respect to (vc − vs )k for clarity. From Rabello and Si [218].

The spectral function A(k; 0), at the chemical potential, has also been calculated as a function of temperature. Representative plots are shown in Fig. 31. These are to be contrasted with the delta-function in a Landau Fermi liquid. The reader will further note that in Fig. 31, the energy distribution curves are much broader than the momentum distribution curves. This is a general occurrence in one-dimensional systems and is a consequence of the fact that an injected electron of momentum and energy (k; !) disintegrates (while conserving energy and momentum) into two independent spin and charge excitations having energies !c; s = vc; s |k |. 4.5. Correlation functions Since the Hamiltonian is separable in charge and spin and as is a product of independent charge and spin degree of freedom, all real-space correlation functions are products of independent charge and spin factors. We show the most important correlation functions in the illustrative examples below, and refer for a summary of the various exact expressions to [202].

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Fig. 31. (Left panel) Momentum distribution curves at ! = 0 (i.e., the spectrum at Axed ! = 0 as a function of k) for a spin rotationally invariant Tomonaga–Luttinger liquid, plotted as a function of vs k=T ; (right panels) energy distribution curves at k = 0 (the spectrum at Axed k = 0) as a function of !=T . In both panels, vc =vs = 3 and c = 0 in (a), c = 0:25 in (b), and c = 0:5 in (c). From Orgad [202].

Sometimes, the Bosonization method in its elegance obscures the underlying physics of these correlation functions. The genesis of the power-law dependence of the correlations exhibited below is the nesting in both charge and spin particle–hole channels and the Cooper channel. The logarithmic singularities evident in the simplest calculation turn into power laws on summing the singularities exactly. To obtain the exact values of the exponents one requires an exact method, for example Bosonization.

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The most important feature of the large distance behavior of the charge and spin correlators is their algebraic decay at zero temperature Kc cos(2kF (x − x )) −3=2 H(x)H(x ) + B ln |x − x | 1; c ((x − x ))2 |x − x |1+Kc cos(4kF (x − x )) +B2; c + ··· |x − x |4Kc 1 cos(2kF (x − x )) 1=2 (111) + B ln |x − x | + · · · ˜S(x) · ˜S(x ) 1; s ((x − x ))2 |x − x |1+Kc at asymptotically long distances and Ks = 1. For not very repulsive interactions, so that Kc ¡ 1, the 2kF Muctuations are dominant. We have previously seen that such a CDW=SDW instability may arise due to the special 2kF nesting wavevector in one dimension. 29 The amplitudes {Bi; c } and {Bi; s } are non-universal while the exponents are determined by the stiKness of the free charge and spin Aelds. While the above expressions are for Ks = 1, in the general case Ks = 1, the spin correlator decays asymptotically with the exponent (Ks + Kc ). At non-zero temperatures, it is found that the Fourier transforms of these correlation functions scale as &CDW ≈ T Kc −1 |ln T |−3=2 ; &SS ≈ T Kc −1 |ln T |−3=2 ;

&SDW ≈ T Kc −1 |ln T |1=2 ; &TS ≈ T Kc −1 |ln T |1=2 :

(112)

The “phase diagram” shown in Fig. 27 is of course consistent with the dominant singularities of (112). The reader will note that the quantities on the left-hand side of Eq. (112) diKer from those on the right-hand side by a factor of |ln T |−2 , but that if any quantity on the right-hand side diverges, then so does its counterpart (with the same power of T ). The dominant and the subdominant divergences are marked in each sector of the phase diagram depicted in Fig. 27, with the subdominant behavior indicated between brackets. These results also lead, in principle, to clear experimental signatures. X-rays, which couple to the charge density waves, should peak at low temperatures with intensities given by I2kF ∼ T Kc ;

I4kF ∼ T 2Kc −1 :

(113)

The NMR probe couples to the spin degrees of freedom and the theoretically computed nuclear relaxation time scales as T1 ∼ T −Kc : 29

(114)

We also show a 4kF modulation of the charge-density correlation which arises due to Umklapp scattering near (q independent couplings gi ) we half-Alling, i.e, 4kF = a reciprocal vector. If instead of point contact interactions augment the system by additional long-range Coulomb interactions via d x d x V (x − x )9x .c 9x .c with a Coulomb like kernel V (x) ∼ [x2 + d2 ]−1=2 then singular density correlations at 4kF are triggered. This is of course related to the physics of Wigner crystallization. Owing to the one-dimensional character of the system no true long-range order can be found;√however, the 4kF component of the charge–charge correlations decays in an extremely slow fashion ∼ (exp[ − A ln x]) (slower than algebraic).

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4.6. The Luther–Emery model The Luther–Emery model extends the Tomonaga–Luttinger Hamiltonian by including the backscattering interactions parametrized by H1 , which scatter from (+kF ; ; −kF ; ) to (−kF ; ; kF ; ) and vice versa. The Umklapp processes (H3 ) continue to be discarded. The backscattering term † † H1 = d x g1 (115) r; =+1 −r; =−1 r; =−1 −r; =1 r=±1

written in terms of the bosonic variables introduces a non-trivial sine-Gordon like interaction [113] √ vs (9x .s )2 2g1 2 Hs = dx Ks (9x Ls ) + + cos( 8.s ) (116) 2 Ks (2a)2 with rescaled values of the spin and charge velocities and stiKness constants. 30 When g1 ¿ 0 (repulsive interactions), g1 is renormalized to zero in the long wavelength limit. Since along the RG Mow trajectories Ks − 1 ≈ g1 =(vs ), this means that Ks renormalizes to 1. The physics corresponding to this case is in the Tomonaga–Luttinger model that we just discussed. When symmetry breaking backscattering interactions are attractive and favorable (g1 ¡ 0) the Tomonaga–Luttinger SU (2)L ⊗ SU (2)R symmetry is broken and an associated spin gap opens up. On a more formal level, the non-trivial cosine term in Eq. (116) leads to diKerent minimizing values of the spin Aeld .s dependent on the sign of g1 . Consequently, when the backscattering interactions are attractive, a spin gap of magnitude vs g1 1=(2−2Ks ) @s ∼ (117) a 22 vs opens up. The attractive backscattering leads to the formation of bound particle–hole pairs which form a CDW. The spin correlation length is then Anite vs Os = : (118) @s In the spin gapped phase, 31 the Hamiltonian can be conveniently expressed in terms of new refermionized spin Aelds Pr (x). (119) Pr = Fr exp[ − i =2(Ls − 2r.s )] : Luther and Emery observed that at the point Ks = 1=2, the Hamiltonian in terms of these new spin Aelds becomes that of non-interacting free fermions having a mass gap @s = g1 =(2a). Hs = dx [ − ivs rPr† 9x Pr + @s Pr† P−r ] (120) r=±1 30

If Umklapp√ scattering were included (it is not in the present section), then an analogous term 2g3 =(2a)2 cos( 8.c ) would be generated. The spin and charge Aelds then take similar roles for the backscattering (H1 ) and Umklapp (H3 ) interactions. 31 Gaps in the charge spectrum also develop when Umklapp scattering is relevant.

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leading to the spin excitation spectrum Es = @2s + vs2 (k − kF )2 :

(121)

We will now discuss how the spin gap may fortify both superconductivity and the 2kF chargedensity wave order. When expressed in terms of Bose operators, the superconducting gap operator @SC =

† † r=−1; =+1 r=1; =−1

+

† † r=1; =+1 r=−1; =−1

(122)

turns, as all correlators do, into a product of the spin and charge degrees of freedom. For @SC , the relevant product is amongst the cosine of the spin Aeld .s and an exponential of the Lc operator (i.e., the Aeld dual to the charge =eld). Thus the cosine of the spin Aeld plays the role of the amplitude. It follows that when spin Muctuations are frozen (by opening a gap), superconducting correlations may be consequently enhanced. The appearance of the dual Aeld Lc in an expression for the superconducting gap @SC should come as no surprise as superconducting (phase) and charge (number operator) order are conjugate and dual to each other. According to Eq. (112), in the presence of a spin gap, the superconducting susceptibility scales as &SC ∼ @s T (1=Kc )−2 : The 2kF charge density wave order † Hr2kF = r; −r;

(123)

(124)

is associated, as it must, with the charge Aeld .c (in lieu of its dual Lc ). Consequently, a computation shows that &CDW ∼ @s T Kc −2 :

(125)

Note that the appearance of Kc in the exponent by contrast to the appearance of 1=Kc in &SC associated with the dual charge Aeld Lc . This is once again a part of the old maxim that “CDW (or number) ordering is conjugate and dual to superconducting (or phase) order” in action. In conclusion: The charge Aeld or its dual (the phase Aeld) may condense, under the umbrella of the spin gap, to a 2kF charge density wave or to a superconducting gap. This completes our compendium of the essential properties of one-dimensional interacting fermions. We will now critically examine the results from several diKerent points of views. 4.7. Spin–charge separation As in many other physical problems, the availability of an exact solution to the onedimensional electron gas problem is intimately linked to the existence of additional conservation laws or symmetries. One may attack the Luttinger liquid problem by looking for its symmetries.

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The U (1)L ⊗ U (1)R symmetry present in the absence of Umklapp scattering may be exhibited by considering the eKect of the separate left and right rotations by angles -L; R on the fermion variables L (x; t)

→ ei-L

L (x; t);

R (x; t)

→ ei-R

R (x; t)

:

(126)

All of the currents are trivially invariant under this transformation as the † Aelds transform with opposite phases. Physically, this corresponds to the conservation of the number and net spin of left and right moving particles. As discussed by Metzner and Di Castro [179], these separate conservation laws for the left and right moving charge and spin currents lead to Ward identities which enable the computation of the single-particle correlation functions. In the absence of Umklapp scattering, charge is conserved about each individual Fermi point. The net total charge density H ≡ H+ + H− and charge density asymmetry H˜ ≡ H+ − H− in the Tomonaga–Luttinger Hamiltonian satisfy the continuity equations 9 H = [H; H] = − qj;

9 H˜ = [H; H] ˜ = − qj˜ ;

(127)

where j(q) = uc [H+ − H− ];

j˜ = u˜ c [H+ + H− ]

(128)

and where the velocities are given by uc = vF +

g4c − g2c ;

u˜ c = vF +

g4c + g2c :

(129)

These results follow straightforwardly from the form of HT−L in combination with the fact that the only non-zero commutator is qL [Hr; (q); Hr ; (−q )] = qq rr : (130) 2 Let us illustrate simply how many of the results derived via bosonization may also be directly computed by employing these conservation laws. The existence of gapless charge modes is a direct consequence of the right–left charge conservation laws. The two Arst-order continuity equations given above lead to [92 + uc u˜ c q2 ]H = 0

(131)

from which we can read oK a linear charge dispersion mode ! = vc |q|

(132) √

with velocity vc = uc u˜ c , in agreement with the earlier result (102). Thus, collective charge excitations propagate with a velocity vc . A similar relation may be found for the spin velocity vs which in general is diKerent from vc . This spin–charge separation also becomes clear from

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the explicit form of the expectation values of the charge and spin densities 0| r (x0 )Hr (x; t)

† r (x0 )|0 = (x

− x0 − rvc t) ;

0| r (x0 )Srz (x; t)

† r (x0 )|0 = (x

− x0 − rvs t) ;

(133)

where |0 denotes the ground state. The separate right–left conservation laws cease to hold if (backscattering) impurities are present. Accordingly, as shown by Giamarchi and Schulz, [108] spin–charge separation then no longer holds. 4.8. Spin–charge separation in more than one dimension? Spin–charge separation in one dimension requires extra conservation laws. Can something analogous occur in more than one dimension? No extra conservation laws are discernible in the generic Hamiltonians in two dimension, although such Hamiltonians can doubtless be constructed. Are there conditions in which generic Hamiltonians become dynamically equivalent to such special Hamiltonians (because the unwelcome operators are “irrelevant”)? No deAnite answers to these questions are known. In Section 5.2 and later in this section we shall brieMy review some interesting attempts towards spin–charge separation in higher dimensions. First we present qualitative arguments pointing out the diPculty in this quest. There is a simple caricature given by Schulz [233] for qualitatively visualizing charge–spin separation for a special one-dimensional case: the U → ∞ Hubbard model. This model is characterized (at half-Alling) by the algebraic decay of spin-density correlations, which at short distances appear as almost antiferromagnetic alignments of spins. Let us track the motion of a hole introduced into an antiferromagnetically ordered chain. The hole is subject to only the lattice kinetic term which enables it to move by swapping with a nearby spin. An initial conAguration will be : : : ⇓⇑⇓⇑⇓ O ⇓⇑⇓⇑⇓ : : :

(134)

After one move the conAguration is : : : ⇓⇑⇓⇑ O ⇓⇓⇑⇓⇑⇓ : : :

(135)

After two additional moves to the left the conAguration reads : : : ⇓⇑ O ⇓⇑⇓⇓⇑⇓⇑⇓ : : :

(136)

Thus the initial hole surrounded by two spins of the same polarization has broken into a charge excitation (“holon” or “chargon” — a hole surrounded by antiferromagnetically aligned spins) and a spin excitation (“spinon”) composed of two consecutive parallel spins in an antiferromagnetic environment. The statistics of the localized spinons and holons in this model must be such that their product is fermionic. The feasibility of well-deAned spin and charge excitations hinges on the commuting nature of the right and left kinetic (hopping) operators TRight ; TLeft which are the inverse of each other.

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Any general term of the form Right

Left

Left

Right

NL −NR NL −NR (TLeft )n1 (TRight )n1 (TLeft )n2 (TRight )n2 : : : = (TLeft TRight )NR TLeft = I × TLeft ; (137) R; L nL where NR; L = i ni . We have just shown that terms of the form TL lead to a representation of the sort depicted above which gives rise to spin–charge separation and therefore our result holds for the general perturbative term. The proof of spin–charge separation for the one-dimensional electron gas rests on the existence of separate conservation laws for the left and right moving domain walls, as a result of the fact that the operators TRight and TLeft commute. Such a simple “proof ” cannot be extended to higher dimensions. In higher dimensions this suggestive illustration for spin–charge separation is made impossible by the non-commuting (frustrating) nature of the permutation operators TUp ; TDown ; TRight ; TLeft ; : : : Moreover, even if the nLeft nUp exchange operators commuted we would be left with terms of the form TLeft TUp which when acting on the single hole state will no longer give rise to states that may be seen as a direct product of localized holon and spinon like entities. Let us simply illustrate this by applying a sequence of various exchanges on the planar state | : −+−+−+−+ +−+−+−+− −+−+−+−+ ; (138) +−+ 0 +−+− −+−+−+−+

where + and arrive at | −+− +−+ − −+ ++− −+−

2 2 − denote up and down spins, respectively. By applying TDown TRight TUp TLeft we

+ − + 0 +

− + − + −

+ − + − +

− + − + −

+ − + − +

(139)

2 2 | = | . Unlike the one-dimensional a state which obviously diKers from TDown Tup TRight TLeft case, damage is not kept under check. Note the extended domain wall neighboring the hole, enclosing a 2 × 2 region of spins of the incorrect registry. Note also that the hole is now surrounded by a pair of antiferromagnetically aligned spins along one axis and ferromagnetically aligned spins along the other. A path closing on itself does not lead to the fusion of the “holon” and “spinon” like entities back into a simple hole. As the hole continues to further explore both dimensions, damage is continuously compounded. The state 2 3 T 2 | = | ˜ contains a string of eight spins of incorrect orientation TDown TRight TDown TRight TUp Left surrounded by a domain wall whose perimeter is 16 lattice units long − −++−+−+ ++−++−+− − − −+−+−+ : (140) ++− 0 +−+− −+−+−+−+

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As seen, the moving electron leaves a string of bad magnetic bonds in its wake. The energy penalty of such a string is linear in its extent. It is therefore expected that this (magnetic string) potential leads, in more than one dimension, to a conAning force amongst the spin and charge degrees of freedom. As this caricature for the single hole makes clear, the notion of localized “spinons” and “holons” is unlikely to hold water for the U → ∞ Hubbard model in more than one dimension. The well-deAned SFL solutions for special models with nested Fermi-surface in two dimensions should, however, be noted [96]. A certain form of spin–charge separation in two dimensions may be sought in the very special hole aggregates (or stripes) that have been detected in some of the cuprates [257] and the nickelates [59]. Here, holes arrange themselves along lines which concurrently act as antiferromagnetic domain walls (i.e., behave like holons) in the background spin texture. Charge and spin literally separate and occupy diKerent regions of space. In eKect, the two-dimensional material breaks up into one-dimensional lines with weak inter-connections. 32 Related behavior is also found in numerical work [174] in the so-called, t –t–J model. 33 An important question for such models is the extent to which the interconnections between stripes are “irrelevant”—i.e., the coupled chains problem, which we brieMy allude to in Section 4.10. 4.9. Recoil and the orthogonality catastrophe in one dimension and higher Here we show how the SFL behavior in one dimension is intimately tied to the issue of orthogonality which we discussed in Sections 2.2 and 3.4. This line of thinking is emphasized by Anderson [20,24] who has also argued that this line of reasoning gives SFL behavior in two dimensions for arbitrary small interactions. We consider the eKect of interactions through the explicit computation of our old friend from Section 2.2, the quasiparticle weight Zk1=2 =

N +1 † N |ck | k

:

(141)

As we have seen, this indeed vanishes in all canonical one-dimensional models. Consider the model [50] of a Hamiltonian describing N fermions interacting with an injected particle via a delta function potential H =−

N

N

i=1

i=1

1 92 1 92 − + U (xi − x0 ) : 2m 9xi2 2m 9x02

(142)

32 This observation has led to a line of thought which is of some interest in the context of the issues discussed here. If one focusses on the quantum mechanics of a single line of holes by formulating it as a quantum-mechanical lattice string model [87], the string traces out a two-dimensional world sheet in space–time. Quantum-mechanical particles in one dimension, on the other hand, trace out world lines in space–time. It is claimed that one can recover most of the power-law correlation functions of one-dimensional interacting fermions from the classical statistical mechanics of Muctuating lines, and along these lines approach stripe formation as some form of spin–charge separation in two dimensions [300]. 33 In this paper it was further observed that the kinetic motion of single holes may scramble the background spin environment in such a way that, on average, the holes may become surrounded by antiferromagnetically ordered spins on all sides (i.e., both along the horizontal and along vertical axis)—this is claimed to be a higher dimensional generalization of the holons encountered so far.

