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Physics Reports 394 (2004) 1 – 40 www.elsevier.com/locate/physrep
Magnetic quantum dots and magnetic edge states S.J. Leea; b;∗ , S. Soumab , G. Ihmc , K.J. Changd a Department of Physics, State University of New York, Bualo, NY 14260, USA Quantum-Functional Semiconductor Research Center, Dongguk University, Seoul 100-715, South Korea c Department of Physics, Chungnum National University, Taejon 305-764, South Korea d Department of Physics, Korea Advanced Institute of Science and Technology, Taejon 305-701, South Korea b
Accepted 1 November 2003 editor: A.A. Maradudin
Abstract Starting with de/ning the magnetic edge state in a magnetic quantum dot, which becomes quite popular nowadays conjunction with a possible candidate for a high density memory device or spintronic materials, various magnetic nano-quantum structures are reviewed in detail. We study the magnetic edge states of the two dimensional electron gas in strong perpendicular magnetic /elds. We /nd that magnetic edge states are formed along the boundary of the magnetic dot, which is formed by a nonuniform distribution of magnetic /elds. These magnetic edge states circulate either clockwise or counterclockwise, depending on the number of missing 6ux quanta, and exhibit quite di8erent properties, as compared to the conventional ones which are induced by electrostatic con/nements in the quantum Hall system. We also /nd that a close relation between the quantum mechanical eigenstates and the classical trajectories in the magnetic dot. When a magnetic dot is located inside a quantum wire, the edge-channel scattering mechanism by the magnetic quantum dot is very di8erent from that by electrostatic dots. Here, the magnetic dot is formed by two di8erent magnetic /elds inside and outside the dot. We study the ballistic edge-channel transport and magnetic edge states in this situation. When the inner /eld is parallel to the outer one, the two-terminal conductance is quantized and shows the features of a transmission barrier and a resonator. On the other hand, when the inner /eld is reversed, the conductance is not quantized and all channels can be completely re6ected in some energy ranges. The di8erence between the above two cases results from the distinct magnetic con/nements. We also describe successfully the edge states of magnetic quantum rings and others in detail. c 2003 Elsevier B.V. All rights reserved. PACS: 73.23.Ad; 73.20.Dx Keywords: Two-dimensional electron gases; Magnetic edge states; Magnetic quantum dots; Ballistic transport
∗
Corresponding author. Tel.: +82-2-2260-3953; fax: +82-2-2260-3945. E-mail address:
[email protected] (S.J. Lee).
c 2003 Elsevier B.V. All rights reserved. 0370-1573/$ - see front matter doi:10.1016/j.physrep.2003.11.004
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Contents 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Chronological survey . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3. Magnetic edge states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1. Two di8erent magnetic domains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2. Energy levels of the magnetic quantum dot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3. Correspondence between the quantum mechanical eigenstates and the classical trajectories in the magnetic quantum dot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4. The resonant tunneling through the magnetic quantum dot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4. Quantum wires with magnetic quantum dots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1. Edge state transport . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2. Magnetic edge states of magnetic quantum dots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3. Edge-channel scattering by magnetic quantum dots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4. Conductance and local density of states of quantum wire with two magnetic quantum dots . . . . . . . . . . . . . . 5. Modi/ed magnetic quantum dot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1. Formulation of modi/cation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2. Angular momentum transitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6. Magnetic quantum ring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1. Electronic structures of magnetic quantum ring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2. Modi/ed magnetic quantum ring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7. Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2 4 6 6 6 9 10 12 12 13 15 18 23 23 25 28 28 34 37 38 38
1. Introduction In past decades, advances in semiconductor nano-technology have enabled researchers to fabricate low dimensional nano-scale structures with great control [1–16]. This has made an enormous interest in the study of the mesoscopic systems for novel phenomena, especially those relating to electron transport [17]. The two-dimensional electron gas (2DEG) created in the interface of semiconductor heterostructures such as GaAs=Alx Ga1−x As is one of the important source of the low dimensional systems. Quantum dots, rings, wires and antidots are the typical examples obtained through the additional con/nements on 2DEG. In these systems, the quantum mechanical e8ects such as level quantization due to the con/nement potential and the magnetic /eld, quantum interference, strong electron–electron interaction, as well as single-electron charging e8ects in6uence the electron transport. The Landauer–BLuttiker formalism [18,19] is one of the central tool to understand electron transport in these mesoscopic systems. The conductance of large samples is well known to obey an ohmic scaling low. But as we go to smaller dimension, the conductance does not decrease linearly with the width of the sample, instead, it depends on the number of transverse modes in the conductor. Moreover, there is an interface resistance independent of the length of the sample. In the Landauer–BLuttiker formalism, these features are incorporated. First, we brie6y study the chronological survey. Second, we study the nature of magnetic edge states in magnetic quantum dot, which is formed by spatially nonuniform magnetic /elds on 2DEG.