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The calculation for the small s-wave phase shifts for all single-particle states {.i (xi )} is relatively straightforward. The quasiparticle weight Z reduces to an overlap integral between two (N + 1)-particle Slater determinants, and one Ands 2

Z ∼ N −2(F =)

(143)

F = − tan−1 [UkF =2] :

(144)

with In the thermodynamic limit Z = 0 and no quasiparticles exist. As we see, the scattering phase shifts must conspire to give rise to anomalous behavior (exponents) for the electronic correlation function in such a way that they lead to a vanishing density of states at the Fermi level. We have already given explicit expressions for the anomalous exponent(s) under the presence of general scattering terms. As indicated in Sections 2 and 3, an identically vanishing overlap integral between two (N +1) particle states could be a natural outcome of the emergence of additional quantum numbers labeling orthogonal states. These states could correspond to diKerent topological excitation sectors (e.g. solitons). Each quantum number corresponds to some conserved quantity in the system. In the one-dimensional electron gas this may be derived as we saw as a consequence of separate conservation law for left and right movers. An illustrative example of how singular Fermi liquid behavior due to orthogonality of the wavefunction is robust in one dimension but easily destroyed in higher dimensions, is provided by the X-ray edge singularity problem, already discussed in Section 3.5. As sketched in Fig. 19, we consider the transition of an electron from a deep core level to the conduction band through absorption of a photon. This problem is essentially the same as that of optical absorption in degenerate semiconductors, and from this point of view it is natural to analyze, following NoziZeres [199], the eKect of dispersion in the hole band, the analogue of the deep level state. For optical absorption in a semiconductor, the transition conserves momentum; hence in the absence of Anal state interactions, the threshold absorption is associated with the transition indicated with the arrowed line in Fig. 19, and absorption starts discontinuously above the threshold energy !D provided that the hole mass is inAnite. For Anite hole mass, the threshold gets rounded on the scale of the dispersion of the hole band. However, in one dimension, the edge singularity does survive because low-energy electron–hole excitations in one dimension have momenta only near 0 and near 2kF (see Fig. 10); electron–hole pairs cannot carry away arbitrary momenta. This is seen in the following calculation [199]. Assume a simple featureless Anal state potential V , and consider Arst the case without recoil. The relevant quantity to calculate is the transient propagator for the scatterer G(t) = 0|deiHt d† |0

(145)

as the spectrum is the Fourier transform of G(t). In (145), the potential V is turned on at time 0 and turned oK at time t. In a linked cluster expansion, we may write G(t) = eC(t) , where C(t) is the contribution of a single closed loop. In lowest order perturbation theory, C(t) becomes t t C(t) = d d V 2 g(0; − )g(0; − ) ; (146) 0

0

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where g(0; ) is the free conduction electron propagator at the origin. For large times, one has g(0; ) ≈ −iN (0)=, and when this is used in (146) one immediately And that for large times C(t) = V 2 N (0)2 ln t;

⇒

G(t) ˙ 1=t n ;

(147)

where n = V 2 N (0)2 = 2 =2 + 1

(148)

is the phase shift exponent due to the orthogonality eKect, compare Eq. (56). A power-law decay 2 2 of G(t) ∼ 1=t n at long times corresponds to power-law dependence ∼ (! −!D )n−1 = (! −!D ) = just above the absorption edge. If we now take into account the recoil eKect, then the dispersion of the lower band implies that the hole in this band can hop from site to site. The propagator G(t) is then obtained as a sum over all trajectories R() of the scattering hole which begin and end at R = 0. For a given history, we can extend the above analysis to lowest order by replacing the propagator g(0; ) by g(H; ), where H() = R() − R( ). For positive time diKerence, we can then write g(H; ) = eik −ik·H : (149) k¿kF

For small hopping rates and large times, the integration over the modulus is dominated by the energy term, and this yields a term proportional to −1= as in the recoilless case. The FS over the Fermi sphere. A trajectory of the hole enters through the average exp(−ik · H) simple calculation yields  cos(kF H); d = 1 ;     FS d=2 ; J0 (kF H); exp(−ik · H) = (150)   H) sin(k F   ; d=3 : kF H In order to calculate the large time behavior of the Green’s function, we Anally have to average the square of this result over the distribution function of the trajectories H for large times. Using the large-H behavior of the expressions found above, one then Ands [199] that for large times 1 2 2 d=1 ;  2 N (0)= ;  !  2 FS g2 (H; ) ≈ N 2 (0)−2 exp(−ik · H) (151) = ˙ ln Htyp ()=(Htyp ()2 ); d = 2 ;   H  ˙ 1=(H2typ ()2 ); d=3 ; where Htyp is the typical distance the hole trajectory moves away from the origin in time . In one dimension, we see that g2 still falls oK as 1=2 and hence in analogy to (147) that G(t) has power-law long time behavior: in the presence of recoil, an edge singularity persists but the exponent n is now only half of that in the absence of dispersion of the lower state (a consequence of the averaging over H).

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Since Htyp () grows diKusively as 1=2 for large , the integrand in the expression (146) for C(t) converges faster than 1=t 2 , and hence C(t) converges to a Anite limit for large times. The singular X-ray edge eKect is washed out in two and three dimensions due to the recoil. If the above argument is extended to arbitrary dimension by analytically continuing the angular average over the Fermi surface to continuous dimensions, one Ands that the orthogonality and concomitant singular behavior is destroyed for any dimension d ¿ 1. Nevertheless, the subdominant behavior of the integrals will contain non-integer powers of time, and this gives rise to subdominant non-analytic terms in the spectrum for non-integer d. This behavior is completely in agreement with an analysis of the dimensional crossover from Luttinger liquid behavior to Fermi-liquid behavior as a function of dimension [51]. 4.10. Coupled one-dimensional chains The two coupled chain problem has been thoroughly considered [90,95,159,232,33,194] following earlier perturbative RG calculations [264] on a related model. The two-chain or ladder is especially interesting both theoretically and experimentally. In general inter-chain coupling is a relevant parameter, changing the behavior qualitatively. In the model of coupled Luttinger chains, the weight of the massless bosons characteristic of one dimension goes down with the number of chains. The general lesson to be drawn is that inter-chain coupling is always a relevant parameter, but that for a small number of coupled chains special features of the one-dimensional problem persist. In the passage to two dimension by increasing the number of chains to thermodyanmic values, features of the one-dimensional problem such as charge–spin separation are lost. SpeciAcally for models in which the one-chain problem can be bosonized, the approach to two dimension by increasing the number of chains appears to lead to a Fermi liquid in two dimensions. The two-chain problem presents some interesting new features. One of them is the “d-wave” type superconductivity and the other is the presence of phases of “orbital antiferromagnetism” for some range of parameters [203,173]. 4.11. Experimental observations of one-dimensional Luttinger liquid behavior There has, of course, been a long-standing interest to observe the fascinating one-dimensional Luttinger liquid-type SFL behavior experimentally, but the possibility of clear signatures has arisen only in the last few years. The clearest way to probe for Luttinger liquid behavior is to measure the tunneling into the one-dimensional system. Associated with the power-law behavior (110) in one dimension, one has a power-law behavior for the single-electron tunneling amplitude into the wire. For Axed voltage, this leads to a diKerential conductance dI=dV ∼ V , with the exponent determined by the charge stiKness Kc , the geometry, and the band structure. Hence, from the measurement of the tunneling as a function of temperature or voltage, and thus Kc can be extracted. Recent experiments on resonant tunneling [32] of small islands embedded in one-dimensional quantum wires in semiconductors, grown with a so-called cleaved edge overgrowth method, do indeed yield a power-law temperature behavior of the conductance [106] which is consistent with Luttinger liquid behavior, but the value of the exponent is substantially diKerent from the one expected theoretically.

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It has recently also been realized that nature has been kind enough to give us an almost ideal one-dimensional wire to study one-dimensional electron physics: the wavefunctions of carbon nanotubes turn out to be coherent over very large distances [74]. Although the circumference of the nanotubes is rather large, due to the band structure of the graphite-like structure the conduction in nanotubes can be described in terms of two gapless one-dimensional bands. Moreover, it was realized by Kane et al. [141] that due to the special geometry the backscattering in nanotubes is strongly suppressed, so that they are very good realizations of the Tomonaga– Luttinger model of Section 4.2, with an interaction constant Kc which is determined by the Coulomb energy on a cylinder. Their calculation based on this idea gives a value Kc ≈ 0:2. Fig. 32 shows recent nanotube data [211] which conArm the predictions by Kane et al. [141]: the diKerential conductance dI=dV is found to vary as V as a function of the bias voltage at low but Axed temperatures, or as T as a function of temperature at Axed bias. For tunneling into the bulk of a carbon nanotube, the relevant density of states Nbulk (!) ∼ !bulk

with bulk = c :

(152)

The fact that the exponent here is only half the value attained in the simple Luttinger Liquid— see Eq. (110)—is due to the fact that there are two Luttinger Liquid-like bands present in the carbon nanotubes. 34 Only one linear combination of the two associated charge modes attains a non-trivial stiKness Kc = 1 [141]. By contrast, at the tips of the nanotubes, surplus electronic charge can propagate in only one direction and, as a consequence, the tunneling is more restricted Ntip (!) ∼ !tip

with tip = (Kc − 1)=4 :

(153)

By Fermi’s golden rule, the relevant exponents for (tip–tip) or (bulk–bulk) tunneling are t−t = (Kc − 1)=2 and b−b = 2c , respectively. Bulk–bulk tunneling is achieved by arranging the nanotubes according to the crossing geometry depicted in the inset of Fig. 32 above. By 34

The electrostatic charging energy depends √ only on the symmetrized band mode, which in bosonic variables can be written as Lc; + = (Lc; band=1 + Lc; band=2 )= 2. The essential reason that exponents can change depending on √ the number of bands is that the normalization factor 1= 2 in this bosonic variable enters in the exponent when the electron variables are written in terms of the bosonic modes, as discussed in Section 4.2. A simple way to illustrate the halving of the exponents in the context of the various results we have discussed is by considering the diKerence between the spinful Luttinger liquid that we have discussed and the spinless Luttinger liquid (a system only having charge degrees of freedom): the spinful model has a density of states exponent which is half of the spinless one. A calculation proceeds √along the following lines: for the spinful case, the relevant √ √ i(KcR (x; t)+KsR (x; t))= 2) −i(KcR (0; 0)+KsR (0; 0))= 2) Green’s function is GR=+1 (x; t) = e e . Note the factor 1= 2 in the exponent, coming from the projection√ onto the proper bosonic variables. This expected value can be written as √ √ √ GR=+1 (x; t) = eiKcR (x; t)= 2 e−iKcR (0; 0)= 2 eiKsR (x; t)= 2 e−iKsR (0; 0)= 2 , which with the aid of the results of Section 4.4 becomes GR=+1 (x; t) = |x−v1 t |1=2 |x2 −v12 t 2 |c × |x−v1 t |1=2 |x2 −v12 t 2 |s . From this result, one immediately obtains the den

c

c

s

s

sity of states N (!) = dtG(x = 0; t)ei!t : A simple integration then yields N (!) = [vc−2c vs−2s ] dt t −1 t −2(c +s ) ei!t ∼ !2(c +s ) : Since Ks renormalizes to 1 for repulsive interactions (see the remark just after (116)), one usually has s = 0 and so in this case, the density of states exponent is simply 2s . This is precisely (110). Now consider what one √ gets for the spinless Luttinger liquid. In this case, neither the bosonic spin modes nor the projection factor 1= 2 are present in the above expression for G. This results in a spatial decay with an exponent 2c instead of c , and hence an exponent 4c in the density of states. In other words, for s = 0 the density of states exponent in the spinful case is half of what it is in the spinless case. The same mechanism is at work in the nanotubes.

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Fig. 32. DiKerential conduction dI=dV measured by Postma et al. [211] for carbon nanotubes. At low voltages or temperatures, Coulomb blockade eKects dominate, but at higher temperatures or bias voltages, one probes the one-dimensional SFL behavior. Panel (a) shows the diKerential conductance as a function of bias voltage for various temperatures (note that these temperatures are relatively high, reMecting the fact that the electronic energy scales of the nanotubes is high). The eKective exponent for the large V behavior is 0.48; since these data are for tunneling between two nanotubes, = 2c , so c ≈ 0:24 and Kc ≈ 0:27. The predicted value is Kc ≈ 0:2 [141]. The data in panel (b) show the diKerential conductance as a function of temperature at Axed bias for two nanotubes which cross, as well as for a single nanotube with a bend.

extracting the value of the charge stiKness Kc from each of the independent measurements of t−t ; b−b for the two diKerent geometries, a single consistent stiKness Kc ≈ 0:27 was found [211], in good agreement with theoretical prediction. The data shown in Fig. 32 corroborate the predicted scaling dI=dV ∼ V over about one decade at voltages larger than a few kB T . The problem in obtaining data over a larger range is that at a Axed temperature one has a crossover to linear behavior at small voltages due to thermal eKects, while when performing measurements at Axed voltage as a function of temperature, Coulomb blockade eKects reduce the conductance at low T to a value where they beset probing the intrinsic Luttinger liquid eKects. Nevertheless, other datasets exhibit scaling over a range of up to three decades in V , and moreover, all experiments in diKerent sample layouts with a variety of contact and defect structures yield a similar Luttinger stiKness around 0.23. In conclusion, therefore, taken together experiments on nanotubes yield very good experimental evidence for Luttinger liquid behavior. Other, older, canonical realizations of Luttinger liquids include the Quantum Hall edge states. These represent a chiral spinless Luttinger liquid. Here, low-lying energy states can only prevail at the edge of the sample, and, concurrently, disperse linearly about the Fermi energy. Edge states can attain macroscopic linear (perimeter) extent, and the tunneling experiments between such states [182,114] have observed several features predicted theoretically [287,140]. We refer

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the interested reader to the review articles by Schulz et al. [233] and Fisher and Glazman [98] for further details. We note here recent ingenious experiments [76] which demonstrate, in eKect (among other things) the separate conservation of left and right going electrons in a one-dimensional wire by showing that a separate chemical potential can be ascribed to the left-going electrons and to the right-going electrons. These may set the stage for the observation of spin–charge separation as well. 5. Singular Fermi-liquid behavior due to gauge &elds 5.1. SFL behavior due to coupling to the electromagnetic =eld Almost 30 years ago, Holstein et al. [127] (see also Reizer [221]) showed that the coupling of electrons to the electromagnetic Aelds gives rise to SFL behavior. Since the typical temperatures where the eKects become important are of the order of 10−15 K, the eKect is not important in practice. However, the theory is of considerable general interest. If we work in the Coulomb gauge in which ∇ · A = 0 for the electromagnetic A Aeld, then the transverse propagator Dij0 in free space is given by Dij0 (k; !) = Ai Aj (k; !) =

ˆ ˆ

ij − k i k j 2 c k 2 − !2 −

i

;

(154)

where is an inAnitesimal positive number. The interaction of the electrons with the electromagnetic Aeld is described by the coupling term j · A + HA2 ;

(155)

where j is the electron current operator and H the density operator. Quite generally in the Coulomb gauge, one Ands from perturbation theory, or phenomenologically from the Maxwell equations, that the electromagnetic propagator in a metal can be written as Dij−1 = (Dij0 )−1 − M (ij − kˆi kˆj )−1 :

(156)

The perturbative diagrammatic expansion of M is indicated in Fig. 33. The Arst term leads to a term proportional to the density while the second term is the Arst correction due to particle–hole excitations. In formulas, these terms yield ˆ 2 ][fp−k=2 − fp+k=2 ] 4 1 d dp [p2 − (p · k) M (k; !) = n+ : (157) m m (2)d ! − (p−k=2 − p+k=2 ) Here n is the electron density, and = 1=137 is the Ane structure constant. For !=k → 0, the two terms combined yield

! 3n ! 2 M (k; !) ≈ 4 + 2i : (158) m kvF kvF

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Fig. 33. (a) The diagrams for M; (b) the diagram for the self-energy from the discussion of the SFL eKect arising from the coupling of the electrons to the electromagnetic gauge Aelds; (c) anharmonic interaction of Muctuations such as (c) are non-singular in the SFL problem of coupling of the electrons in metals to electromagnetic Aelds.

At small frequencies, for the propagator D this yields Dij (k; !) ≈

ij − kˆi kˆj ; ! i6n + c2 k 2 − i mkvF

(159)

which corresponds to an overdamped mode with dispersion ! ∼ k 3 . Before discussing how such a dispersion gives SFL behavior in three dimensions, it is instructive to point out that although (159) was obtained perturbatively, Maxwell’s equations ensure that the Aeld propagator must generally be of this form at low frequencies and momenta. Indeed, for a metal we can write the current j as j = (k; !)E; if we combine this with the Maxwell equation ∇ × H = j + 9E= 9t we easily And that the general form of the propagator is Dij (k; !) =

ij − kˆi kˆj : 4i!(k; !) + c2 k 2

(160)

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Here, is the diamagnetic permeability. For pure metals, the low-frequency limit is determined by the anomalous skin eKect [5] and (k; 0) ∼ k −1 . According to expression (160), this 1=k behavior then implies that the dispersion at small frequencies generally goes as ! ∼ k 3 for a pure metal. For dirty metals, considering the fact that (k; 0) approaches a Anite limit 0 at small wavenumbers, according to (160), there is a crossover to a behavior ! ∼ k 2 at small wavenumbers. Gauge invariance of the theory requires that the photon cannot acquire a mass (a Anite energy in the limit k → 0) in the interaction process with the electrons, and hence the form of Eq. (159) remains unchanged. Thus the anharmonic corrections to the photon propagator due to processes such as those shown in Fig. 33(c) do not change the form of Eq. (159). The self-energy of the electrons due to photon exchange, Fig. 33(b), may now be calculated with conAdence given the small coupling constant in lowest order. The leading contribution in d = 3 is (kF ; !) ∼ (! ln ! + i! sgn !) :

(161)

The momentum dependence, on the other hand, is non-singular as a function of k − kF . The non-analytic behavior of the self-energy as a function of frequency implies that the resistivity of a pure metal in d = 3 is proportional to T 5=3 at low temperatures. 35 The simplicity and strength of the above example lies in the fact that the theory has no uncontrolled approximations, and the gauge-invariance of the photon Aeld dictates the low-energy low-momentum behavior of the photon propagator Dij (k; !). Moreover, vertex corrections are not important because the Migdal theorem is valid [209] when the frequency of Muctuations is very small compared to their momenta, as in ! ∼ q3 . However, the SFL behavior as a result of the coupling to the electromagnetic Aeld is not relevant in practice. This can most easily be argued as follows. For a Fermi gas, the entropy per particle is S 2 mkB2 T ; = N ˝2 kF2

(163)

while for the entropy for the electrons interacting with the electromagnetic Aeld one Ands from the above results [127] S 22 kB2 T !0 ≈ ln (164) N 3 ˝ckF T with !0 =

cF : vF

(165)

1 This follows from the fact that in the quasielastic approximation, the transport relaxation rate − is related to tr −1 the single-particle relaxation rate (L) due to scattering through an angle L near the Fermi surface 35

1 − tr =

d/(1 − cos(L))−1 (L) :

(162)

1=3 For small T , ! ∼ k 3 ∼ T , and hence the characteristic angle of scattering, L ∼ (k=kF ) ∼ (T=E Upon 3F ) 2 is small. 2 expanding (1 − cos L) L =2, one Ands that the eKective transport scattering rate goes as d k k f(k 3 =T ) ∼ T 5=3 .

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Upon comparing these two results, one concludes that the SFL eKects start to become important for temperatures kB T . !0 e−3m

∗ c=2˝2 k

F

:

(166) 105

−1

for ordinary metals, Since the numerical factor in the exponent is typically of order −15 the temperature range one Ands from this is of order 10 K for values of of order unity, according to this estimate. Note, however, that for pure ferromagnetic metals can be as large as 104 . Possibly, in some ferromagnets, the eKects can become real. 5.2. Generalized gauge theories The example of coupling to the electromagnetic Aeld identiAes one possible theoretical route to SFL behavior, but as we have seen that the crossover temperatures that one estimates for this scenario are too small for observable physical properties. The smallness of the estimated crossover temperature is essentially due to the fact that the coupling to the electromagnetic Aeld is determined by the product vF =c, where = 1=137 and typically vF =c = O(10−2 ). Motivated by this observation, many researchers have been led to explore the possibility of obtaining SFL eKects from coupling to diKerent, more general gauge bosons which might be generated dynamically in strongly correlated fermions. For a recent review with references to the literature, see [158,192,193]. The hope is that if one could consistently And such a theory in which the small factor vF =c arising in the electromagnetic theory is replaced by a term of order unity, realistic crossover temperatures might arise. Much of the motivation in this direction comes from Anderson’s proposal [24] of spin–charge separation and resonating valence bonds in the high-temperature superconductivity problem which we discuss in Section 7. The essence of approaches along these lines is most easily illustrated by considering electrons on a lattice in the case in which strong on-site (Hubbard-type) repulsions forbid two electrons to occupy the same site. Then, each site is either occupied by an electron with an up or down spin, or by a hole. If we introduce Actitious fermionic creation and annihilation operators f† and f for the electrons and Actitious bosonic hole creation and annihilation operators b† and b for the holes, we can express the constraint that there can only be one electron or one hole on each lattice site by 2 † fi fi + b†i bi = 1 for each i : (167) =1

With this convention, the real electron Aeld and boson operators i

= fi b†i :

i

can be written as a product of these fermion (168)

This expresses the fact that given constraint (167), a real fermion annihilation at a site creates a hole. This way of writing the electron Aeld may be motivated by the physics of the one-dimensional Hubbard model: there a local excitation may indeed be expressed in terms of a charged spinless holon and an uncharged spinon. In general, in a transformation to boson and fermion operators as in (168) there is some freedom as to with which operator we associate the charge and with which the spin.