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Magnetic quantum structures such as magnetic quantum dots have attracted much attention recently. With the application of spatially inhomogeneous magnetic /eld, a number of alternative magnetic structures were proposed on the 2DEG, such as magnetic quantum dots using a scanning tunneling microscope lithographic technique [3], magnetic superlattices by the patterning of ferromagnetic materials integrated by semiconductors [4], type-II superconducting materials deposited on conventional hetero-structures [1], and non-planer 2DEG systems grown by a molecular beam epitaxy [20]. The electron transport features through these magnetic quantum structures are very di8erent from the electrostatic quantum structures, thus, a variety of new phenomena associated with the magnetic structures are expected. Motivated from these features, we calculate exactly the single electron eigenstates and energies of a magnetic quantum dot as a function of magnetic /eld. We propose magnetic edge states, which are the states along the boundary between two di8erent magnetic domains, and /nd two types of magnetic edge states which circulate in opposite directions to each other along the boundary of the magnetic dot. The properties of magnetic edge states largely depend on the missing magnetic 6ux quanta inside the dot and the angular momentum of the states. We also /nd a close relation between the quantum mechanical eigenstates and the classical trajectories in the magnetic quantum dot. Very recently, the e8ects of magnetic edge states on magnetoresistance have been reported experimentally in transverse magnetic steps [21]. Third, we study the ballistic electrons transport and magnetic edge states in quantum wires with a magnetic quantum dot. It is easily expected from classical trajectories that the scattering of conventional edge channels [22] by magnetic quantum structures is quite di8erent from those by electrostatic quantum structures. The study of such a scattering mechanism is important to understand electron transport in magnetic structures and to suggest future device application. The magnetic dot, considered here, is formed by two di8erent magnetic /elds inside and outside the dot. When the inner /eld is parallel to the outer one, we /nd that the two-terminal conductance is quantized like the electrostatic quantum point contact [23] and shows the features of a transmission barrier and a resonator. This feature results from the harmonic potential-like magnetic con/nements and is similar to those of electrostatic dots or antidots. On the other hand, when the inner /eld is reversed and conventional edge states interact with magnetic edge states, the conductance is not quantized and all incident channels can be completely re6ected by the dot in some energy ranges, so that the conductance G oscillates between 0 and 2e2 =h with G = 0 plateaus. In this case, the magnetic con/nements are the types of double wells and merged single wells, which are caused by the /eld reversal at the dot boundary. The di8erence of the conductance and magnetic edge states between the two cases of parallel and reversed /elds results from the distinct magnetic con/nements. The calculation method to evaluate the conductance of quantum wires with magnetic quantum dot is improved and generalized to take into account the presence of two or more magnetic quantum dots, where the use of the region-dependent functional form of the vector potential and the gauge transformation is essential. Applying the proposed calculation method, we found the presence of magnetic edge states extended spatially over two magnetic quantum dots. Fourth, we study the modi/ed magnetic quantum dot. In modi/ed magnetic quantum dots, electrons are magnetically con/ned to the plane where the magnetic /elds inside and outside the dot are di8erent from each other. The energy spectrum exhibits quite di8erent features depending on the directions of the magnetic /elds inside and outside the dot. In particular, the case of opposite directions of the /elds is more interesting than that of the same direction. An electrostatic potential