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Whenever we split a single electron operator into two, as in (168), then there is a gauge invariance, as the product is unchanged by the transformation fi (t) → ei1(t) fi (t);

bi (t) → ei1(t) bi (t) :

(169)

We can promote this invariance to a dynamical gauge symmetry by introducing a gauge Aeld a and writing the Hamiltonian in the continuum limit as i∇ + a 2 † H = dr f − f + (. − f )f† f 2m f i∇ + a 2 † b + (. − b )b† b : (170) + dr b − 2mb Here, . is a Lagrange multiplier Aeld which is introduced to implement constraint (167). Note that the a Aeld enters in much the same way in the Hamiltonian as the electromagnetic Aeld usually does—indeed, a change in the individual f’s and b’s by a space-dependent phase factor as in (169) can be reabsorbed into a change of a. Note also the presence of the chemical potentials f and b to enforce that for a deviation of the (average) density n from one per site, the density of holes is 1 − n and of fermions is n. The next step in the theory is to And the Muctuation propagator for the a Aeld, as a function of f and b . For Anite f and negative b (bosons uncondensed), the Muctuation propagator is similar to that of the previous section but with (vF =c) replaced by a term of O(1). Spinon and holon self-energies can now be calculated and composition laws [134] are derived to relate physical correlation functions to correlation functions of spinons and holons. Unfortunately, this very attractive route has turned out to be less viable than had been hoped. It is not clear whether the diPculties are purely technical; they are certainly formidable. The essential reason is that while photons have no mass and are not conserved, and hence cannot Bose condense, in a gauge theory obtained by introducing additional bosons, the bosons generally can and will Bose condense because they do have a chemical potential. Bose condensation leads to a mass term in the propagator for the gauge Aelds. The singularities in the fermions due to the gauge Aelds then disappear. It is the analogue of the fact that superconductivity leads to the Meissner eKect—there the emergence of the superconducting Aeld breaks gauge invariance and leads to the expulsion of the magnetic Aeld from the superconductor. The latter eKect can also be thought of as being due to the generation of a mass term for the gauge Aelds. Several variants of these ideas have been proposed with and without attempts to suppress the unphysical condensation through Muctuations [188]. The trouble is that such Muctuations tend to bind the spinons and holons and the happy situation in one dimension where they exist independently—being protected by (extra) conservation laws, see Section 4.9—is hard to realize. As usual, it appears that the introduction of new quantum numbers requires new symmetries. Interesting variants using the idea of spin–charge separation have recently appeared [235,188]. In passing, we note that the idea to split an electron operator into a boson and fermion operator, as in (167), is not limited to gauge theories like the ones discussed in this section. In the form used here, where the boson is carrying the hole charge, such theories are usually

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known as slave-boson approaches. Now, the Heisenberg spin one-half problem can also be formulated as a hard-core boson problem, and higher-order spins can be formulated in terms of so-called Schwinger bosons [31]. However, since one generally has some freedom in introducing slave variables for interacting electron problems, mixed variants also exist. For example, the t–J -model can be rewritten in terms of a model with Schwinger bosons carrying the spins and spinless slave fermions carrying the charge [31]. 6. Quantum critical points in fermionic systems As mentioned in Section 1, quantum critical behavior is associated with the existence of a T = 0 phase transition; of course, in practice one can only experimentally study the behavior at non-zero temperatures, but in this sense, the situation is no diKerent from ordinary critical phenomena: one never accesses the critical point itself, but observes the critical scaling of various experimentally accessible quantities in its neighborhood. In practice, the most common situation in which one observes a quantum critical point is the one sketched in Fig. 1, in which there is a low-temperature ordered state—a ferromagnetic state, antiferromagnetic state or charge-density wave-ordered state, for instance—whose transition temperature to the disordered state or some other ordered state goes to zero upon varying some parameter. In this case, the quantum critical point is then also the end point of a T = 0 ordered state. However, sometimes the “ordered” state really only exists at T = 0, for example in metal–insulator transitions and in quantum Hall eKect transitions [247,212]. A well-known example of this case in spin models is in two-dimensional antiferromagnetic quantum Heisenberg models with “quantumness” as a parameter g [56], which do not order at any Anite temperature, but which show genuine ordered phases at T = 0 below some value g ¡ gc . Although the question concerning the origin of the behavior of high-temperature superconductors is not settled yet, there are strong indications, discussed in the next section, that much of their behavior is governed by the proximity to a quantum critical point. One of the Arst formulations of what we now refer to as quantum critical behavior was due to Moriya [189,190] and Ramakrishnan [186] who did an RPA calculation for a model of itinerant fermions with a Stoner-type instability to a ferromagnetic state. In modern language, their approach amounts to a 1=N expansion. Various other important contributions were made [38,79]. The standard more modern formulation now, which we will follow, is due to Hertz [123]. A nice introduction can be found in the article by Sondhi et al. [247], and for a detailed expose, we refer to the book by Sachdev [225]. See also the review by Continento [67]. 6.1. Quantum critical points in ferromagnets, antiferromagnets, and charge-density waves A clear example of quantum critical behavior, and actually one for which one can compare with theoretical predictions, is summarized in Figs. 34 –37. The Agures show various data from [254] on the magnetic compound MnSi [205,206,160,170,254]. Fig. 34 shows that for low pressures and temperatures, this compound exhibits a magnetic phase whose transition temperature Tc vanishes as the pressure is increased up to pc = 14:8 kbar. This value of the pressure then identiAes the quantum critical point. Fig. 35 shows that when the same data are plotted as Tc4=3

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Fig. 34. Magnetic phase diagram as a function of pressure of MnSi [205,206,254]. From [254].

versus pressure, the data fall nicely on a straight line except close to the critical pressure. This observed scaling of Tc4=3 with p − pc away from pc is in accord with the behavior predicted by the theory discussed below. Actually, the transition is weakly Arst order near pc ; so a very detailed veriAcation of the theory is not possible. Fig. 36 shows data for the temperature dependence of H − H0 near pc , where H0 is the residual low-temperature resistivity. In the presence of a Aeld of 3 T, one observes the usual H − H0 ∼ T 2 Fermi-liquid scaling, but at zero Aeld the results are consistent with H − H0 ∼ T 5=3 behavior predicted by the theory. However, if we write the low-temperature resistivity behavior as H = H0 + AT L

(171)

then both the residual resistivity H0 and the amplitude A are found to show a sharp peak at pc as a function of pressure — see Fig. 37. This behavior is not understood nor is the fact that the exponent L does not appear to regain the Fermi-liquid value of 2 for signiAcant values of p ¿ pc (at H = 0) and in a temperature regime where the theory would put the material in the quantum-disordered Fermi-liquid regime. 6.2. Quantum critical scaling Before discussing other experimental examples of quantum critical behavior, it is expedient to summarize some of the essential quantum critical scaling ideas. As is well known, at a Anite temperature transition, the critical behavior is classical and we can use classical statistical mechanics to calculate the correlation functions. This is so

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Fig. 35. Power-law dependence of the Curie temperature as a function of pressure for MnSi. From [254].

because due to critical slowing down, the characteristic time scale diverges with the correlation length, ∼ Oz :

(172)

Near a critical point the correlation length O diverges as O ∼ |T − Tc |− :

(173)

The combination of these two results shows that critical slowing down implies that near any Anite temperature critical point the characteristic frequency scale !c goes to zero as !c ∼ |T − Tc |z :

(174)

Therefore, near any phase transition !c Tc , and as a result the phase transition is governed by classical statistical physics; the Matsubara frequencies are closely spaced relative to the temperature, the thermal occupation of bosonic modes is large and hence classical, etc. In classical statistical mechanics, the dynamics is slaved to the statics; usually, the dynamical behavior is adequately described by time-dependent Landau-Ginzburg type of equations or Langevin equations which are obtained by building in the appropriate conservation laws and equilibrium scaling behavior [117]. At a quantum critical point, on the other hand, the dynamics must be determined a priori from the quantum-mechanical equations of motion.

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Fig. 36. T -dependence of (H(T ) − HT =0 ) near the critical pressure pc with and without an external magnetic Aeld. From [254].

The general scaling behavior near a T = 0 quantum critical point can, however, be discussed within the formalism of dynamical scaling [117,164], just as near classical critical points. Consider for example the susceptibility for the case of MnSi that we considered above. The scaling ansatz for the singular part of the susceptibility &(k; !; p) = MM (k; !; p)

(175)

implies that near the critical point where the correlation length and time-scale diverge, the zero-temperature susceptibility & is a universal function of the scaled momentum and frequency &(k; !; p) = O−dM T(kO; !Ot ) ;

(176)

where now O ∼ |p − pc |− ;

Ot ≡ = Oz :

(177)

This is just like the classical scaling with T − Tc replaced by p − pc . The reason for writing Ot instead of is that in quantum statistical calculations, the “timewise” direction becomes like an additional dimension, so that Ot plays the role of a correlation “length” in this direction. However, the time-direction has both a long-time cutoK given by 1=kB T and a short-time cutoK

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Fig. 37. Evolution of H0 and A under pressure, when the temperature dependence of the resistivity is Atted to (H = H0 + AT 2 ). From [254].

given by the high-energy cutoK in the problem—exchange energy or Fermi energy, whichever is smaller in the ferromagnetic problem. The short-time cutoK has its analog in the spatial scale. The long-time cutoK, which determines the crossover from classical to quantum behavior, plays a crucial role in the properties discussed below. The crucial point is that when z = 1, there is an anisotropic scaling between the spatial and time-wise directions, and as we shall discuss below, this implies that as far as the critical behavior is concerned, the eKective dimensionality of the problem is d + z, not d + 1. The exponent dM in (176) reMects that a correlation function like & has some physical dimension which often is inevitably related to the spatial dimension. The dependence of critical properties on spatial dimensions must be expressible purely in terms of the divergent correlation length O. Often, dM is Axed by dimensional considerations (in the language of Aeld theory, it is then given by the “engineering dimension” of the Aeld), but this may not be true in general. It must be so, however, if & is a correlation function of a conserved quantity. 36 Let us now address the Anite temperature scaling, taking again the case of MnSi as an example. The various regimes in the T –p diagram discussed below are indicated in Fig. 38.

36 E.g. if we consider the free energy per unit volume at a classical transition, the energy scale is set by kB T , and dM = d; likewise, if we consider the surface tension of an interface, whose physical dimension is energy per unit surface area, dM = d − 1.

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Fig. 38. Generic phase diagram and crossovers for quantum critical points with the various regimes indicated. O is the correlation length at T = 0; LT is the “thermal length” given in Eq. (178).

To distinguish these regimes it is necessary to deAne an additional quantity, the thermal length LT ≡

˝

kB T

:

(178)

LT corresponds dimensionally to a time-scale. It marks the crossover between phenomena at long time scales which can be treated essentially classically from those on a shorter scale which are inherently quantum mechanical. Whenever Ot ¡ LT , the correlation length and time are Anite and quantum mechanics begins to dominate. This is the regime on the right in the Agure. Fermi-liquid behavior is expected in this regime. However, if one approaches the critical point (T = 0; p = pc ) from above along the vertical line, then LT diverges but so does Ot . Moreover, Ot diverges faster than LT since z is usually larger than 1. This means that the characteristic Muctuation energy and temperature are similar. So the behavior is quasiclassical throughout each correlated region down to zero temperature (In a path integral formulation [247,225] one considers the model on a inAnite strip whose width is Anite in the timewise direction and equal to LT . Hence, for Ot ¿ LT the model is fully correlated across the strip in this direction). This regime is therefore characterized by anomalous T dependence in the physical quantities up to some ultra-violet cutoK. It is important to stress that this so-called “quantum critical scaling behavior” 37 is expected in the observable properties in the complete region between the dashed 37

The term is somewhat problematic; it refers to the quasiclassical Muctuation regime around a quantum critical point.

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lines joining together at T = 0; p = pc up to the high temperature scale in the problem, which is usually several times larger than the highest transition temperature to the ordered phase as a function of p. So SFL is observable over a whole range of parameter p for temperatures between the left and right crossover lines. If one approaches the line of phase transitions to the ordered phase, which is marked with a solid line in the Agure, one has a region with SFL properties dominated by classical Muctuations close to the transition. Millis [183] has corrected Hertz’s results [123] on this point, and has found that the critical temperature of the phase transition scales as Tc ∼ |p − pc |z=(d+z−2) . Estimates of the classical critical region are also given. Results along similar lines may also be found in [66,67]. If we include both the temperature and the parameter p, the scaling ansatz for the imaginary part of & becomes ! & (k; !; p; T ) ∼ O−dM T1 kO; !Oz ; : (179) T This can be rewritten in other forms depending on which experiment is being analyzed. For example, the above form is especially suitable for analysis as a function of (p − pc ). For analyzing data as a function of temperature, we may instead rewrite LT & (k; !; p; T ) = LT−dM =z T2 kL1=z ; !L ; (180) T T Ot and for analyzing data as a function of frequency k ! 1 −dM =z T3 ; ; : & (k; !; p; T ) = T T 1=z T TOt

(181)

Moreover, the scaling of the free energy F can be obtained from the argument that it is of the order of the thermal energy kB T per correlated volume Od . Moreover, since LT acts as a Anite cutoK for Ot in the timewise direction, we then get the scaling F ∼ TO−d ∼ T (Ot )−d=z ∼ T 1+d=z :

(182)

By diKerentiating twice, this also immediately gives the speciAc heat behavior at low temperatures. In writing the above scaling forms, we have assumed that no “dangerously irrelevant variables” exist, as these could change !=T scaling to !=T @ scaling. 38 In order to get the critical exponents and the crossover scales, one has to turn to a microscopic theory. The theory for this particular case of the quantum critical point in MnSi is essentially

38

“Dangerously irrelevant variables” are irrelevant variables which come in as prefactors of scaling behavior of quantities like the free energy [97]. Within the renormalization group scenario, the hyperscaling relation d = 2 − is violated above the upper critical dimension because of the presence of dangerously irrelevant variables. Presumably, dangerously irrelevant variables are more important than usually at QCPs, since the eKective Muctuation dimension is above the upper critical dimension for d = 3 and z ¿ 1. Some examples are discussed in [225].

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a random phase approximation and proceeds along the following lines: (i) one starts with a model of interacting fermions; (ii) an ordering Aeld M (k; !) is introduced; (iii) one assumes that the fermions can be eliminated near the critical point to get a free energy in terms of M of the form F = d! d d k&−1 (k; !)|M (k; !)|2

+

d {! }

d {k }VM (k1 ; !1 )M (k2 ; !2 )M (k3 ; !3 )M (k4 ; !4 )

×(!1 + !2 + !3 + !4 )(k1 + k2 + k3 + k4 ) + · · · :

(183)

Note that this is essentially an extension of the usual Landau–Ginzburg–Wilson free energy to the frequency domain. Indeed, from here on one can follow the usual analysis of critical phenomena, treating the frequency ! on an equal footing with the momentum k. The important result of such an analysis is that the eKective dimension as far as the critical behavior is d + z, not d + 1 as one might naively expect. Since z ¿ 1 in all known examples, the fact that the eKective dimension is larger or equal than d + 1 reMects the fact that the correlation “length” Ot in the timewise direction grows as Oz , i.e., at least as fast as the spatial correlation length. 39 Moreover, the fact that z ¿ 1 implies that the eKective dimension of a d = 3 dimensional problem is always larger or equal than four. Since the upper critical dimension above which mean Aeld behavior is observed equals four for most critical phenomena, one thus arrives immediately at the important conclusion that most quantum critical points in three dimensions should exhibit classical Muctuations with mean Aeld scaling exponents! It also implies that the critical behavior can typically be seen over a large parameter or temperature range—the question of the width of the critical region, which normally is determined by the Ginzburg criterion, does not arise. On the other hand, questions concerning the existence of dangerously irrelevant variables, due for example to the scaling of the parameters V in Eq. (183), do arise. In order to judge the validity and generality of these results, it is important to keep in mind that they are derived assuming that the coePcients of the M 2 ; M 4 terms are analytic functions of k; ! and the pressure p, etc. This is completely in line with the usual assumption of analyticity of the bare coupling parameters in a renormalization group approach. This assumption may well be violated—in fact none of the impurity models discussed earlier can be treated along these lines: the fermions cannot be integrated out there, and if one attempts to apply the above procedure, one Ands singular contributions to the bare coupling parameters. Later on we shall discuss a three-dimensional experimental example where this assumption appears to be invalid. Secondly, it is inherently an expansion about the non-magnetic state, which cannot apply in the ordered phase: In the ordered phase with non-zero magnetization, M = 0, there is a gap for some momenta in the fermionic spectra. This gap cannot be removed perturbatively. 39 This has important consequences for a scaling analysis of numerical data, aimed at determining the critical behavior. For it implies that the Anite size scaling has to be done anisotropically, with the anisotropy depending on the exponent z which itself is one of the exponents to be determined from the analysis.

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In a ferromagnet, the ground state susceptibility on the disordered side is given by i! −1 2 & (k; !) = (p − pc ) + k + : kvF

351

(184)

In the Arst two terms, we recognize the usual mean Aeld type behavior with a correlation length that diverges as O ∼ (p − pc )1=2 , hence the critical exponent = 1=2. The last term, which describes Landau damping of the spin wave modes, is very special here as it arises from Muctuations of magnetization, a quantity which is conserved (commutes with the Hamiltonian). Therefore, the characteristic damping rate must approach zero as k → 0. Since at the critical point (184) leads to a damping ! ∼ k 3 , the critical exponent z = 3. According to the theory described above, the critical behavior at the quantum critical point (T = 0; p = pc ) is therefore of the mean Aeld type for any physical dimension d ¿ 1, since the eKective dimension d+z ¿ 4 (with only logarithmic corrections to mean Aeld theory when d = 1). The scattering of electrons oK the long-wavelength spin waves is dominated by small angle scattering, and it is easy to calculate the resulting dominant behavior of the self-energy of the electrons. Near the critical point, the behavior of & is very similar to the electromagnetic problem that we discussed in Section 5. Analogously, one also Ands SFL behavior here: in d = 3, (kF ; !) ∼ ! ln ! + i|!| while in d = 2 one obtains (kF ; !) ∼ !2=3 + i|!|2=3 . Furthermore, for the resistivity one Ands in three dimensions H ∼ T 5=3 —this is consistent with the behavior found in MnSi in the absence of a Aeld, see Fig. 36. Moreover, as we mentioned earlier, according to the theory, near the critical point Tc should vanish as |p − pc |z=(d+z−2) ; with d = z = 3 this yields Tc ∼ |p − pc |3=4 . As we saw in Fig. 35, this is the scaling observed over a large range of pressures, except very near pc . ZrZn2 [115] is an example in which the ferromagnetic transition is shifted to T = 0 under pressure continuously. The properties are again consistent with the simple theory outlined. There is, however, trouble on the horizon [161]. The asymptotic temperature dependence for p ¿ pc is not proportional to T 2 , as expected. We will return to this point in Sections 6.4. For antiferromagnets or charge density waves the critical exponents are diKerent. In these cases, the order parameter is not conserved, and the inverse susceptibility in these cases is of the form i! −1 2 & (k; !) = (p − pc ) + (k − k0 ) + ; (185) where k0 is the wavenumber of the antiferromagnetic or charge-density wave order. From this expression we immediately read oK the mean Aeld exponents z = 2 and = 1=2. Since the eKective dimension d + z is above the upper critical dimension for d = 3, the mean Aeld behavior is robust in three dimensions. In d = 2, on the other hand, the eKective dimension d+z = 4 is equal to the upper critical dimension, and hence one expects logarithmic corrections to the mean Aeld behavior. Indeed, in two dimensions one Ands for the self-energy [126] (kˆ F ; !) ∼ ! ln ! + i|!| (only) for those kˆ F from which spanning vectors to other regions of the Fermi surface separated by k0 can be found; the resistivity goes as H(T ) ∼ T 2 ln T in this case. However, if several bands cross the Fermi surface, as often happens in heavy fermions, Umklapp-type scattering may enforce the same temperature dependence in the resistivity as in the single-particle self-energy,

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Fig. 39. (Left panel) Temperature–pressure phase diagram of high-purity single-crystal CeIn3 . A sharp drop in the resistivity consistent with the onset of superconductivity below Tc is observed in a narrow window near pc the pressure at which the N]eel temperature TN tends to absolute zero. (upper inset) This transition is complete even below pc itself. (lower inset). Just above pc , where there is no N]eel transition, a plot of the temperature dependence of d(ln ^H)=d(ln T ) is best able to demonstrate that the normal state resistivity varies as T 1:6±0:2 below about 3 K. ^H is the diKerence between the normal state resistivity and its residual value (which is calculated by extrapolating the normal state resistivity to absolute zero). For clarity, the values of Tc have been scaled by a factor of ten. (right panel) Temperature–pressure phase diagram of high-purity single-crystal CePd 2 Si2 . Superconductivity appears below Tc in a narrow window where the N]eel temperature TN tends to absolute zero. The inset shows that the normal state a-axis resistivity above the superconducting transition varies as T 1:2±0:1 over nearly two decades in temperature. The upper critical Aeld Bc2 at the maximum value of Tc varies near Tc at a rate of approximately −6T=K. For clarity, the values of Tc have been scaled by a factor of three, and the origin of the inset has been set at 5 K below absolute zero. Both plots are from Mathur et al. [170].

except at some very low crossover temperature. The physical reason for this dependence despite the fact that the soft modes are at large momentums (and therefore vertex corrections do not change the temperature dependence of transport relaxation rates) is that the set of kˆ F usually covers a small portion of the Fermi surface. Fig. 39 shows the phase diagram of two compounds that order antiferromagnetically at low temperatures. The Arst one, CeIn3 , is a three-dimensional antiferromagnet. A superconducting phase intervenes at very low temperatures (note the diKerent scale on which the transition to the superconducting phase is drawn), covering the region around the quantum critical point at a pressure of about 26 kbar. At this pressure, the normal state resistivity is found to vary as H(T ) ∼ T 1:6±0:2 which is consistent with the theoretical prediction that at a quantum critical point dominated by antiferromagnetic Muctuations the resistivity should scale as H ∼ T 1:5 . The right panel in Fig. 39 shows the phase diagram and resistivity data of the three-dimensional antiferromagnet CePd 2 Si2 ; the data in this case are best Atted by H ∼ T 1:2 ; this is consistent with the theoretical prediction H ∼ T 1:25 which results if one has a (k − k0 )4 dispersion around

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the AFM vector k0 in one direction and the usual (k − k0 )2 dispersion in the other two. However, no independent evidence for such dispersion is available yet. In both these cases, part of the region of superconductivity, in a region bounded by a line emanating from the QCP and going on to the transition line from the antiferromagnetic to the normal state, is expected to be antiferromagnetic as well. These two compounds are also of interest because the phase diagram bears a resemblance to the phase diagram of the high-Tc copper-oxide based superconductors in which the conduction electron density is the parameter varied—see Fig. 49. Unlike the heavy fermion compounds where the ordered phase is antiferromagnetic, the order in copper-oxides near the QCP is not AFM. Its nature is in fact unknown. In the heavy fermion compounds, superconductivity promoted by antiferromagnetic Muctuations is expected to be of the d-wave variety [187] as it is in the high-Tc copper-oxide compounds. 6.3. Experimental examples of SFL due to quantum criticality: open theoretical problems We have discussed the observed quantum critical behavior in some system which is largely consistent with the simple RPA-like theoretical predictions. There are, however, quite a few experimentally observed signatures of singularities near QCPs, especially in heavy fermion compounds, which are not understood theoretically by the simple RPA theory of the previous subsection. In this section, we present some prominent examples of these. RPA-type theories work when the dissipation of Muctuations is given very simply. The failures below show that dissipation in the quantum to classical crossover regime in actual physical systems is quite often much more interesting; it has singularities not anticipated in RPA. The diPculties are almost certainly not just mathematical. While dissipation in classical mechanics is intoduced ex cathedra, in quantum problems we need to understand it in a fundamental way. The experimental observations fall into two general categories, in both of which the lowtemperature resistivity does not obey the power laws expected of Fermi liquids: Compounds in which resistivity decreases from its limit at T = 0 and those in which it increases. In both cases, the Cv =T is singular for T → 0. It is reasonable to associate the former with the behavior due to impurities and the latter with the QCP properties of the pure system. However, as we discussed in Section 3.9, the QCP due to impurities requires tuning to special symmetries unlikely to be realized experimentally. As we will discuss, the eKect of impurities without any special symmetries but coupling to the order parameter is expected to be quite diKerent near the QCP of the pure system compared to far from it. Under some circumstances, it is expected to be singular and may dominate the observations. We start with experiments in the second category. Figs. 40 – 42 show several datasets for the heavy fermion compound CeCu6−x Aux for various amounts of gold. This compound exhibits a low-temperature paramagnetic phase for x ¡ 0:1 and a low-temperature antiferromagnetic phase for x ¿ 0:1. The speciAc heat data of Fig. 40 and the resistivity data of Fig. 42 show that while without Au, i.e., in the paramagnetic regime, the behavior is that of a heavy Fermi-liquid metal, the alloy near the quantum critical composition CeCu5:9 Au0:1 exhibits a speciAc heat with an anomalous Cv ∼ T ln T

(186)

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Fig. 40. The speciAc heat C=T of CeCu6−x Aux versus log T . From L_ohneysen et al. [282–284]. Fig. 41. Susceptibility data for CeCu5:9 Au0:1 . From L_ohneysen et al. [285].

over a temperature range of almost two decades. At the same composition, the resistivity shows a linear temperature dependence, and the susceptibility data √ in Fig. 41 which have been Atted to a deviation from a constant as T → 0 varying as a T cusp. The anomalous behavior is replaced by Fermi-liquid properties by both a magnetic Aeld and increasing the substitution of copper by gold or by application of pressure [44,281]. The compound YbRh2 S2 seems to have similar properties [258]. Related properties have also been found in U2 Pt 2 In [88,89] and in UPt3−x Pd x [75], UBe13 , CeCu2 Si2 , CeNi2 Ge2 [250]. The SFL properties observed at the Mott insulator-to-metal transition in BaVS3 [99] are also of related interest. A good example of an antiferromagnetic QCP in itinerant electrons is in the alloy series Cr 1−x Vx for which the magnetic correlations have been measured [122]. None of the quantum critical properties of the CeCuAu compounds is consistent with any of the models that we have discussed. Information on the magnetic Muctuation spectra for CeCu5:9 Au0:1 is available through neutron scattering experiments [229,251]. The data shown in Fig. 43 show rod-like peaks, indicating that the spin Muctuations are almost two-dimensional at this composition. The neutron scattering data can be Atted by an expression for the spin susceptibility of the form &−1 (k; !) = C[f(k) + (−i! + aT ) ]

(187)

with a function f which is consistent with an eKectively two-dimensional scattering f(k) = b(k⊥ )2 + c(k )4 :

(188)

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Fig. 42. Resistivity data for CeCu5:9 Au0:1 . From L_ohneysen et al. [285].

k and k⊥ are the deviations from the AFM Bragg vector parallel and perpendicular to the c-axis in these (nearly) orthorhombic crystals. Further a ≈ 1;

= 0:74 ± 0:1 :

(189)

At present, there is no natural leeway for this anomalous exponent within known theoretical frameworks. However, if one accepts this particular form of & as giving an adequate At, then the observed speciAc heat follows: at the critical composition, we expect the scaling relation −(d−1) −1 O

F ∼ TO⊥

∼ TT (d−1)=2 T =4 ∼ T 1+(d−1=2)=2 ;

(190)

which immediately gives CV ∼ d 2 F=dT 2 ∼ T for = 4=5. A better calculation [229,230] provides the logarithmic multiplicative factor. Even the measured uniform magnetic susceptibility is consistent with the above form of &(q; !). The observed resistivity does not follow directly from the measured &; a further assumption is required. The assumption that works is that fermions couple to the Muctuations locally, as in

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Fig. 43. Neutron scattering data for CeCu5:9 Au0:1 , a compound which is close to a QCP. The Agure shows q-scans along three diKerent crystallographic directions, from top to bottom in the a, b and c directions for ˝! = 0:1 meV. The Agures show that there is only a weak q-dependence along the rods (q⊥ ), while transverse scans (q ) show well-deAned peaks with nearly the same line width. From Stockert et al. [251].

an eKective Hamiltonian ∼ ci;† ci; Si; ; . Then if the measured Muctuation spectra are that of some localized spins Si , the single-particle self-energy is that due to the exchange of bosons with propagator proportional to k

&(k; !) ∼ ln(!) + i sgn(!) :

(191)

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Fig. 44. The phase diagram of UCu5−x Pd x . At low dopings and temperatures, the system is in an antiferromagnetic phase. In the undoped sample (x = 0): TN = 15 K with a magnetization 1B . For doping x = 1 and 1.5 the speciAc heat Cv =T ∼ T (for x = 1) while displaying weak logarithmic characteristics for x = 1:5. Similarly, the susceptibility &(T ) ∼ ln T and & ∼ T −0:25 (for x = 1; 1:5, respectively). Courtesy of M. C. Aronson.

This ensures that the single-particle relaxation rate as well as the transport relaxation rate 40 is proportional to T . A major theoretical problem is why the non-local or “recoil” terms in the interaction of itinerant fermions are irrelevant — i.e., why is the eKective Hamiltonian not † † k; q ck+q; ck; (Sq + S−q ) ; ;

(192)

or, in other words, why has momentum conservation been legislated away? The singularity of & also raises the question whether the anharmonic processes, Fig. 33(c), which are benign and allow the elimination of fermions in the RPA theory, give singular contributions to &. Also, can fermions really be eliminated in calculating the critical behavior? At a more mundane level, what is the form of the microscopic magnetic interactions in the problem which lead to the observed two-dimensional nature of the correlations? As an example in the second category, in Fig. 44 we show the phase diagram of some of the resistivity data of the heavy fermion compound UCu5−x Pd x . There are several other compounds in this category also; for a review we refer to [169,171]. For a theoretical discussion of the scaling properties of some of this class of problems see [27]. For x ¡ 1, there is an (antiferromagnetic) ordered state at low temperatures, while for x ¿ 2, a spin-glass phase appears. At Arst sight, one would therefore expect possible SFL behavior only near the critical composition x = 1 and near x = 2. The remarkable observation, however, 40

In this calculation, the k4 dispersion is neglected entirely, so that the problem is two dimensional. The inclusion of this term changes the result to T 5=4 .

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Fig. 45. The temperature dependence of the static susceptibility of &(T ) for both UCu4 Pd and UCu3:5 Pd 1:5 , showing that for both compounds, &(T ) has a low temperature divergence as T −1=3 . The measuring Aeld is 1 T. From Aronson et al. [29,30].

is that over a whole range of intermediate compositions, one observes anomalous behavior of the type [29,30] H = H0 − BT 1=3 ;

& ∼ T −1=3 :

(193)

The data for & that show this power-law behavior for UCu3:5 Pd 1:5 and UCu1 Pd are shown in Fig. 45. Note that the anomalous scaling is observed over a very large temperature range, and that it is essentially the same for the compound with x = 1 which as Fig. 44 shows is a good candidate for a composition close to a QCP, and the compound with x = 1:5 which is right in the middle of the range where there is no phase transition. The fact that this is genuine scaling behavior is independently conArmed [29,30] from the fact that the frequency-dependent susceptibility, measured by neutron scattering, shows a very good collapse of the data with the scaling assumption 41 ! & (!; T ) = T −1=3 T : (194) T Again, none of this behavior Ands a clear explanation in any of the well-studied models. One is tempted to use the critical points of impurity models (see for example [70] and references therein), but runs into the diPculty of having to tune to special symmetries. The ideas of critical 41

We note that this as well as the result for & in CeCuAu are examples of an anomalous dimension, as the engineering dimension of the susceptibility & is 1=energy—see the remark made just after Eq. (177).

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Fig. 46. Schematic occupation number function nk for UPt 3 and for UPd 3 .

points of metallic spin-glasses (see for example [224,234]), although theoretically appealing, are also not obviously applicable over such a wide range of composition. It must be mentioned however, that NMR does show clear evidence of the inhomogeneity in the singular part of the magnetic Muctuations in several heavy fermion compounds [42]. This has inspired models of varying sophistication (see [185] and references therein, and also [55]) in which the Kondo temperature itself has an inhomogeneous distribution. It is possible to At the properties with reasonable distributions but there is room for a deeper examination of the theoretical issues related to competition of disorder, Kondo eKects and magnetic interactions between the magnetic moments.This point is reinforced by recent measurements of local magnetic Muctuation spectra through -relaxation measurements [165]. Singular Muctuations are observed with a power law in agreement with (194). What is new is that the Muctuations are deduced to be independent of spatial location, indicating that they are a collective property and cannot be attributed to inhomogeneous local scales such as the Kondo temperatures. 6.4. Special complications in heavy fermion physics In heavy fermion compounds, there is often an additional complication that besets treating a QCP as a simple antiferromagnetic transition coupled to itinerant electrons. Often, such materials exhibit magnetic order of the f-electrons (with magnetic moments of O(1B ) per f-electron). Thus, such materials have local moments in the ordered phase; so the disappearance of the (anti)ferromagnetic order at a quantum critical point is accompanied by a metal–insulator transition of the f-electrons. This means that the volume of the Fermi surface changes in the transition. We may illustrate the above scenario by comparing U Pt3 , a heavy fermion compound with eKective mass of the order of 100, with UPd 3 , an “ordinary” metal with eKective mass of O(1) in which the f-electrons are localized. A schematic summary of the momentum occupation nk for the two cases is shown in Fig. 46: in the former nk is shown with two discontinuities, one small O(10−2 ) representing the large renormalization in the eKective mass of “f-electrons” while the other is close to 1 representing the modest renormalization of s- and d-electrons. The other case, representative of UPd 3 has just one Fermi surface with a jump in nk close

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to 1. The Fermi surface in the former encloses the number of electrons equal to the sum of the f and s–d electrons while the latter includes only the s–d electrons. This is consistent with de Haas-van Alphen measurements as well as the band structure calculations of the two compounds; but the band structure calculation must be carried out with the f-electrons assumed as being itinerant in the former and as part of the localized core in the latter. The magnetic transitions in heavy fermion compounds (with ordered moment of O(1B )) occurs through the conversion of itinerant f-electrons to localized electrons. So the Fermi surface on the two sides of the transition must switch between the two schematic representations in Fig. 46. The problem couples the “metal–insulator transition” of the f-electrons to the magnetic Muctuations—those of itinerant electrons on one side and of interacting local moments on the other. The Muctuations of the metal–insulator transition and the Fermi surfaces is an important part of the problem. Some theoretical work with these ideas in mind is available [242]. Another possible approach is to generate an eKective Hamiltonian for the heavy fermion lattice from a pairwise sum of the eKective Hamiltonian deduced from the two Kondo impurity problems discussed in Section 3.9 and study its instabilities. The two impurity problems contain the rudiments of some of the essential physics. In connection with the data in CeCu6−x Aux , we have discussed two important puzzles: the non-trivial exponent dM =z measured by &(k; !), and the coupling of fermions to the local Muctuations alone for transport properties. In the other category (impurity-dominated), the Arst puzzle reoccurs; the second puzzle may be explained more easily since the measured Muctuation spectrum &(k; !) is in fact k-independent. Both puzzles reoccur in the SFL phenomena in the cuprate compounds to be discussed in Section 7. 6.5. ECects of impurities on quantum critical points As is well known, randomness can have an important eKect on classical phase transitions. Two classes of quenched disorder are distinguished: First, impurities coupling quadratically to the order parameter [121] or, equivalently, impurities which may be used to deAne a local transition temperature Tc (r); the second class concerns impurities coupling linearly to the order parameter [131]. The so-called Harris criterion, for the Arst class, tells us that the disorder is relevant, i.e., changes the exponents or turns the transition to a crossover, if the speciAc heat exponent of the pure system is positive or, equivalently, if d − 2 ¡ 0 :

(195)

For application to QCP phenomena, the value of to be used is diKerent in the quantum Muctuation regime and the quasiclassical regime. 42 For the latter, should be deAned by the correlation length O ∼ (T − Tc )−1 for a Axed (p − pc ) while in the former, near T = 0 it should be deAned by O ∼ (p − pc )−2 . Accordingly, the eKect of disorder depends on the direction from which one approaches the QCP. Similarly, the celebrated Imry-Ma argument [131] for linearly coupled disorder can be generalized to QCPs. 42 Actually, the Harris criterion is derived in the form (195) for and d, not in terms of the exponent . This is particularly important at QCPs, since as we discussed in Section 6.2 for QCPs, one is often above the upper critical dimension where the hyperscaling relation d − 2 = breaks down.

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In problems of fermions, additional eKects of disorder arise because the vertices coupling the impurity to the fermions can be renormalized due to the singularity in the Muctuation of the pure system [271]. Not too much work has been done along these lines. A simple example is the eKect of magnetic impurities near a ferromagnetic transition [155]. The growth of the magnetic correlation length converts a single-channel Kondo eKect to a multichannel Kondo eKect with a regime in which the singularities discussed in Section 3.8 for the degenerate multichannel Kondo eKect may be realized without tuning any parameters [166]. This may be relevant to the deviations from the predictions of the pure case discussed here in the properties near the QCP in MnSi. Extensions of these ideas to antiferromagnetic and other QCPs would be quite worthwhile. 7. The high-Tc problem in the copper-oxide-based compounds About 105 scientiAc papers have appeared in the Aeld of high-Tc superconductors since their discovery in 1987. For reviews, see the Proceedings [175] of the latest in a series of Tri-annual Conferences or [110]. Although no consensus on the theory of the phenomena has been arrived at, the intensive investigation has resulted in a body of consistent experimental information. Here, we emphasize only those properties which are common to all members of the copper-oxide family and in which singular Fermi-liquid properties appear to play the governing role. The high-Tc materials are complicated, and many fundamental condensed matter physics phenomena play a role in some or other part of their phase diagram. As we shall see, the normal state near the composition of the highest Tc shows convincing evidence of being a weak form of an SFL, a marginal Fermi liquid. Since the vertices coupling fermions to the Muctuations for transport in the normal state and those for Cooper pairing through an exchange of Muctuations can be derived from each other, the physics of SFL and the mechanism for superconductivity in the cuprates are intimately related. 7.1. Some basic features of the high-Tc materials A wide variety of Cu–O containing compounds with diKerent chemical formulae and diKerent structures belong to the high-Tc family. The common chemical and structural features are that they all contain two-dimensional stacks of CuO2 planes which are negatively charged with neutralizing ions and other structures in between the planes. The minimal information about the structure in the Cu–O planes and the important electronic orbitals of the copper and oxygen atoms is shown in Fig. 47. The structure of one of the simpler compounds La2−x Sr x CuO4 is shown in Fig. 48(a) with the CuO2 plane shown again in Fig. 48(b). For x = 0 the CuO2 plane has a negative charge of −2e per unit cell which is nominally ascribed to the Cu2+ (O2− )2 ionic conAguration. Since O2− has a Alled shell while Cu2+ has a hole in the three-dimensional shell, the Cu–O planes have a half-Alled band according to the non-interacting or one-electron model. However, at x = 0, the compound is an antiferromagnetic insulator with S = 1=2 at the copper sites. This is well known to be characteristic of a Mott-type insulator in which the electron–electron interactions determine the ground state. Actually [297,266,267], copper-oxide compounds at x = 0 belong to the charge-transfer sub-category of Mott-insulators. However, at x = 0, the ground state and low-energy properties of all Mott-insulators are qualitatively

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Fig. 47. (a) Schematic structure of the copper-oxide ab-planes in La2 CuO4 . Ba or Sr substitution for La in the parent compound La2 CuO4 introduces holes in the CuO2 planes. The structure of other high Tc materials diKers only in ways which do not aKect the central issues, e.g. it is oxygen doping in YBa2 Cu3 O6+x that provides planar holes. The magnetic moments of the planar copper atoms are ordered antiferromagnetically in the ground state of the undoped compounds. From [135]. (b) The “orbital unit cell” of the Cu–O compounds in the ab plane. The minimal orbital set contains a dx2 −y2 orbital of Cu and px and a py orbital of oxygen per unit cell.