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is introduced to the system to study the e8ects of an electric con/ning potential on the eigenenergy of a single electron in the modi/ed magnetic quantum dot. The additional potential raises the whole energy spectrum and changes its shape. The ground-state angular momentum transitions occurring in a bare modi/ed magnetic quantum dot disappear on introduction of the additional parabolic potential. Finally, we present the model of a magnetic quantum ring, where electrons are con/ned to a plane, and the magnetic /elds are zero inside the ring and constant elsewhere. The energy states that deviate from the Landau levels are found to form the magnetic edge states along the boundary regions of the magnetic quantum ring. The probability densities of these magnetic edge states are found to be well corresponded to the circulating classical trajectories. In contrast to magnetic or conventional quantum dots, the eigenstates of the magnetic quantum ring show angular momentum transitions in the ground state as the magnetic /eld increases, even without including electron–electron interactions. For a modi/ed magnetic quantum ring with the distribution of nonzero magnetic /elds inside the ring and di8erent /elds outside it, we also /nd similar behaviors such as the angular momentum transitions in the ground state with increasing the magnetic /eld. Our review does not include the conventional magnetic thin /lm, stripes, and nano-particle, e.g. Fe nano-particle on Cu(GaAs) substrate, which can be used for the high density recording media. Some people use the terminology of “magnetic quantum dot (QD)” for this magnetic nano-particle but we do not treat this subject in our paper. We are concentrated in magnetic quantum dot made by the circular dot of two dimensional electron gas (2DEG) due to boundary of the two di8erent region of the inhomogeneous magnetic /eld Bin = B0 (r ¡ r0 ) and Bout = B (r ¿ r0 ). 2. Chronological survey In this section, we survey chronologically the various approaches for the magnetic quantum dot. As previously mentioned, we do not include the magnetic nano-particle for which some people use the terminology of the magnetic quantum dot. In 1990 von-Klitzing group [1] made thin gates of type II superconducting materials on top of the 2DEG in a GaAs/AlGaAs heterostructure to /nd the e8ect of the modulation of an applied magnetic /eld. They found a weak-localization magnetoconductance for small /elds proportional to the magnetic /eld B, in contrast to the B2 homogeneous result. Later on they used similar structure of 2DEG for a 6ux detector of a superconducting /lm [2]. In the same year McCord and Awschalom [3] of IBM found the method of direct deposition of magnetic dots using a scanning tunneling microscope. They used Fe(CO)5 as the source gas for the deposits and gold /lm and SQUID pick up coil as the substrates and got magnetic dots with diameters ranging from 10 to 30 nm and heights from 30 to 100 nm. They estimated that the dots were composed of about 50% iron, the remainder being primarily carbon contamination along with a small amount of oxygen. They claimed that measurements on the particles at low temperatures showed them to be magnetic and reveal macroscopic spin properties. They used the nomenclature of “magnetic dots” or “magnetic particles” instead of “magnetic quantum dots”. After that many di8erent groups made various kinds of structures to study the behavior of electrons in inhomogeneous magnetic /elds [4–6]. Leadbeater et al. [4] studied single crystal /lms of -MnAl grown by MBE on GaAs substrates and showed a large hysteresis loop by the extraordinary
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Hall e8ect. Krishnan et al. [5] reported, in 1992, the structure and properties of the thermodynamically stable -phase Mn1−x Gax single crystal thin /lms grown on GaAs and suggested that this set of materials is a very promising one for magneto-optic recording with the additional potential of integrating semiconductor/magnetic devices. The /rst realization of a spatially modulated periodic magnetic /eld was performed by Carmona et al. [7] by putting superconducting stripes on the surface of the heterostructure with a 2DEG. They observed oscillatory magnetoresistance due to a commensurability e8ect between the classical cyclotron diameter and the period of magnetic modulation. Other realization was through magnetic superlattices or periodic magnetic modulations by the patterning of ferromagnetic materials integrated by semiconductors [8]. Authors of Ref. [7] investigated the e8ect of the parallel ferromagnetic stripes of Dy on 2DEG of AlGaAs/GaAs heterostructure and found that the longitudinal resistance of the 2DEG displays, as a function of the externally applied /eld, the magnetic commensurability oscillations which result from the interplay between the two characteristic length scales of the system, the classical cyclotron radius Rc of the electrons and the period a of the magnetic /eld modulation, 2Rc = ( + 1=4)a, where = 0; 1; : : : is an integer oscillation index. Motivated by these experiments a number of theoretical investigations was followed pursuing the transport properties of these magnetic structures [9–14]. Peeter’s group [9,10,13] studied systems of magnetic quantum steps, barriers, and magnetic wells and found the energy spectrum and the nature of the bound and/or scattered states. They showed the interesting features of inhomogeneous magnetic-/eld can bind the electrons. This is essentially di8erent from elastic