Fig. 48. (left) The crystal structure of La2 CuO4 . From [201]. Electronic couplings along the c direction are weak; (right) schematic of the CuO2 plane. The arrows indicate the alignment of spins in the antiferromagnetic ground state of La2 CuO4 . Speckled shading indicates oxygen p orbitals; coupling through these leads to a superexchange in the insulating state.

the same. By substituting divalent Sr for the trivalent La in the above example, “holes” are introduced in the copper-oxide planes with density x per unit cell. Fig. 49 is the generic phase diagram of the Cu–O compounds in the T –x plane. In the few compounds with electron doping which have been synthesized properties vary with doping density in a similar way.

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Fig. 49. Generic phase diagram of the cuprates for hole doping. The portion labeled by AFM is the antiferromagnetic phase, and the dome marked by SC is the superconducting phase. Crossovers to other characteristic properties are marked and discussed in the text. A low-temperature “insulating phase” in Region II due to disorder has not been shown.

Antiferromagnetism disappears for x typically less than 0.05 to be replaced by a superconducting ground state starting at somewhat larger x. The superconducting transition temperature is peaked for x typically between 0.15 and 0.20 and disappears for x typically less than 0.25. We will deAne xm to be the density for maximum Tc . Conventionally, copper-oxides with x ¡ xm are referred to as underdoped, with x = xm as optimally doped and with x ¿ xm as overdoped. Superconductivity is of the “d-wave” singlet symmetry. The superconducting region in the T –x plane is surrounded by three distinct regions: a region marked (III) with properties characteristic of a Landau Fermi liquid, a region marked (I) in which (marginally) SFL properties are observed and a region marked (II) which is often called the pseudo-gap region whose correlations in the ground state still remain a matter of conjecture. The topology of Fig. 49 around the superconducting region is that expected around a QCP. Indeed, it resembles the phase diagram of some heavy fermion superconductors (see e.g. Fig. 39) except that region II has no long-range antiferromagnetic order—the best experimental information is that, generically, spin rotational invariance as well as (lattice) translational invariance remains unbroken in the passage from (I) to (II) in the Cu–O compounds. The quantity (T ) ≡ Cv (T )=T and the magnetic susceptibility &(T ), which are temperature independent for a Landau Fermi liquid begin to decline rapidly [162] in the passage from region I to region II, which we will call Tp (x), but without any singular feature. However, the transport properties—resistivity, nuclear relaxation rate (NMR), etc.—show sharper change in their temperature dependence at Tp (x). The generic deduced electronic contribution to the speciAc heat for overdoped, optimally doped and underdoped compounds is shown in Fig. 50.

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Fig. 50. The electronic contribution to the speciAc heat as a function of temperature for underdoped, optimally doped and overdoped samples of Y0:8 Ca0:2 Ba2 Cu3 O7−x . For optimally doped and overdoped samples, the heat capacity remains constant as the temperature is lowered, then shows the characteristic features at the superconducting transition temperature Tc and rapidly approaches zero in the superconducting state. For underdoped samples, however, the heat capacity starts to fall well above Tc as the temperature is reduced, and there is only a small peak at Tc indicating much smaller condensation energy than the optimal and overdoped compounds [162]. From [35].

The generic behavior for an underdoped compound for the resisitivity, nuclear relaxation rate and Knight shift—proportional to the uniform susceptibility—is shown in Fig. 51. Angle resolved photoemission (ARPES) measurements show a diminution of the electronic density of states starting at about Tp (x) with a four-fold symmetry: no change along the (; ) directions and maximum change along the (; 0) directions. The magnitude of the anisotropic “pseudo-gap” is several times Tp (x). It is important to note that given the observed change in the single-particle spectra, the measured speciAc heat and the magnetic susceptibility in the pseudo-gap region are consistent with the simple calculation using the single-particle density of states alone. Nothing fancier is demanded by the data, at least in its present state. Moreover, the transport properties as well as the thermodynamic properties at diKerent x can be collapsed to scaling functions with the same Tp (x) as a parameter [291]. 7.2. Marginal Fermi liquid behavior of the normal state Every measured transport property in Region I is unlike that of a Landau Fermi liquid. The most commonly measured of these is the dc resistivity shown for many diKerent compounds at

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Fig. 51. Signatures of the pseudo-gap in various transport properties for the underdoped compound YBa2 Cu4 O8 . At high temperatures, the resistivity (solid line) decreases linearly with temperature. In the pseudogap region, it drops faster with temperature before falling to zero at the superconducting transition temperature (about 85 K). Similarly, the NMR relaxation time displays characteristics of the optimum doped compounds above about 200 K (squares on dashed line) but deviates strongly from it in the pseudogap region. The NMR shift (top squares) also deviates from the temperature-independent behavior (not shown) below the inset of the pseudogap. Note that the pseudogap expresses itself as a sharper change with temperature in the transport properties compared to the equilibrium properties—speciAc heat and magnetic susceptibility [46,293,10]. From [35].

the “optimum” composition in Fig. 52 including one with Tc ≈ 10 K. The resistivity is linear from near Tc to the decomposition temperature of the compound. As shown in Fig. 51, in the “under-doped” region, the departure from linearity commences at a temperature ∼ Tp (x) marked in Fig. 49. Similarly, the crossover into region (III) shown in Fig. 49 is accompanied by Fermi-liquid-like properties. Wherever measurements are available, every other measured transport property shows similar changes. The diKerent measured transport properties study the response of the compounds over quite diKerent momentum and energy regions. For example, the Raman scattering studies the longwavelength density and current response at long wavelength but over a range of frequencies from low O(1 cm−1 ) to high, O(104 cm−1 ). On the other hand, nuclear relaxation rate T1−1 depends on the magnetic Muctuations at very low frequencies but integrates over all momenta, so that the short-wavelength Muctuations dominate. In 1989, it was proposed [268,269] that a single hypothesis about the particle–hole excitation spectra captures most of the diverse transport anomalies. The hypothesis is that the density as well as magnetic Muctuation spectrum has an absorptive part with the following property: " = − &o !=T for !T ; & (q; !) (196) = − &o sgn(!) for !c |!|T :

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Fig. 52. Resistivity as a function of temperature for various high-temperature superconductors. From [39].

Here &o is the order of the bare single-particle density of states N (0) and !c is an upper cutoK. The Muctuation spectrum is assumed to have only a weak momentum dependence, except at very long wavelength, where a q2 dependence is required for Muctuations of conserved quantities like density or spin (in the absence of spin–orbit interactions). A form which implements these requirements for the conserved quantities, with a rather arbitrary crossover function to get the diKerent regimes of !=T , is & (q; !) ∼

−xq2

!(!2 + 2 x2 )

√ for vF q !x ;

(197)

where x = ! for !=T T and x = T for !=T 1. Using the Kramers–Kronig relations, one deduces that the real part of the corresponding correlation functions has a log(x) divergence at all momenta except the conserved quantities where the divergence does not extend to vF q ¿ x. Thus compressibility and magnetic susceptibility are Anite. (Aside from the contributions encapsulated in the approximate forms (196) or (197), an analytic background Muctuation spectrum of the Fermi liquid form is of course also present.) The spectral function (196) is unlike that of a Landau Fermi liquid discussed in Section 2, which always displays a scale—the Fermi-energy, Debye-frequency, or spin-wave energy, etc.— obtained from parameters of the Hamiltonian. Such parameters have been replaced by T . As we have discussed in Section 6, this scale-invariance of (196) is characteristic of Muctuations in the

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Fig. 53. (a) Diagram for the singular contribution to the one-particle self-energy with the Muctuating &(q; ). g’s are the vertices which in microscopic theory [270] is shown to have important momentum dependence, but which gives negligible momentum dependence to the self-energy. (b) For &0 (q; ) which is momentum independent, a total vertex - may sometimes be usefully deAned, which has the same frequency dependence as Eq. (198) and which is also q-independent. The lines are the exact single-particle Green’s functions.

quasiclassical regime of a QCP. Eq. (196) characterizes the Muctuations around the QCP: comparing with Eq. (179), the exponent dM =z = 0 and 1=z = 0. These are equivalent to the statement that the momentum dependence is negligibly important compared to the frequency dependence. This is a very unusual requirement for a QCP in an itinerant problem: the spatial correlation length plays no role in determining the frequency dependence of the critical properties. The experimental results for the various transport properties for compositions near the optimum are consistent in detail with Eq. (196), supplemented with the elastic scattering rate due to impurities (see later). We refer the reader to the original literature for the details. The temperature independence and the frequency independence in Raman scattering intensity up to ! of O(1 eV) directly follows from (197). Eq. (197) also gives a temperature independent contribution to the nuclear relaxation rate T1−1 as is observed for Cu nuclei. The transport scattering rates have the same temperature dependence as the single-particle scattering rate. The observed anomalous optical conductivity can be directly obtained by using the continuity equation together with Eq. (197), or by Arst calculating the single-particle scattering rate and the transport scattering rate. The single-particle scattering rate is independently measured in ARPES experiments and provides the most detailed test of the assumed hypothesis. To calculate the single-particle scattering rate, assume to begin with a constant coupling matrix element g for the scattering of particles by the singular Muctuations. We shall return to this point in the section on microscopic theory. Then provided there is no singular contribution to the self-energy from particle–particle Muctuations, the graph in Fig. 53 represents the singular self-energy exactly. It is important to note that for this to be true, Eq. (197) is to be regarded as the exact (not irreducible) propagator for particle–hole Muctuations; it should not be iterated. The self-energy (q; !) can now be evaluated straightforwardly to And a singular contribution x 2 2 (!; q) ≈ g (&o ) ! ln −i x (198) !c 2 for x!c and vF |(q − kF )|!c . The noteworthy points about (198) are: (1) The single-particle scattering rate is proportional to x rather than to x2 as in Landau Fermi liquids. (2) The momentum independence of the single-particle scattering rate.

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Fig. 54. The T = 0 distribution of bare particles in a marginal Fermi liquid. No discontinuity exists at kF but the derivative of the distribution is discontinuous.

(3) The quasiparticle renormalization amplitude x −1 Z = 1 − 1 ln !c

(199)

scales to zero logarithmically as x → 0. Hence the name marginal Fermi liquid. (4) The single-particle Green’s function G(!; q) =

1 ! − (q − ) − (q; !)

(200)

has a branch cut rather than a pole. It may be written as Z(x) ; ! − (˜q − ) ˜ − i= ˜

(201)

where ˜q is the renormalized single-particle energy ˜q − ˜ = Z(q − ) ≈ ZvF · (q − kF )

(202)

for small |q − kF |. Also, ˜−1 (x) = Z Im (!), and the eKective Fermi velocity v˜F = ZvF has a frequency and temperature-dependent correction. (5) The single-particle occupation number has no discontinuity at the Fermi surface, but its derivative does, see Fig. 54. So the Fermi surface remains a well-deAned concept both in energy and in momentum space. The predictions of (200) have been tested in detail in ARPES measurements only recently. ARPES measures the spectral function A(q; !) = −

1 (q; !) : [! − (q − ) − (q; !)]2 + [ (q; !)]2

(203)

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Fig. 55. (a) Momentum distribution curves for diKerent temperatures. The solid lines are Lorentzian Ats; (b) momentum widths of MDCs for three samples (circles, squares, and diamonds). The thin lines are T -linear Ats which show consistency with Eq. (203) and the MFL hypothesis. The resistivity (solid black line) is also shown. The double-headed arrow shows the momentum resolution of the experiment. From Valla et al. [261].

In ARPES experiments, the energy distribution curve at Axed momentum (EDC) and the momentum distribution curve at Axed energy (MDC) can both be measured. It follows from Eq. (203) that if is momentum independent perpendicular to the Fermi surface, then an MDC scanned along k⊥ for ! ≈ should have a Lorentzian shape plotted against (k − kF )⊥ ˆ For a marginal Fermi liquid (MFL), this width with a width proportional to (!)=vF (k). should be proportional to x. The agreement of the measured line shape to a Lorentzian and ˆ is the variation of the width with temperature are shown in Fig. 55. The Fermi velocity v(k) measured through the EDC with the conclusion that it is independent of kˆ to within the experimental errors. Further data from the same group shows that the temperature dependence is consistent with linearity all around the Fermi surface [262] with a coePcient independent of kˆ although the error bars are huge near the (; 0) direction. Besides the MFL contribution, there is also a temperature independent contribution to the width which is strongly angle-dependent, to which we will soon turn. The ambiguity of the temperature (and frequency) dependence near the (; 0) direction is removed by the EDC measurements. In Fig. 56, the EDCs at the Fermi surface in the (; ) direction and the (; 0) directions are shown together with a At to the MFL spectral function with a constant contribution added to . EDCs have the additional problem of an energy-independent experimental background. This has also been added in the At. In both directions, has a contribution proportional to ! with the same coePcient within

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Fig. 56. Fits of the MFL self-energy - + 1˝! to the experimental data, according to (204). Estimated uncertainties are ±15% in - and ±25% in 1. (a) A scan along the (1,0) direction, - = 0:12, 1 = 0:27; (b) a scan along the (1,1) direction, - = 0:035, 1 = 0:35. From Kaminsky and co-workers [139].

the experimental uncertainty. Fig. 57 presents the self-energy for the At at 13 diKerent points on the Fermi surface, showing that the inelastic part is proportional to ! and independent of momentum. In summary, the ARPES experiments give that ∼ -(kˆF ) + (!; T ) : (k; !; T ) = (204) MFL The (!; T )-independent contribution -(kˆ F ) can only be understood as due to impurity scattering [2]. Its dependence on kˆ F can be understood by the assumption that in well-prepared samples, the impurities lie between the Cu–O planes and therefore only lead to small angle scattering of electrons in the plane. The contribution -(kˆ F ) at kˆ F then depends on the forward scattering matrix element and the local density of states at kˆ F which increase from the (; ) direction to the (; 0) direction. This small angle contribution has several very important consequences: (i) relative insensitivity of residual resistivity to disorder, (ii) relative insensitivity of d-wave superconductivity transition temperature to the elastic part of the single-particle scattering rate [142], and (iii) most signiAcantly, relative insensitivity to the anomalous Hall eKect and magneto resistance. Such anomalous magneto-transport properties follow from a proper solution of the Boltzmann equation including both small angle elastic scattering and angle independent MFL inelastic scattering [275]. The momentum independence of the inelastic part of is crucial to the SFL properties of the cuprates. Since the inelastic scattering to all angles on the Fermi surface is the same, i.e., s-wave scattering, the vertex corrections to transport of vector quantities like particle current and energy current are zero. It follows that the momentum transport scattering rate measured in

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Fig. 57. The left panel shows the energy distribution curves measured in optimally doped Bi2212 perpendicular to the Fermi-surface at 13 points shown in the top-right panel. Each of these is Atted to the Marginal Fermi-liquid self-energy plus a frequency independent scattering contribution, i.e., with Im (!; kˆF ) = a(kˆF + b!), with Ats of the quality shown in Fig. 56. The variation of the Atting parameters a and b on the 13 points is shown in the bottom-right panel. The parameter b is seen to be independent of direction to within experimental error while a increases by about a factor of 4 in going from the (; ) direction to the (; 0) direction. (Figure courtesy of A. Kaminsky and J.C. Campuzano, presented at Proceedings of the International Conference on Spectroscopy of Novel Superconductors, Chicago, May 13–17, 2000). Similar results may be found in the work by Valla et al. [262].

resistivity or optical conductivity and the energy transport rate measured in thermal conductivity have the same (!; T ) dependence as the single-particle scattering rate 1=(!; T ). Recently far-infrared conductivity measurements [68] have been analyzed and shown to be consistent with 1=(!; T ) deduced from MFL including the logarithmic corrections.

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As already discussed no singular correction to the magnetic susceptibility is to be expected on the basis of (196). However, the speciAc heat should have a logarithmic correction so that (T ) = 0 [1 + 1 ln(!c =T )] :

(205)

Such a logarithmic correction has not yet been deciphered in the data presumably because the electronic speciAc heat in the normal state is less than O(10−2 ) of the total measured speciAc heat and must be extracted by a subtraction procedure which is not suPciently accurate. 7.3. General requirements in a microscopic theory The MFL self-energy, Eq. (198), has been veriAed in such detail in its (!; T; q) dependence that it is hard to see how any theory of CuO compounds can be relevant to the experiments without reproducing it (or a very close approximation to it) in Region I of the phase diagram of Fig. 49. Such a scale-invariant self-energy is characteristic of the quasiclassical regime of a QCP and indeed the topological features of the phase diagram are consistent with there being a QCP at xc near the composition for the highest Tc (Alternatively, a QCP in the overdoped region where Tc vanishes is predicted in some approaches, like in [235]). To date, no method has been found to obtain Eq. (197) except through the scale-invariant form of Muctuations which is momentum independent (z ≈ ∞) as in Eq. (196). A consistent and applicable microscopic theory of the copper-oxides must show a QCP with Muctuations of the form (197). This is a very unusual requirement for a QCP in a homogeneous extended system for at least two reasons. First, the Muctuations must have a negligible q-dependence compared to the frequency dependence, i.e., z ≈ ∞ and second, the singularity in the spectrum should just have logarithmic form; i.e., there should be no exponentiation of the logarithm giving rise to power laws. Such singularities do arise, as we discussed in Section 3, in models of isolated impurities under certain conditions but they disappear when the impurities are coupled; recoil kills the singularities. The requirement of negligible q=! dependence runs contrary to the idea of critical slowing down in the Muctuation regime of the usual transitions, in which the frequency dependence of the Muctuations becomes strongly peaked at zero frequency because the spatial correlation length diverges. Another crucial thing to note is that any known QCP (in more than one dimension) is the end of a line of continuous transitions at T = 0. Region II (at least at T = 0) must then have a broken symmetry (this includes part of Region III, which is also superconducting). The experiments appear to exclude any broken translational symmetry or spin-rotational symmetry for this region 43 although as discussed in Section 7.1, a sharp change in transport properties is observed along with a four-fold symmetric diminution of the ARPES intensity for low energies at T ≈ Tp (x). If there is indeed a broken symmetry, it is of a very elusive kind; experiments have not yet found it. 43 A new lattice symmetry due to lattice distortions or antiferromagnetism, if signiAcant, would change the fermi surface because the size of the Brillouin zone would decrease. This would be visible both in ARPES measurements as well as in hall eKect measurements.

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Fig. 58. The Cooper-pair vertex and the normal state self-energy, Fig. 53, are intimately related.

A related question is how a momentum independent &(q; !; T ) can be the Muctuation spectrum of a transition which leads to an anisotropic state as in Region II. Furthermore, how can such a spectrum lead to an anisotropic superconducting state? After all, it is unavoidable that &(q; !; T ) of Eq. (197) which determines the inelastic properties in Region I may also be responsible for the superconductive instability. After all, the process leading to the normal self-energy, Fig. 53, the superconductive self-energy, and the Cooper pair vertex, Fig. 58, are intimately related. Given Im &(q; !; T ), the eKective interaction in the particle–particle channel is Vpair (k; k ± q) = g2 Re &(q; !) :

(206)

Re &(q; !) is negative for all q and for all −!c 6 ! 6 !c . So we do have a mechanism for superconductive pairing in the Cu–O problem given by the normal state properties just as the normal state self-energy and transport properties of, say, Pb tell us about the mechanism for superconductivity in Pb. In fact, given that the normal state properties give that the upper cut-oK frequency is of O(103 ) K and that the coupling constant 1 ∼ g2 N (0) is of O(1), the correct scale of Tc is obtained. The important puzzle is, how can this mechanism produce d-wave pairing given that &(q; !) is momentum independent. How can one obtain momentum independent inelastic self-energy in the normal state and a d-wave superconducting order parameter from the same Muctuations? In the next section, we summarize a microscopic theory which attempts to meet these requirements and answer some of the questions raised. 7.4. Microscopic theory There is no consensus on even the minimum necessary model Hamiltonian to describe the essential properties in the phase diagram, Fig. 49, of the CuO compounds. It is generally agreed that, since other transition metal compounds do not share the properties of CuO compounds, a model Hamiltonian with some rather special features is called for. Two such features are: (A) They are two-dimensional with an insulating antiferromagnetic ground state and spin S = 1=2 per unit cell at half-Alling. Although not unique, this feature is rare. If this is the

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determining feature, a two-dimensional Hubbard model is adequate [24]. Even this model is not soluble in d = 2. (B) The copper oxides are the extreme limit of charge transfer compounds [297] in which charge Muctuations in the metallic state occur almost equally on oxygen and copper. Then the longer-range ionic interactions, which in magnitude are comparable to the on-site interactions, have a crucial role to play in the low-energy dynamics in the metallic state through excitonic eKects. A model with both Cu and O orbitals, hopping between them, and the excitonic interactions besides the on-site repulsions is then required [266,267]. This is of course even harder to deal with than the Hubbard model. Numerous attempts have been made using one or the other such models to obtain SFL behavior. We brieMy discuss the motivations for the pursuit of model (A) before summarizing in a little more detail the only attempt to obtain the phenomenological Muctuation spectrum of Eq. (196), and which relies on a model of type (B). 7.4.1. The doped Hubbard model The investigations of the copper-oxide problem from this point of view ask some valid and deep questions [24]. How does a low concentration of holes move through the spin conAgurations in a two-dimensional model with double occupancy strictly prohibited? 44 In Section 4.8, we have sketched the diPculties of connecting to the same problem in one dimension when the ground state at zero doping is an antiferromagnet. In fact, analytic [228] and numerical [172] answers to the question for a single hole show the spectral weight of a heavy particle with an incoherent part composed of multiple spin-wave polaronic cloud. Simply extrapolated (a dangerous thing to do), a Fermi liquid is expected. The larger zero-point Muctuations of the S = 1=2 model, compared to a large-spin model only change the relative weight of the coherent and the incoherent parts. However, more subtle possibilities must be considered. The antiferromagnetic ground state of a Heisenberg S = 1=2 model (or the undoped non-degenerate Hubbard model) in two dimensions is close in energy to a singlet ground state. A possible description of such a state is as a linear combination on the basis of singlet-bonds of pairs of spins. As noted earlier, such itinerant bond states have been termed resonating valence bonds [25] (by analogy to the ground state of benzene like molecules). The massive degeneracy of the singlet bond-basis raises interesting possibilities. If the quantum Muctuations of spins were (signiAcantly) larger than allowed by S = 1=2, such states would indeed be the ground state, as they are in the one-dimensional model or two-dimensional models with additional frustrating interactions [6]. It is possible that by doping with holes in the S = 1=2 Heisenberg model, the additional quantum Muctuations induce a ground state and low-lying excitations which utilize the massive degeneracy of RVB states. Especially intriguing is the fact that resonating valence ground state may be looked up on as the projection of the BCS ground state to a Axed number of particles [24]. Furthermore, in the normal state this line of reasoning is likely to lead to an SFL. 44 Questions of this type have a long history in the Aeld of correlated electron systems going back to the classic work of Nagaoka [191] on the ferromagnetism induced by the motion of one hole in a half-Alled inAnitely repulsive Hubbard model.

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A speciAc proposal incorporating the RVB idea [144] relies on the ground state of the half-Alled model to be localized dimers. Then defects in this state due to deviation from half-Alling can plausibly support excitations which are charged spinless bosons. Further work on this idea may be found in [225,188]. Related ideas were put forth in [82,236]. These are a very attractive set of ideas and no proof exists that they are disallowed. We have already considered an implementation of these ideas in Section 5.2 on generalized gauge theories. As discussed, controlled calculations using these ideas are hard to come by. Moreover, what theoretical results do exist do not correspond in a persuasive way to the experimental results on the copper-oxide materials. One should take special note here of the idea of Anyon superconductivity which besides being a lovely theoretical idea, is founded on the solution to a well-deAned model, and has clear experimental predictions. Laughlin and collaborators [156,138,91] found a speciAc model with long-range four-spin interactions for which his quantum hall wavefunction is the ground state. Therefore, time-reversal and parity are spontaneously broken in this state. This state is superconducting. The predicted time-reversal broken properties have not been observed experimentally [248]. An alternative idea from the microscopic characterization of these materials as doped Hubbard models is that a dilute concentration of holes in the Hubbard model is likely to phase separate or form ordered one-dimensional charge-density wave=spin-density wave structures [298,299,86]. There exists both empirical [257] and computational [288] support for this idea at least for a very dilute concentration of holes. For concentrations close to optimum compositions these structures appear in the experiment to exist only at high energies with short correlation lengths and times and small amplitudes in the majority of copper–oxygen compounds. Their relation to SFL properties is again not theoretically or empirically persuasive. 7.4.2. The excitonic interactions model This relies on a model of type (B). A brief sketch of the calculations leading to a QCP and an MFL spectrum is given here. We refer the reader to [266,268,275] for details. At half-Alling, the ground state and the low-lying excitations of such models are identical to the Hubbard model. However, important diKerences can arise in the metallic state. Consider the one-electron structure of such models. The O–O hopping in the lattice structure with dx2 −y2 orbitals in Cu and px , py orbitals on O, as shown in Fig. 47, produces a weakly dispersing “non-bonding” band while the Cu–O hopping produces “bonding” and “anti-bonding” bands—see Fig. 59. We need consider only the Alled non-bonding band and the partially Alled anti-bonding bands shown in Fig. 59. In the mean Aeld approximation, such an electronic structure together with the on-site interaction and the ionic interactions is unstable to an unusual phase provided the latter, summed over the nearest neighbors, is of the order or larger than the bandwidth. In this phase, translational symmetry is preserved but time-reversal symmetry and the four-fold rotation symmetry about the Cu sites is broken. The ground state has a current pattern, sketched in Fig. 60, in which each unit cell breaks up into four plaquettes with currents in the direction shown. The variation of the transition temperature with doping x is similar to the line Tp (x) in Fig. 38, so that there is a QCP at x = xc . Experiments have been proposed to look for the elusive broken symmetry sketched in Fig. 60 [274].

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Fig. 59. Three bands result from the orbitals shown in Fig. 47(b) in a one-electron calculation; two of these are shown. The chemical potential lies in the “anti-bonding” band and is varied by the doping concentration. The other band shown is crucial for the theory using excitonic eKects as in [270].

The long-range interactions in the model also favor other time-reversal breaking phases which also break translational invariance. This is known from calculations on ladder models [203]. Such states have also been proposed for copper-oxide compounds [148,128,58]. Our primary interest here is how the mechanism of transition to such a phase produces the particular SFL Muctuation spectrum (197) in Region I of the phase diagram. The driving mechanism for the transition is the excitonic singularity, due to the scattering between the states of the partially Alled conduction band c and the valence band v of Fig. 59. This scattering is of course what we considered in Sections 3.5 and 4.9 for the problem of X-ray edge singularities for the case that the interband interaction V in Eq. (55) is small and the valence band is dispersionless (i.e., the no recoil case). Actually, the problem is exactly soluble for the no-recoil case even for large V [65]. For large enough V , the energy to create the exciton, !ex , is less than the v–c splitting @. The excitonic line shape is essentially the one sketched in Fig. 19(b) and given by Eq. (56) for ! ¿ !ex , but is now the phase shift modulo which is the value required to pull a bound state below @. The excitonic instability arises when !ex → 0. The eKect of a Anite mass or recoil on the excitonic spectra is to smoothen the edge singularity on the scale of the valence band dispersion between k = 0 and k = kF . The phase shift or the interaction energy V no longer determines the low-energy shape of the resonance. V does determine its location. The low-energy Muctuation spectra is determined by the following argument. Let us concentrate on q = 0 which is obviously where Im &ex (q; !) is largest. The absorptive part of a particle–hole spectra must be odd in frequency Im &(q; !) = − Im &(q; −!) :

(207)

As V increases, Im &ex (0; !) must shift its weight towards zero-frequency as shown in Fig. 61. Let us continue to denote by |!ex | the characteristic energy of the maximum in Im &. For |!| small compared to |!ex |, Im &(0; !) ∼ ! while for |!| large compared to |!ex |, it is very slowly

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Fig. 60. One of the possible current patterns in the time-reversal breaking phase predicted for Region II of the phase diagram.

Fig. 61. Sketch of the development of the particle–hole spectra in the microscopic model for the cuprates.

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Fig. 62. Singularity of interaction -aaaa between states “a” near the chemical potential generated by the excitonic singularity between the states “a” of the conduction band and states “b” of the valence band. The excitonic singularity is indicated by the shaded block.

varying up to a cut-oK !c on the scale of the Fermi energy. Then by the Kramers–Kronig relation, to leading order Re &(0; !) ∼ ln(!c =max(!x ; !)) :

(208)

For any Anite |!ex |; Re & is Anite and there is no instability. Only for |!ex | → 0, i.e. Im &(0; !) → sgn !; Re &(0; !) is singular ∼ ln |!| and there is an instability. Thus in an excitonic instability of a Fermi sea with a dispersive valence band, the zero-temperature spectrum has the form &(!; 0) ∼ ln |!| + i sgn ! at the instability. Given a parameter p such that the instability occurs only at pc , i.e., !ex (p → pc ) → 0, the generalization for Anite temperature T and momentum q and p = pc is

−1 i! !c &(q; !) = + ln max(!; T; !ex (p)) max(!; T; !ex (p)) #−1 + a2 q2 + (pc (T ) − p) : (209) Here, !c is the cut-oK frequency of O(@). Since the binding energy is O(1 eV), the size of the exciton, a, is of the order of the lattice constant. The q dependence of (209) is negligible compared to the frequency dependence. The exponent z is eKectively inAnite. At p ≈ pc , to logarithmic accuracy, the above expression (209) is identical to the phenomenological hypothesis (197). The eKective low-energy interaction for states near the chemical potential, which is sketched in Fig. 62, is singular when the excitonic resonance is at low frequency. Here is an example of the mechanism mentioned under (v) in Section 2.6 where the irreducible interaction obtained by integrating over non-perturbed high-energy states is singular. This is, of course, only possible when the interactions represented by the shaded block in Fig. 62 are large enough. In relation to some of the questions raised about the phenemenology at the end of the last subsection, the momentum dependence of the vertex coupling the low-energy fermions to the Muctuations in Figs. 53 and 58 has been evaluated [273]. It is non-local with a form depending

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on the wavefunctions of the conduction and valence bands and the leading result is g(k; k ) ∼ (sin(kx a=2) sin(kx a=2) − sin(ky a=2) sin(ky a=2)) :

(210)

(k − k ) = 0

this is proportional to [cos(kx a) − cos(ky a)]. This is intimately related to Note that at the d-wave current distribution in the broken-symmetry phase predicted for Region (II), shown in Fig. 60. Eq. (210) is such that when the diagram of Fig. 53 is evaluated, the intermediate state momentum integration makes the self-energy depend very weakly on the incoming momentum. But when the pairing kernel of Fig. 58 is evaluated, it is momentum dependent and exhibits attraction in the d-wave channel. Similarly, as has been shown [273], the vertex of Eq. (210) leads to an anisotropic state with properties of the pseudo-gap state of Region II below a temperature Tp (x). The principal theoretical problem remaining with this point of view is that a transition of the Ising class occurs at Tp (x) at least in mean Aeld theory. This would be accompanied by a feature in the speciAc heat unlike the observations. 45 The microscopic theory reviewed above reproduces the principal features of the phase diagram Fig. 49 of the copper-oxide superconductors, and of the SFL properties. It also gives a mechanism for high-temperature superconductivity of the right symmetry. Further conAdence in the applicability of the theory to the cuprates will rest on the observation of the predicted current pattern of Fig. 60 in Region II of the phase diagram. 46 8. The metallic state in two dimensions The distinction between metals and insulators and the metal–insulator transition has been a central problem in condensed matter physics for seven decades. Despite the accumulated theoretical and empirical wisdom acquired over all these years, the experimental observation made in 1995 of a metal–insulator transition in two dimensions [149] was a major surprise and is a subject of great current controversy. The theoretical work in the 1980s [11,92–94,157,14] on disordered interacting electrons pointed to a major unsolved theoretical problem in two dimensions. Infrared singularities were discovered in the scattering amplitudes which scaled to strong coupling where the theory is uncontrolled (The situation is similar to that after the singularities in the one- or two-loop approximations in the Kondo problem were discovered, revealing a fascinating problem without providing the properties of the asymptotic low-temperature state). However, the problem was not pursued and the Aeld lapsed till the new experiments came along. The 1980s theoretical work shows that this problem naturally belongs as a subject in our study of singular Fermi liquids. We will Arst summarize the principal theoretical ideas relevant to the problem before the 1995 experiments. We then brieMy summarize the principal results of these and subsequent experiments. Reviews of the experiments have appeared in [3,16,17]. 45

One might appeal to disorder to round oK the transition, but this appears implausible quantitatively. More likely, the nature of the transition is strongly aKected by the Muctuation spectra of the form of Eq. (196) and is unlikely to be of the Ising class. 46 As already mentioned in Section 2.6, ferromagnetism in some compounds has an excitonic origin. The dynamics near such a transition should also exhibit features of the edge-singularity as modiAed by recoil.

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Fig. 63. Sketch of a MOSFET. Holes (or electrons) are trapped at the interface of the semiconductor and the insulator due to the band gap diKerence between them, the dipole layer and the applied electric Aeld. Two-dimensional electrons (holes) may also be found by layered structures (heterostructures) of semiconductors with diKerent band gaps such as GaAs and AlGaAs.

There are two types of theoretical problems raised: the nature of the metallic state and the mechanism of the metal–insulator transition. We will address the former and only brieMy touch on the latter. 8.1. The two-dimensional electron gas We consider an electron gas with a uniform positive background with no complications arising from the lattice structure—this is how the many-electron problem was originally formulated: the Jellium model. This situation is indeed realized experimentally in MOSFETS (and heterostructures) in which an insulator is typically sandwiched between a metallic plate and a semiconductor—see Fig. 63. By applying an electric Aeld, a two-dimensional charge layer accumulates on the surface of the semiconductor adjacent to the insulator, whose density can simply be changed by varying the Aeld strength (For details see [26].). Similar geometries have been used to observe the quantum Hall eKects and the metal–insulator transition by varying the density. Typically we will be interested in phenomena when the average inter-electron distance is O(10) − O(102 ) nm. The thickness of the insulating layer is typically more than 100 nm, so that the positive (capactive) charge on the insulator provides a uniform background to a Arst approximation. In Si samples, surface roughness is the principal source of disorder at high densities, while at low densities (in the regime where the transition takes place) ionized impurity scattering dominates due to the fact that there is much less screening. In GaAs, remote

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impurity scattering dominates, and this scattering is mostly small angle. This is the main reason that the mobility in these samples is large. Neglecting disorder, the problem is characterized by rs , deAned as the ratio of the potential energy to the kinetic energy rs =

me2 √ : 4˝2 n

(211)

Here n is the electron density, m the band mass, and the background static dielectric constant. We can also write rs2 a20 =

1 ; n

(212)

which expresses that rs is the radius of the circle whose area is equal to the area per conduction electron, measured in units of the Bohr radius a0 . For a two-valley band structure, as on the (110) surface of Si, the kinetic energy is reduced and rs is twice that deAned by Eqs. (211) and (212). For rs 1, (the dense electron limit) the kinetic energy dominates and metallic behavior is expected. For rs 1, the potential energy dominates and a crystalline state (Wigner crystal) is expected. The best current numerical estimates place the transition to the crystalline state at rsc ≈ 37 [253]. The entropy at the transition is tiny, indicating that the radial distribution function for the liquid state at low densities is similar to that of the crystal for distances up to a few times rs a0 . It is important to note for our subsequent study that magnetism is always lurking close by. Reliable numerical calculations show that the magnetic state in the Wigner crystal near the critical density is determined by multiple-particle exchanges [47]. On the metallic side, the energy of the ferromagnetic state is only a few percent above the unpolarized metallic or crystalline states for rs ≈ rsc [253]. Disorder is expected to make the metal–insulator state continuous. On the insulating side at T → 0, the disordered Wigner crystal is expected to be glassy and have low-energy properties of a Coulomb glass [239]. On the metallic side Muctuations in the local density of electrons might be expected to lead to locally polarized magnetic states or possibly to some unusual frustrated magnetic states [57]. The perturbative calculations with disorder and interactions, already alluded to [92–94,54] also hint at the formation of magnetic moments in the metallic state. It is the interplay of such magnetic Muctuations with itinerancy which is one of the principal theoretical problem in understanding the metallic state. 8.2. Non-interacting disordered electrons: scaling theory of localization Detailed reviews on the material in this section may be found in [256,157,14,133]. The concept of localization of non-interacting electrons for strong enough disorder was invented in 1958 by Anderson [18]. In one dimension, all electronic states are localized for arbitrarily small disorder while in three dimension a critical value of disorder is required. That d = 2 is the marginal dimension in the problem was discovered through the scaling theory of localization.

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The conceptual foundations for the scaling theory of localization were laid by Thouless and co-workers [255,256] and by Abrahams et al. [1], and were developed formally by Wegner [286]. Abrahams et al. [1] also made predictions which could be tested experimentally. Thouless noted Arst of all that the conductance G of a hypercube of volume LD in any dimension d is dimensionless when expressed in units of (e2 =h), thus deAning a scale independent quantity g = G=(e2 =h) :

(213)

Next, he argued that g for a box of linear size 2L may be obtained from the properties of a box of size L and the connection between the two of them. The conductance of a box of size L itself increases with the transition amplitude t between energy levels in the two boxes and decreases with the characteristic width of the distribution of the energy levels in the boxes ^W (L) due to the disorder ^t(L) g(L) ≈ f : (214) ^W (L) For weak-Gaussian disorder, the bandwidth may be expected to be proportional to the square root of the number of impurities in the box, so ^W (L) ∼ Ld=2 . The transition amplitude t is obtained by the hopping between near-neighbors near the surface of the boxes of size L. It is therefore proportional to the surface area Ld−1 . Thus 47 g(L) = f(L(d−2)=2 ) :

(215)

Now, in three dimensions the conductivity should approach a constant for large L (Ohm’s law!), and hence the conductance should scale as L. This implies that the scaling function f(x) should go for large L as f(x) ∼ x2 . Note that for d ¿ 2 the g therefore increases with increasing L while for d ¡ 2 the large L behavior is determined by the small argument behavior of the scaling function; clearly d = 2 is the marginal dimension. In a very inMuential paper, Abrahams et al. [1] analyzed the B-function of the RG Mow B(g) ≡ d(ln g)=d(ln L)

(216)

and showed by a perturbative calculation in 1=g that B(g) = (d − 2) −

1 1 ; 2 g

(217)

where the Arst part comes from Eq. (215) with f(x) ∼ x2 . For small enough g (i.e., for large disorder) we expect exponential localization g(L) ∼ e−L=D , where D is the localization length, so that B(g) ∼ (−L=D). The smooth connection between the perturbative result (217) for large g and the exponentially localized solution at small g is shown in Fig. 64. While for d = 3 (or any d ¿ 2), a critical disorder gc is required for localization, for d = 2 states are asymptotically localized for any disorder for non-interacting fermions. The 47

This line of reasoning of course breaks down when we include electron–electron interactions.