potential steps, which always act repulsive. In 1994 Chang and Niu [11] studied the energy spectrum of a 2DEG in a 2D periodic magnetic /eld. Both a square magnetic lattice and a triangular one were considered. They found that a general feature of the band structure was bandwidth oscillation as a function of the Landau index. This theory could be applied to a triangular magnetic lattice on a 2DEG which was realized by the vortex lattice of a superconductor /lm coated on top of a heterojunction. Later on You et al. [12,14] also investigated the transport properties of the nanostructures consisting of magnetic barriers produced by the deposition of ferromagnetic stripes on heterostructures and found that the electron tunneling through multiple-barrier magnetic structures exhibits complicated resonant features. For this study they took two types of magnetic barriers, which were produced by the deposition, on top of a heterostructure, of a ferromagnetic stripe with magnetization (a) perpendicular and (b) parallel to the 2DEG located below the stripe. They used a logarithmic function and a arctangent function for the vector potential, for cases (a) and (b), respectively. In 1997, Nogaret et al. [15] studied similar subject as what von Klitzing’s group did [8], but they put ferromagnets di8erent way and got a periodic magnetic /eld that alternates in sign. They observed a giant low-/eld magnetoresistance due to electrons propagating in open orbits along lines of zero magnetic /eld. The main contribution came from the open orbits among the three types of electron orbits, open, intermediate, and closed. They could explain the observed form and magnitude of the magnetoresistance in a semiclassical model. However, most of the previous investigations overlooked the importance of magnetic edge states which can be formed at the boundary of di8erent magnetic domains in close analogy with conventional edge states formed by the electrostatic con/nements. Some of present authors [16] presented a model of a magnetic quantum dot and explained the magnetoresistance in terms of the magnetic edge state which will be explained in detail in next section.
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3. Magnetic edge states 3.1. Two dierent magnetic domains For the two-dimensional electron gas (2DEG) applied by inhomogeneous magnetic /elds, which provide two di8erent magnetic domains, as shown schematically in Fig. 3.1, the current-carrying states (hereafter referred to the magnetic edge states in close analogy with electrostatically induced conventional ones) exist near the boundary between the two domains [24]. These magnetic edge states have quite di8erent properties from the conventional ones, thus, a variety of new phenomena associated with the magnetic structures are expected in the electron transport. However, to our knowledge, only a little attention has been paid to this problem [25]. In this chapter, we investigate the nature of magnetic edge states in a magnetic quantum dot which is formed by inhomogeneous magnetic /elds; electrons are apparently con/ned to a plane and within that plane the magnetic /eld is zero within a circular disc and constant B outside it [26]. We calculate exactly the single electron eigenstates and energies of a magnetic quantum dot as a function of magnetic /eld, using a single scaled parameter s=r02 B=0 , which represents the number of missing magnetic 6ux quanta within the dot, where r0 is the radius of the quantum dot and 0 (=h=e) is the 6ux quantum. We /nd two types of edge states which circulate in opposite directions to each other along the boundary of the magnetic dot and exhibit quite di8erent energy dependences on angular momentum. We /nd a close relation between the quantum mechanical eigenstates and the classical trajectories in the magnetic quantum dot; the quantum mechanical eigenstate corresponds to a certain ensemble average of the classical motions which consist of straight line paths in the dot region and cyclotron orbits with a quantized radius in the outside region. These radius and central positions of the cyclotron orbits critically depend on the value of s. For a narrow two-dimensional conductor with a magnetic quantum dot at the center, the calculated magnetoconductances show aperiodic oscillations instead of the Aharonov–Bohm type of periodic oscillations [27], and this behavior is attributed to the characteristics of the magnetic edge states, which is absent in the conventional ones. 3.2. Energy levels of the magnetic quantum dot The single particle SchrLodinger equation for a two-dimensional magnetic quantum dot is (˜ p+ 2 ∗ ∗ ˜ eA) =(2m ) (˜r) = E (˜r), where m is the e8ective mass of electron and e is the absolute value of
Fig. 3.1. Schematic diagram of classical trajectories of electrons for the magnetic edge states on the magnetic domain boundary.