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Fig. 64. The scaling function for non-interacting electrons with disorder deduced by Abrahams et al. [1].

characteristic value of the localization length in d = 2 is estimated from the perturbative solution: 1 L ; (218) g(L) = g0 − 2 ln ‘ where g0 is the dimensionless conductance at L ≈ ‘. In conventional Boltzmann transport theory g0 = (e2 =2˝)kF ‘. The localization length D is of the order of the value of L at which the correction term is of order g0 , so that (219) D ≈ ‘ exp kF ‘ : 2 At T → 0, the sample size of a sample with kF ‘1 has to be very large indeed for weak localization to be observable. The theory described above must be modiAed at Anite temperatures due to inelastic scattering. −1 If the inelastic scattering rate is much less than the elastic scattering rate, −1 in , localization eKects are cut-oK at a length scale LTh (T ), the Thouless length scale LTh = (Din )1=2 ;

(220)

where D = (vF2 =d) is the (Boltzmann) diKusion constant. However, as noted by Altshuler et al. [12,13], the correct scale for the cut-oK is −1 . , the phase breaking rate. In an individual collision the energy change ^E may be such that the phase changes only by a very small amount, in ^E 2. The phase breaking time is then longer and is shown to be given by . ∼ (^Ein )−2=3 in . The T = 0 theory with the “phase length” L. = (D. )1=2

(221)

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Fig. 65. Interfering (time-reversed) parts in elastic scattering oK a Axed set of impurities. The probability for the particle to arrive at B is reduced because of the enhanced probability for the particle to arrive back at A, as a result of interference.

replacing L then gives the Anite temperature scaling behavior to which experiments may be compared. The characteristic temperature Tx at which weak-localization eKects become prominent may be estimated in a manner similar to (218) Tx . (Tx ) = exp(−kF ‘) :

(222)

This expression puts useful bounds on the temperatures required to observe weak localization. Eq. (217) is derived microscopically by considering repeated backward scattering between impurities. It can also be derived by considering quantum interference between diKerent paths to go from one point A to another B [41]. The total probability / for this process is 2 /= ai = |ai |2 + a∗i aj ; (223) i

i

i=j

where ai is the amplitude of the ith path. The second term in Eq. (223) is non-zero only for classical trajectories which cross, for example at the point O in Fig. 65. The probability of Anding a particle at the point O is increased from 2|ai |2 to |a1 |2 + |a2 |2 + 2 Re a∗1 a2 = 4|a1 |2

(224)

because the two paths are mutually time-reversed. Increasing this probability of course leads to a decrease in the probability of the particle to arrive at B, and hence to a decrease in the conductivity. This argument makes it clear as to why the interfering paths must be shorter than the phase relaxation rate due to inelastic processes and why magnetic impurities or a magnetic Aeld which

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introduces phase shift between two otherwise time-reversed paths suppress weak localization. In two dimensions [14] e2 1 1 (H; T ) − (O; T ) = 2 + ln x ; (225) + 2 ˝ 2 x where

is the digamma function and x = 4L2. eH= ˝c ≡ (L. =LH )2 :

(226)

The quantity in brackets in (225) is equal to x2 =24 for x → 0 and to ln(x=4) − for x → ∞. Spin–orbit scattering preserves time-reversal symmetry but spin is no longer a good quantum number. The spins are rotated in opposite directions in the two self-intersecting paths of Fig. (65) if the impurities are spin–orbit scatters [125,14,157]. This has been shown to lead to an average overlap of the spinfunction of − 12 (because a rotation by 2 of wavefunction of a spin 1=2 particle leads to a wavefunction of opposite sign). The correction to the B-function of Eq. (217) due to this eKect is 1 1 : 22 g

(227)

This eKect tends to an enhancement of the conductivity. 8.3. Interactions in disordered electrons Fermi liquid theory for interacting electrons survives in three dimensions in the presence of a dilute concentration of impurities [43]. Some noteworthy diKerences from the pure case are: (1) Owing to the lack of momentum conservation, the concept of a Fermi surface in momentum space is lost but it is preserved in energy space, i.e., a discontinuity in particle occupation as a function of energy occurs at the chemical potential. The momentum of particles may be deAned after impurity averaging. General techniques for calculating impurity-averaged quantities are well developed; see for example [4,43]. Here and subsequently in this chapter the self-energies, vertices, etc. refer to their form after impurity averaging. (2) In the presence of impurities, the density–density correlation (and spin-density correlation, if spin is conserved) at low frequencies and small momentum must have a diKusive form (this is required by particle-number conservation and the continuity equation) (q; !) = %

Dq2 ; i! + Dq2

q‘−1

and

!−1 :

(228)

Here % = dn=d is the compressibility and D is the diKusion constant. For non-interacting electrons, D = 13 vF2 . Interactions renormalize D and % [43]. In the diagrammatic representation used below, the diKusive propagator is shown by a cross-hatched line connecting a particle and a hole line as in Fig. 66. (3) Owing to statement 1, the impurity-averaged single-particle spectral function at a Axed k is spread out over an energy -, so that for frequencies within a range - of the chemical

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Fig. 66. Elementary processes important in the problem of two-dimensional disordered interacting electrons and referred to in the text. (a) Representation of the diKusion propagator due to impurity scattering vertices and corresponding self-energy. The particle lines and hole lines should be on opposite sides of the chemical potential. (b) Singular second-order interactions. (c) Singular vertex in the density channel (and in the spin-density channel for the spin-conserving problem). (d) Singular (irreducible) Arst-order interactions. (e) Elementary singular polarization propagator.

potential, it has both a hole part (for ! ¡ ) and a particle part (for ! ¿ ). This is an important technical point in microscopic calculations. (4) The Ward identities relating the coupling of vertices to external perturbations change for the coupling to unconserved quantities (for the pure case they are given in Section 2.6). For example, no Ward identity can be derived for the vertex needed for the conductivity calculation, i.e., Lim!→0 Limq→0 0impure , because current is not conserved. = Lim Lim 0impure q→0 !→0

k 9(k; !) − m 9k

(229)

holds because after impurity averaging momentum is conserved. However, microscopic calculations show that, at least when Fermi-liquid theory is valid (cf. Section 2.4), Lim Lim 0impure = Lim Lim 0impure : !→0 q→0

q→0 !→0

(230)

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Indeed, if this were not so, one would not get a Anite dc conductivity at T = 0 for a disordered metal in d = 3. An argument for this is as follows: Normally, we calculate the conductivity by Arst taking the limit q → 0 and then the limit ! → 0, as on the left-hand side of (230). In practice, however, even when we apply a homogeneous Aeld to a system, the electrons in a disordered medium experience a Aeld which varies on the scale of the distance between the impurities, and so the physically relevant limit is the one on the right-hand side of (230), where the limit ! → 0 is taken Arst. But the validity of (230) appears not to extend to the case of singular Fermi liquids, at least for the present case where the singularities are q-dependent. This is one of the important diPculties in developing a consistent theory for disordered interacting electrons in d = 2. The diKusive form of the density correlation function and spin-density correlation is the culprit of the singularities which arise due to interactions in two dimensions. For example, the elementary eKective vertex in Fig. 66 due to a bare frequency independent short-range interactions in two dimensions is v

2

0

‘ −1

1 dq q i! + Dq2

≈ v2 N (0) ln(!) :

(231)

The singularity arises because (!; q) = f(!=Dq2 ). Recall that for pure electrons (!; q) = f(!=vF q) leading to a logarithmic singularity for the second-order vertex in one dimension and regular behavior in higher dimensions. Similarly, (!; q) = f(!=q3 ) leads to a logarithmic singularity in the second-order vertex in three dimensions, as we saw in Section 5.1 on SFLs due to gauge interactions. Note that in Eq. (231) and other singular integrals in the problem have ultra-violet cutoKs at q ≈ ‘−1 and ! ≈ −1 since the diKusive form is not applicable at shorter length scales or time scales. It also follows that Boltzmann transport theory is valid at temperatures larger than −1 . Actually, even the Arst-order interaction dressed by diKusion Muctuations is singular. Consider Arst the diKusion correction to the vertex shown in Fig. 15 0 1 = (i! + Dq2 )−1 ; 00

(232)

provided ¡ 0; − ! ¿ 0 or vice versa. The restriction is a manifestation of point (3) and arises because in the diKusion process, only intermediate states with one line above (particle) and the other below (hole) the chemical potential contribute as they alone deAne the physical density. This leads to the Arst-order irreducible interaction and the polarization graph shown in Fig. 67 to be logarithmically singular. For the small q of interest for singular properties, one need consider interactions only in the s-wave channel. One then has two interaction parameters, one in the singlet channel and the other in the triplet channel. Consider the problem with Coulomb interactions. Then the eKective interaction in the singlet channel sums the polarization bubbles connected by Coulomb interactions. Using (228) for the polarization bubble, it is shown [14] that for small momentum transfer the interaction in the

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Fig. 67. EKective interactions can be split into singlet and triplet channels. In the singlet-only channel (a), the density–density interaction is screened by the Coulomb interaction and is universal at long wavelengths. In the triplet channel and in the singlet channel for large momentum, the screened density–density interaction appears only in the cross channel and is therefore non-universal.

Fig. 68. Simplest processes contributing to the singular self-energy. (a) Exchange process; (b) Hartree process.

spin singlet (S = 0) channel, Fig. 67, becomes Vsinglet = 2% :

(233)

In the non-interacting limit % = N (0), independent of density. Consider now the ladder-type interactions illustrated in Fig. 68. These involve both the singlet and the triplet interactions. The momentum carried by the interaction lines is, however, to be integrated over. Therefore, the triplet interactions do not have a universal behavior, unlike the singlet interactions. Altshuler, Aronov and collaborators [14] (see also [103]) calculated the logarithmic corrections to Arst order in the interactions for various physical quantities. To these one can add the contribution already discussed due to weak localization. The corrections to the single-particle density of states, the speciAc heat and the conductivity over the non-interacting values

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are, respectively:

N ! 1 ; = ln|!| ln −2 2 N 4F (Ds ) C 3 1 1 − F ln|T| ; = C F 2 3 1 = 2 2 − F ln|T| : 4 2

(234) (235) (236)

The compressibility has no logarithmic corrections. In these equations, s is the screening length and F is a parameter which is of the order of the dimensionless interaction rs . The Arst terms in (235) and (236) are due to exchange processes and the second due to the Hartree processes. The exchange process, of which the contribution to the self-energy is shown in Fig. 66, use the interaction in the singlet channel; hence the universal coePcient. The second contribution uses both the triplet and the (large part of the q) singlet interactions. In Arst order of interaction, the diKerence in signs of the two processes is natural. In pure systems, the Hartree process does not appear as it involves the q = 0 interaction alone which is exactly canceled by the positive background. For disordered systems, due to the Muctuation in the (ground state) density, a Arst-order Hartree process, Fig. 68 contributes. In the presence of a magnetic Aeld, the Sz = ± 1 parts of the triplet interactions acquire a low-energy cut-oK. Therefore, the logarithmic correction to the resistivity is suppressed leading −1 to negative magnetoresistance proportion to F(H=kT )2 for small H=kT but gB H −1 so ; s where −1 −1 so and s are spin–orbit and spin-scattering rates, respectively, for appropriate impurities. 8.4. Finkelstein theory Finkelstein [92] has used Aeld-theoretical methods to generalize Eqs. (234) – (236) beyond the Hartree–Fock approximation. His results have been rederived in customary diagrammatic theory [52,40,53]. The interference processes leading to weak localization are again neglected. The theory may be regarded as Arst order in 1=kF ‘. In eKect, the method consists in replacing the parameter F by a scattering amplitude t for which scaling equations are derived. The equivalent of the Fos parameter is Axed by imposing that the compressibility remains unrenormalized, i.e., does not acquire logarithmic corrections. A second important quantity is a scaling variable z, which is analogous to the dynamical scaling exponent z which we discussed in Section 6, which gives the relative scaling of temperature (or frequency) with respect to the length scale. A very unusual feature of the theory is that z itself scales! Scaling equations are derived for t and z to leading order 1=kF ‘. As T → 0, both t and z diverge. The divergence in z (see the discussion in Section 7) usually means that the momentum dependence of the Muctuations is unimportant compared to their frequency dependence. The divergence in t as T → 0 in such a case has been interpreted to imply divergent spatially localized magnetic Muctuations; in other words, it implies the formation of local moments [93,54]. At the same time, the scaling equations show conductivity Mowing to a Anite value.

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Fig. 69. Schematic renormalization group Mow for the disordered interacting electron problem according to the Finkelstein theory. The dashed lines represent the eKect on the solid lines on applying a magnetic-Aeld which couples to spins alone.

The scaling trajectories of Finkelstein’s theory are shown schematically in Fig. 69. While in the non-interacting theory with disorder, one always has an insulator, this theory always Mows towards a metal. However, the theory cannot be trusted beyond t ∼ 1, as then it is uncontrolled. The theory also cannot be trusted for large disorder, kF ‘ ∼ O(1), even for small interactions. It is worth emphasizing that Finkelstein’s theory gives an eKect of the interactions in a direction opposite to the leading perturbative results. The perturbative results themselves of course are valid only for small rs while Finkelstein theory is strictly valid only for rs ¡ O(1). One possibility is that the Finkelstein result itself is a transient and the correct theory scales back towards an insulator (the dashed lines in Fig. 69). Another possibility is that it correctly indicates (at least for some range of rs and disorder) a strong-coupling singular Fermi liquid metallic Axed line. The new experiments discussed below can be argued to point to the latter direction. It is hard except in very simple situations (the Kondo problem, for instance) to obtain the approach to a strong-coupling Axed point analytically. In that case, one may usually guess the nature of the Axed point and make an expansion about it to ascertain its stability. 48 48

Although the theory breaks down in the strong coupling regime, this situation is somewhat comparable to the hints that the weak-coupling expansion gave in the early phase of the work on the Kondo problem: these weak-coupling expansions broke down at temperatures comparable to the Kondo temperature, but did hint at the fact that the low-temperature regime was a strong coupling regime.

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In making such a guess, the SFL properties towards which the Finkelstein solution Mows should be kept in mind: (i) The conductivity Mows towards a Anite value in the theory as T → 0. (ii) The density of single-particle states Mows towards zero N (!) ∼ ! :

(237)

(iii) The magnetic susceptibility diverges at a Anite length scale (the eKect of a diverging z) indicating the formation of local moments. The last point appears to be crucial. As may be seen from Eq. (244) below, the growth of the triplet scattering overrides the exchange processes which favor the insulating state. Indeed, if the triplet divergence is suppressed by an applied magnetic Aeld, the theory reverts to the perturbative form of Eq. (225). The scale of the magnetic Aeld for this eKect is given by the temperature. The formation of localized regions of moments may be linked to the fact already discussed that the ferromagnetic state is close in energy to the paramagnetic Muid (and the crystalline states) as density is decreased. The experiments discussed below have a signiAcant correspondence with this picture, although there are some crucial diKerences. A possible strong coupling Axed point 49 is a state in which the local moments form a singlet state with a Anite spin stiKness of energy of O(Hc ) in the limit T → 0. This eliminates any perturbative instability of the triplet channel about the Axed point. The state is assumed to have zero density of single-particle states at the chemical potential. This eliminates the localization singularity as well as the singularity due to the singlet channel. This state is then perturbatively stable. The conductivity of such a state can be shown to be Anite. The occurrence of a characteristic scale Hc observed in the magnetoresistance experiments discussed below with Hc → 0 as the metal–insulator transition in zero Aeld as n → nc is also in correspondence with these ideas. 8.5. Compressibility, screening length and a mechanism for metal–insulator transition Suppose the metallic state in two dimensions is described by a Axed point hinted by the Finkelstein theory and an expansion about it. Such a description must break down near the critical rs where a Arst-order transition to the Wigner transition must occur in the limit of zero disorder. General arguments suggest that the transition for Anite disorder must be continuous [132]. A suggestion for the breakdown of the Finkelstein regime follows from the calculation of the correction to the compressibility due to disorder [241]. As already mentioned, no perturbative singularity is found in the compressibility due to interactions. However, the correlation energy contribution of the zero-point Muctuations of plasmons is altered due to disorder with a magnitude which also depends on rs . The leading order contribution in powers of (kF ‘)−1 can be calculated for arbitrary rs . Including this contribution, the compressibility % may be written 49

This paragraph is based on the unpublished work of Q. Si and C.M. Varma.

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in the form %0 %0 + 0:11rs3 =(!0 ) + O((rs4 )=(!0 )2 ) : = % %pure

(238)

Here %pure is the compressibility for zero disorder, %0 = N (0), and !0 is the Rydberg. In the Hartree–Fock approximation √ %0 = 1 − ( 2=)rs : (239) %pure The best available numerical calculations also give %0 =% varying slowly enough with rs that the correction term (238) dominates for rs of interest near the metal–insulator transition even for a modest disorder. For example for !0 ≈ 10, the disorder contribution in Eq. (238) is larger than the pure contribution for rs ¿ 10. This has an important bearing on the metal–insulator transition because the screening length s is given by s=s0 = %0 =% ;

(240)

where s0 = a0 =2. Strictly speaking, s is the screening length for an external immobile charge and the screening of the electron–electron interactions is modiAed from (238) due to vertex renormalizations. However, in this case they do not change the essential results. From Eq. (238) it follows that the screening length s(‘) ¿ ‘, the mean free path, for rs ¿ 3(!0 ) :

(241)

Suppose the condition s(‘) ¿ L ¿ ‘ is satisAed. Here L again is the size of the box for which the calculation is carried out, deAned through DL−2 ≈ T . The assumption of screened short-range interactions, with which perturbative corrections leading to results of Eqs. (234) – (236) are obtained, is no longer valid. In this regime, the calculations must be carried out with unscreened Coulomb interactions. The correction proportional to F in Eqs. (234) – (236) is not modiAed but the singlet contributions are more singular (due to the extra q−1 in the momentum integrals). For instance, Eq. (236) is modiAed to √ L = = − ( 2=2 )rs : (242) ‘ This implies a crossover to strong localization. It is therefore suggested that the metallic state ceases to exist when condition (241) is satisAed. The above line of reasoning is of particular interest because as discussed below, a sharp variation in the compressibility is indeed observed to accompany the transition from the metallic-like to insulating-like state as density is decreased (as shown in Fig. 79 below). 8.6. Experiments Soon after the publication of the theory of weak localization, its predictions were seemingly veriAed in experiments on Si-MOSFETS [48,260]. The experiments measured resistivity on not very clean samples of high density with resistivity of O(10−2 h=e2 ). In a limited range of temperature, the predicted logarithmic rise in resistivity with decreasing temperature with about the right prefactor was found [41]. In view of the perturbative results of Altshuler and Aronov [14] and the knowledge that electron–electron interactions alone lead to a Wigner insulator at low

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Fig. 70. Resistivity data on a Ane scale for the two highest densities in Fig. 75 below, showing correspondence with the theory of weak localization at such high densities. From Pudalov et al. [216]. See text.

densities, one was led to the conviction that the metallic state does not exist in two dimensions. It was expected that samples with larger rs will simply show logarithmic corrections to the resistivity at a higher temperature and pure samples at a lower temperature. Not too much attention was paid to Finkelstein’s results which pointed to the more interesting possibility of corrections in the opposite direction. The more recent experiments on a variety of samples on a wider range of density and of higher purity than earlier have refocused attention on the problem of disorder and interactions in two dimensions and, by implication, in three dimensions as well. Several reviews of the experiments are available [3,16,17]. We will present only a few experimental data to highlight the theoretical problems posed, and will focus on the behavior of the data as a function of temperature. The scaling of the data as a function of the electron density n − nc or Aeld E will not be discussed; there is a considerable body of data on non-linear E-dependence (see e.g. [238] and references therein) but the signiAcance of the data is not clear at present. The Arst thing to note is that results consistent with the earlier data [48,260] are indeed obtained for high enough densities. Fig. 70 shows the resistance versus temperature in Si for rs ∼ O(1). The magnitude of the temperature dependence is consistent with the predictions of weak localization corrections. As we will show below in the same region of densities, the

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Fig. 71. Resisitivity as a function of temperature for a wide range of densities (and Fermi energy) in a disordered Si MOSFET. The inset shows accurate measurements of H(T ) close to the separatrix for another sample. From Sarachik and Kravchenko [226,3].

negative magneto-resistance predicted as the correction to weak localization, discussed above, is also observed. Fig. 71 shows the resistivity as a function of T over a wide range of densities. Similar data from [150] over a large small of densities is shown in Fig. 72, and data over a large range of densities are plotted as a function of T=EF in Fig.73. The resistivity clearly shows a change of sign in the curvature as a function of density at low temperatures. The resistivity at the crossover density as a function of temperature is shown down to 20 mK in the inset of Fig. 72 and is consistent with temperature independence. The true electron temperature in these samples is a question of some controversy [16,3], but more recent experiments, whose data are shown in Fig. 74, have corroborated these results by studying this issue very carefully down to 5 mK.

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Fig. 72. Resistivity versus temperature at Ave diKerent electron densities in the experiments of Kravchenko and Klapwijk [150]. The inset shows that the middle curve (ns = 7:25 × 1010 cm−2 ) changes by less than ±5% in the entire temperature range.

In the high-density region the resistivity does rise with decreasing temperature logarithmically, consistent with earlier measurements. The consistency of these datasets for two very diKerent types of samples therefore gives strong evidence that these are genuine eKects in both types of systems. The data shown in Figs. 71–73 is for Si-MOSFET samples. The data for GaAs heterostructures, and Si in other geometries is qualitatively similar [120,62,63,210,119,184]. Fig. 74 shows data on high-quality gated GaAs quantum wells with densities on the metallic side of the metal–insulator “transition” taken to temperatures as low as 5 mK. The resistivity is essentially temperature independent at low temperatures. The logarithmic corrections expected from weak-localization (calculated using the measured resistivity and the theoretically expected . ) is also shown.

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Fig. 73. Plot of the resistivity as a function of the scaled temperature T=EF . The encircled region indicates the range of parameters explored in Fig. 72 and in [150]. The dash–dotted vertical line depicts the empirical temperature TQ = 0:007EF below which the logarithmic temperature dependence like that of Fig. 70 sets in. From Prinz et al. [214].

8.6.1. Experiments in a parallel magnetic =eld A magnetic Aeld applied parallel to the plane couples primarily to the spin of the electrons. For small Aelds and for nnc , a positive magnetoresistance proportional to H 2 is observed as expected from perturbative calculations in the interactions. For Aelds such that B H ≈ EF , the electrons are fully polarized and the resistivity saturates as expected. The temperature dependence of the resistivity begins to become insulating-like at low temperatures with the crossover temperature increasing as n decreases [243]. This is an indication that the metallic state becomes unstable as the spins are polarized. A complete set of data is shown in Fig. 75 where resistivity versus temperature in an Si-MOSFET with density varying across nc is shown

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Fig. 74. (a) Temperature dependence of the longitudinal resistivity of a two-dimensional hole gas for various gate biases and associated densities in experiments on GaAs. The solid curves are estimated weak localization predictions; (b) – (f): MagniAed view of the data in (a) averaged over a 5% temperature interval. The estimated weak localization prediction has been shifted to coincide with the data curve at T = 50 mK. From Mills et al. [184].

together with the resistivity versus magnetic Aeld at the lowest temperature for some densities on the n ¿ nc side. It is noteworthy that the temperature dependence of the high-Aeld data (not shown) appears to fall on the curve of resistivity versus temperature (at H = 0) which the high Aeld (low temperature) data saturates asymptotically. The parallel magnetoresistance has been examined carefully for n close to but larger than nc , and is shown in Figs. 76 for p-type GaAs [294]. 50 It is discovered that a critical Aeld 50

Roughly similar results are found in Si-MOSFETS, but a unique crossover Aeld Bc as in Fig. 76 is not found [223].

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Fig. 75. Results for resistivity versus temperature and versus magnetic Aeld applied in the plane for a few densities on either side of nc . The magnetic Aeld is shown on the upper axis and the data is taken at the lowest temperature for some of the densities shown in the resistivity versus temperature plots. From Pudalov et al. [215].

as a function of density Hc (n) exists such that for H ¡ Hc (n) the resistivity continues to be metallic-like dH=dT ¿ 0 and for H ¿ Hc (n) it is insulating like dH=dT ¡ 0. The Aeld Hc (n) tends to zero as n → nc . The low-temperature data on the high Aelds side is puzzling and should be re-examined to ensure that the electron temperature is indeed the indicated temperature.

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Fig. 76. Plot of the magnetoresistance. In (a) the T dependence of H in the zero Aeld metallic phase is shown on a semilog plot for a hole density 3:7 × 1010 cm−2 for varying B values. As B increases from zero, the strength of the metallic behavior measured by the total change in H from about 1 K to 50 mK weakens progressively, and for B ¿ Bc dH=dT becomes negative (i.e., the system becomes insulating). An alternate way of demonstrating the existence of a well-deAned Bc is to plot H against B at several diKerent temperatures. In (b) H is plotted versus B at a hole density 1:5 × 1010 cm−2 . Bc is read oK the crossing point marked by the arrow. In (c), the diKerential resistivity dV=dI measured at 50 mK across the B induced metal–insulator transition is shown at magnetic Aeld strengths similar to those in (a). From Yoon et al. [294].

8.6.2. Experiments in a perpendicular =eld The behavior of a resistance in all but very small perpendicular Aelds, is dominated by the quantum Hall eKect (QHE). The connection of the quantum Hall transitions to the metal– insulator transition at n = nc and H = 0 is an interesting question which we will not touch on. At low Aelds and for n ¿ nc , outside the QHE regime, negative magnetoresistance predicted by weak-localization theory are observed. Data for nnc is shown in Fig. 77 and agrees quite well with the theoretical curves as shown; similar results for n close to nc are also reported [119]. More recent low-temperature data in GaAs heterostructures [184] is reproduced in Fig. 78 for

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Fig. 77. Plot of the magnetoresistance. The change in the resistivity ^H(B) = Hxx (B) − Hxx (0) versus magnetic Aeld B at an electron density of 1:05 × 1012 cm−2 at various temperatures. The open circles denote the measurements and the full line is the best least square At according to the single electron weak localization correction to the conductivity. From Brunthaler et al. [49].

densities n ¿ nc but close to nc . A magnetoresistance two orders of magnitude smaller than the weak localization theory is estimated although the width of the negative magnetoresistance region is not inconsistent with the weak-localization correction. 8.6.3. Compressibility measurements Compressibility (%) measurements [81,129] in the region around n = nc show a rapid change in %−1 from the negative value characteristic of high rs metallic state to positive values—see Fig. 79. These are very important measurements which show that a thermodynamic quantity has a very rapid variation near n ≈ nc . We have already discussed that such changes were predicted [241] to occur through perturbative corrections due to disorder in the energy of interacting electrons. Some recent ingenious measurements [130] of the local electrostatic potential show that in the region n ≈ nc large-scale density Muctuations (puddles) occur with weak connections between them. Such density Muctuations become more numerous with weaker contacts between them as the density is lowered into the insulating phase. These show up in the experiments as local Muctuation in which %−1 approaches 0. Completely isolated puddles (Coulomb dots) of-course must have %−1 = 0. 8.7. Discussion of the experiments in light of the theory of interacting disordered electrons In comparing the experimental results with the theory, it is necessary to separate out the eKects due to “customary-physics”—for instance electron–phonon interactions, creation of ionized impurities with temperature [15], change of screening from its quantum to its classical form as a function of temperature [73] change of single-particle wavefunctions with a magnetic

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Fig. 78. Variation of the longitudinal resistance with perpendicular magnetic Aeld for two-dimensional sample at T = 9 mK and at various indicated densities. The weak-localization correction is estimated to be O(102 ) larger than the observations at these densities. From Mills et al. [184].

Aelds [145], inter-valley scattering [292], etc.—from the singular eKects due to impurities and interactions. The separation is at present a matter of some debate. However, it seems that the following features of the experimental data in relation to the theoretical ideas summarized in Sections 8.3, 8.4 are especially noteworthy. These must be read bearing in mind our earlier discussion that most of the interesting experiments are in a range of rs and disorder where the theoretical problems are unresolved and only hints about the correct form of a theory are available. • At rs 6 O(1) and kF ‘1, a logarithmic increase in resistance with decreasing T consistent

with weak localization as well as with the perturbative interaction correction is observed.

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Fig. 79. Compressibility data. In this experiment, at low frequencies, Ix is directly proportional to Rx , the dissipation of the two-dimensional hole system, while Iy is proportional to the inverse compressibility. Ix and Iy are shown as a function of density for Ave diKerent temperatures ranging from 0.33 to 1:28 K at an excitation frequency of 100 Hz. The crossing point of the Ave dissipation channel curves corresponds to the metal–insulator transition at B = 0. The minimum of the inverse compressibility occurs at the same hole density of 5:5 × 1010 cm−2 . From Dultz et al. [81].

A positive magnetoresistance consistent with the latter is also observed. Also observed is the correction to weak localization due to phase-breaking of backscattering in a perpendicular magnetic Aeld. The latter yields sensible values and temperature dependence for the phase relaxation rate given by the theory. It appears that at high enough density, the weak localization theory supplemented by the perturbative theory of interactions is in excellent agreement with the experiments in the range of temperatures examined. • As rs is increased (and kF ‘ decreased), the logarithmic resistance is lost in the observed temperature range, whereas weak-localization theory predicts that the coePcient of such terms (as well as the onset temperature for their occurrence) should increase. For rs not too large, the decreased logarithmic term may be associated with the perturbative corrections (225) due to interactions. • Upon further increasing rs , the derivative dH=dT becomes positive in the low-temperature region as in a metal. The magnetoresistance in a parallel Aeld is positive ∼ H 2 as is predicted by Finkelstein [although the variation is closer to H 2 =T rather than as (H=T )2 ] [216]. The phase-breaking correction in a perpendicular Aeld continues to be observed. However, quite curiously the deduced . is larger than deduced from resistivity—by deAnition a phase-breaking rate serves as a cutoK only if . ¡ . • In the “metallic” regime for intermediate rs , a strongly temperature dependent contribution for T 6 EF is found which may be Atted to the form H (n) exp(−Ea (n)=T ). The magnitude of this

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term rapidly decreases as the density n decreases. No accepted explanation for this contribution has been given. In Si, the change of resistivity at n ≈ 10nc due to this contribution is an order of magnitude larger than in GaAs. It has been proposed [14,215] that this contribution together with the weak-localization contribution may well account for all the data in the “metallic” regime since it pushes the minimum of the resistivity below which the logarithmic temperature dependence is visible to lower temperature than the available data at lower densities. This issue can be resolved by experiments at lower temperatures. At this point, especially in view of the consistency of the recent results of Mills et al.—see Figs. 74 and 78—with the earlier experiments, one can say that it requires an unlikely conspiracy of contributions to remove the temperature dependence over a wide range for diKerent materials and with diKerent degrees of disorder. A quite diKerent scenario also consistent with the existing data is that the logarithmic upturn in the resistivity observed in high-density samples is a transient that on further decreasing the temperature disappears to be replaced by dH=dT → 0 as T → 0, at least above some characteristic density which is a function of disorder. We will come back to this issue when we discuss the possible phase diagram. • As rs approaches rsc ; dH=dT tends to zero (through positive values). A separatrix is observed with dH=dT ≈ 0 over about two orders of magnitude in temperature for Si and over an order

of magnitude in GaAs. For rs ¿ rsc ; dH=dT is negative beAtting an insulator. rsc appears to be smaller for dirtier samples but not enough systematic data is available for drawing a functional relation. • The electronic compressibility rapidly changes near the “transition” and rapidly becomes small on the insulating side. Its value on the insulating side is consistent with approaching zero in the limit of zero temperature. Although this is in qualitative accord with the theoretical suggestion [241], further experiments simultaneously measuring the compressibility and the conductivity at low temperatures are necessary to correlate the metal–insulator transition with the rapid variation of compressibility or the screening length. Note that it follows from the Einstein relation = D% that if % is Anite in the metallic state ( Anite) and zero in the insulating state ( = 0), % must go to zero at the transition. Otherwise, we would have the absurd conclusion that D → ∞ at the transition. Interesting phenomenological connections between the transport properties and formation of “puddles of electron density” of decreasing size as the metal–insulator “transition” is approached have been drawn [178]. The important question is why such behavior begins to dominate as the density is lowered to nc . Evidence that the formation of “puddles” is a result of disorder strongly augmented by electronic correlations is available in recent measurements [130]. • Near rs = rsc , the resistivity as a function of temperature on the insulating side appears to be a

reMection of that on the metallic side about the dH=dT = 0 line if the data is not considered at low temperatures [149]. Now with more complete data, we know that the resistivity Mattens to zero slope at low temperatures on the “metallic” side of nc . The most likely behavior appears to be that the resistivity approaches a Anite value at low temperature on one side of nc and an inAnite value on the other. A one-parameter scaling ansatz [77] for the problem with interaction and disorder gave H → ∞ for n ¡ nc and H → 0 for n ¿ nc as T → 0 and

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reMection symmetry just as at any second-order transition with one scaling parameter. 51 Does this necessarily imply that multidimensional scaling is required near this transition? However, another important point to bear in mind is that since resistance does not depend on a length scale in two dimensions, it need not be a function, in particular, of the correlation length near the transition. The resistivity is allowed to be Anite on one side of a metal–insulator transition and inAnite on the other even though the transition may be continuous and the correlation length diverges on either side with the same exponent. The glassy nature (Coulomb glass) of the insulating state is also expected to change the critical properties. • The resistivity at low temperatures for nnc has been Atted to an activated form ˙ exp(@=T ) with ≈ 1=2 and with @ → 0 as n → nc . This is characteristic of a Coulomb glass [239]. Whether this is indeed the asymptotic low temperature form is not completely settled. • The resistance at n ≈ nc appears to vary from sample to sample but is within a factor of 3 of the quantum of resistance. It is worth emphasizing that nc is close to the density expected for Wigner crystallization. With Coulomb interactions and disorder, the insulating state is indeed expected to be Wigner glass. In that case, one might expect singular frequency-dependent properties and hysteretic behavior near the transition. • For rs ≈ rsc the resistance in a parallel Aeld is especially noteworthy. In a parallel Aeld dH=dT decreases until at a Aeld H = Hc (n) it changes sign. Hc vanishes at nc , the density where dH=dT = 0 for H = 0. In this regime, H(H; T; n) can be scaled as [294] (n − nc ) (H − Hc (n − nc )) ; : (243) H T TB This means that the transition from the metallic state to the insulating state can be driven by a magnetic Aeld. It appears that the “metallic” state owes its existence to low-energy magnetic Muctuations which are quenched by a magnetic Aeld. This is in line with Finkelstein theory and the Mow diagram of Fig. 69 yet the existence of a scale Hc is not anticipated by the calculations of Finkelstein (nor, of course, is the mere existence of nc ). As H is further increased dH=dT approaches the insulating behavior characteristic of n ¿ nc at H = 0. At a Axed temperature, the resistivity saturates for gB H ¿ EF , i.e., for a fully polarized band. For small perpendicular Aelds, negative magnetoresistance of the form of (225) continues to be observed at least for Si for nnc . In GaAs, this contribution at least in the range nc ¿ n ¿ 2nc is negligible. • The Hall coePcient RH is continuous across the transition, obeying the kinetic theory result RH ∼ 1=n. On the “metallic” side this is not surprising. On the “insulating” side this is

reminiscent of the properties of Wigner glasses [61,109].

8.8. Phase diagram and concluding remarks It is worthwhile to try to guess the T = 0 phase diagram of interacting disordered electrons on the basis of the data and the available theory, inadequate though it is. A convenient set of 51

The data also led to suggestions for a superconducting ground state on the metallic side [207], and to an anyonic state [301]!

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Fig. 80. A tentative phase diagram at T = 0 for two-dimensional disordered electrons with interactions.

axes is rs (or n−1=2 ), as it parameterizes the dimensionless interaction, and the resistance in units of h=e2 as it parametrizes the dimensionless disorder, see Fig. 80. Reliable theoretical results are found only along the two axes of Fig. 80. States are localized all along the horizontal axis. Localized states at T = 0 must be organized into one or another kind of magnetically ordered state. On the vertical axis a Fermi-liquid gives way via a Arst-order transition to a Wigner crystal which may have various magnetic phases. In light of the experiments, the assumption that the entire region in Fig. 80 is an insulator made too long, has to be abandoned in all likelihood. There does appear to be a “metallic state”. With disorder, a crossover to a Wigner glass must occur at large rs . It is also clear that at high densities weak-localization theory supplemented by perturbative corrections due to interactions works quite well in the range of temperatures examined. At moderate rs for small disorder the Finkelstein correction appears to take over and a “metallic” state takes over. The best evidence for this, paradoxically, is the magnetic Aeld (parallel to the plane) dependence of the resistivity which appears to eliminate the “metallic” state. Based on these considerations, the phase diagram Fig. 80 is put forth. It is surmised that the weak localization correction Mows to strong localization for suPciently strong disorder and small enough rs , but that it gives way to a metallic state at weak-disorder and larger rs . What determines the boundary? A possible criterion is that on one side, the Finkelstein renormalization is more important and on the other side localization due to disorder is more important. The crossover to strong localization occurs at a length scale D given by Eq. (219) where the resistivity doubles. The scaling equation for the triplet interaction parameter is [92,51,157] dt =d ln L = kF ‘(1 + 2t )2

(244)

so that for small initial value 0t at L = ‘ one gets t (L) = 0t +(kF ‘) ln(L=). As L = D, the triplet interaction parameter t ≈ 0t + 1. The boundary between the “metallic” and the “insulating”

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regions on this basis is linear at small rs and small g−1 , as shown in Fig. 80. This is highly conjectural but the existence of the phase boundary at the point rs → 0 and 1=g → 0 is more robust. This scenario can be tested in high density, low disorder samples by measurements of resistivity at very low temperatures. If correct in some regime of parameters near the boundary, the logarithmic weak-localization correction should appear at high temperatures and disappear at lower temperatures. We have stressed that the “metallic state” in two-dimensions is likely to be a singular Fermi liquid with an interesting magnetic-ground state. 52 Direct or indirect measurements of the magnetic susceptibility through, for instance the magnetic Aeld dependence of the compressibility should yield very interesting results. Also interesting would be measurements of the single-particle density of states through tunneling measurements. Further systematic and careful measurements of the compressibility are also required to correlate the transition from the “metallic” state to the increase in susceptibility. The basic theoretical and experimental problem remains the characterization of the “metallic state” its low-temperature entropy, magnetic susceptibility, single-particle density of states, etc. The experimental and theoretical problems are many but one hopes not insurmountable. Acknowledgements This article is an outgrowth of lectures delivered in spring 2000 by C.M. Varma during his tenure as Lorentz Professor at the Universiteit Leiden. He wishes to thank the faculty and staK of the Physics department and the deep interest shown by the attendees of the lectures. Special thanks are due to numerous colleagues who provided the experimental data and who explained their ideas and clariAed countless issues. ZN also wishes to take this opportunity to greatly thank his former mentor, S.A. Kivelson, for coaching in one-dimensional physics. Finally, ZN and WvS also wish to express their thanks to Debabrata Panja, Michael Patra, Kees Storm and especially Carlo Beenakker from the Instituut-Lorentz for all their help in generating, scanning, and compressing the numerous Agures. References [1] E. Abrahams, P.W. Anderson, D.C. Licciardo, T.V. Ramakrishnan, Scaling theory of localization: absence of quantum diKusion in two dimensions, Phys. Rev. Lett. 42 (1979) 673. [2] E. Abrahams, C.M. Varma, What angle-resolved photoemission experiments tell about the microscopic theory for high-temperature superconductors, Proc. Nat. Acad. Sci. 97 (2000) 5714. [3] E. Abrahams, S.V. Kravchenko, M.P. Sarachik, Metallic behavior and related phenomena in two dimensions, cond-mat=0006055, Rev. Mod. Phys. 73 (2001) 251. 52

In this connection recent Shubnikov deHaas measurements [276] are of great interest. The results are, however, not uncontroversial, see [217].

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CONTENTS VOLUME 361 J.D. Vergados. The neutrinoless double beta decay from a modern perspective P. Reimann. Brownian motors: noisy transport far from equilibrium

1 57

C.M. Varma, Z. Nussinov, W. van Saarloos. Singular or non-Fermi liquids

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Contents of volume 361

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FORTHCOMING ISSUES* P. Tabeling. Two-dimensional turbulence: a physicist approach C.-I. Um, K.-H. Yeon, T.F. George. The quantum damped harmonic oscillator T. Yamazaki, N. Morita, R. Hayano, E. Widmann, J. Eades. Antiprotonic helium J.K. Basu, M.K. Sanyal. Ordering and growth of Langmuir–Blodgett ﬁlms: X-ray scattering studies G.E. Brown, M. Rho. On the manifestation of chiral symmetry in nuclei and dense nuclear matter S. Nussinov, M.A. Lampert. QCD inequalities C.A.A. de Carvalho, H.M. Nussenzveig. Time delay L.S. Ferreira, G. Cattapan. The role of the D in nuclear physics C. Chandre, H.R. Jauslin. Renormalization-group analysis for the transition to chaos in Hamiltonian systems S. Stenholm. Heuristic ﬁeld theory of Bose–Einstein condensates G. Baur, K. Hencken, D. Trautmann, S. Sadovsky, Yu. Kharlov. Coherent gg and gA interactions in very peripheral collisions at relativistic ion colliders

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