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the electron charge. In polar coordinates (r; ) on the plane, the vector potential ˜A can be chosen as 0 for r ¡ r0 and (r 2 − r02 )B=(2r)ˆ for r ¿ r0 , so that B = 0 for r ¡ r0 and nonzero Bzˆ otherwise. The wave functions and the energies are easily determined by the continuity of the wave functions and their derivatives at the boundary of the dot. Since the wave functions are separable, i.e., nm (˜r) = Rnm (r)eim ; where m is the angular momentum quantum number and n (=0; 1; 2; : : :) is the radial quantum number (the number of nodes in the radial wave function), the equation for the radial part is written as 2 m2 d 1 d − 2 + 2E Rnm (r) = 0 (r ¡ r0 ) ; + (3.1) dr 2 r dr r 2 (m − s)2 d 1 d 2 − + − r + 2[E − (m − s)] Rnm (r) = 0 (r ¿ r0 ) : (3.2) dr 2 r dr r2 √ 2 Here Rnm (r) = C1 J|m| ( 2Er) for r ¡ r0 and Rnm (r) = C2 r |m−s| e−r =2 U (a; b; r 2 ) for r ¿ r0 . It is convenient to express all quantities in dimensionless units by letting ˝!L [ =√ ˝eB=(2m∗ )] and the inverse 2 ∗ 2 ∗ length $ = m !L =˝ be 1. Then, since ˝ =m = ˝!L =$ → 1 and r0 → s, s = Br02 e=h is only the relevant parameter. The function Jm is the Bessel function of the order m, and U is the con6uent hypergeometric function with a = −(E − me8 − |me8 | − 1)=2, b = |me8 | + 1, and me8 = m − s. It is noted that Eq. (3.2) has the same form as that of the uniform magnetic /eld case, except that the angular momentum m is replaced by the e8ective angular momentum me8 . The meaning of this replacement of m to me8 will be discussed in the next section. In the magnetic quantum dot, the Landau level degeneracy is lifted for the states near the dot. From Eqs. (3.1) and (3.2), if the e8ective potential Ve8 (r) is de/ned as 2 m (r ¡ r0 ) ; 2 2r (3.3) Ve8 (r) = 2 2 me8 + r + me8 (r ¿ r0 ) ; 2r 2 2 √ the minimum of Ve8 (r) always occurs at r = r0 (= s) for the states with |me8 | ¡ s, i.e., 0 ¡ m ¡ 2s, which correspond to the magnetic edge states circulating counterclockwise, as we will see below. The m = 0 state is widely distributed over the dot due to the lack of the centrifugal force, and the minimum of Ve8 (r) for the states with |me8 | ¿ s, i.e., m ¡ 0 or m ¿ 2s, is located at r = |me8 | outside the quantum dot, similar to the case of uniform magnetic /elds. The states with m ¡ 0, which exist near the dot, give rise to the magnetic edge states circulating clockwise. Fig. 3.2 shows the energy levels of the magnetic quantum dot for di8erent values of m at s = 5, U for magnetic /elds of teslas. The lowest energy state occurs the radius of which is about 500 A at m = 0 and the degeneracy of the Landau levels are removed, as shown in Fig. 3.2. This result indicates that the inhomogeneity of magnetic /elds mostly perturbs the states near the boundary of the quantum dot, and this perturbation is caused by the missing of s 6ux quanta. From the wave functions, the probability current density Jnm carried by the state nm can be calculated as
˝˜ 1 ∗ ∇ + e˜A Jnm = ∗ Re nm ; (3.4) nm m i
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Fig. 3.2. Dependence of the energy eigenvalues Enm on the angular momentum m for s = 5. Dashed lines represent the bulk Landau levels.
Fig. 3.3. Dependence of the probability current Inm (in units of !L ) on the angular momentum m for s = 5 and n = 0 case.
and the probability current Inm of the state nm is related to the derivative of Enm with respect to m as follows [9]: ∞ 1 9Enm Inm = : (3.5) Jnm dr = h 9m 0 In Fig. 3.3, the probability current Inm for s=5 and n=0 is drawn as a function of m. The probability currents for the perturbed states are found to have nonzero, resulting in the magnetic edge states. For m ¿ 0, Inm have positive values for counterclockwise circulations whereas for m ¡ 0, Inm have negative values for clockwise ones. In Fig. 3.4, the energy levels are plotted as a function of magnetic /eld s for di8erent values of (n; m), with the energy ˝!L set to one at s = 5 and the radius r0 /xed. As the magnetic /eld
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Fig. 3.4. Energy spectra as a function of s. The energy unit of ˝!L = 1 at s = 5 is used. Dotted lines represent the Landau levels.
increases, the deviations of energies from the bulk Landau levels become signi/cant, which lead to the magnetic edge states near the boundary of the quantum dot. In the limit of B → ∞, we /nd that the energies approach to those for the conventional circular dot which is electrostatically con/ned by hard walls without magnetic /elds. 3.3. Correspondence between the quantum mechanical eigenstates and the classical trajectories in the magnetic quantum dot To see the signi/cance of me8 , let us consider momentarily the 2DEG in a uniform magnetic /eld, in which the eigenstates are described by the degenerate Landau levels, Ei = ˝!c (i + 1=2), where !c = eB=m∗ . When the symmetric gauge is chosen, n and m remain good quantum numbers and the probability density of the eigenstate (n = 0; m) has a maximum at r = |m| in dimensionless units. In this case, the quantum mechanical eigenstate (n; m) with the eigenvalue Enm corresponds to the ensemble average of the classical cyclotron motions [28] with the radius ri and its center located at rj from the origin, which satisfy the following relations from the conservations of energy and angular momentum;
Enm 2n + |m| + m + 1 ri = = ; rj = ri2 − m : (3.6) 2 2 However, because of the uncertainty principle, the central position of the cyclotron orbit cannot be determined quantum mechanically. From the analysis used for the uniform /eld case, we can also show that the (n; m) state exactly corresponds to the ensemble average of the classical motions which consist of the straight line paths in the dot region and the cyclotron orbits with the radius ri and the center located at rj outside the dot. These straight lines and cyclotron orbits intersect each other at the dot boundary. In this
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case, the relations between the (n; m) states and the corresponding ri and rj values are determined from the conservations of energy and angular momentum for the magnetic quantum dot and are written as
Enm ; rj = ri2 − me8 : (3.7) ri = 2 Eq. (3.7) has the same form as Eq. (3.6) except that Enm calculated from Eqs. (3.1) and (3.2) is lifted from the bulk Landau level in Eq. (3.6) and m is replaced by me8 due to the inhomogeneity of magnetic /elds. The replacement of me8 from m is due to the missing magnetic 6ux quanta s. For example, the (0; m ¡ 0) state locates at rj ∼ |m| in the uniform /eld case and encloses |m| magnetic 6ux quanta. When the s 6ux quanta are missed inside the dot, this state moves outside to keep enclosing |m| magnetic 6ux quanta and locates at rj ∼ |m| + s in result. The classical trajectories for the (0; 0), (0; −1), and (0; 1) states are drawn in Fig. 3.5, showing a clear correspondence between the quantum eigenstates and the classical motions; the probability densities |Rnm (r)|2 and the directions of the probability currents Inm correspond to the classical motions. The classical trajectory corresponding to the (0; 0) state carries no current because it always passes through the origin, and the classical motions of the (0; −1) and (0; 1) states correspond to the probability currents of the states in the clockwise and counterclockwise directions, respectively. We /nd that our correspondence analysis may answer to the important question whether the classical motions corresponding to the quantum eigenstates are periodic or not. In the magnetic quantum dot, periodic motions occur if the angle ) made by two lines connected from the origin to the centers of two successive orbits [see Fig. 3.5(c)] is 2p=q, where p and q are integers. From a simple geometrical argument, ) is found to satisfy the relation cos()=2) = (ri2 + rj2 − r02 )=(2ri rj ). However, at this moment, it is diVcult to make a de/nite answer because of the numerical errors for evaluating Enm . 3.4. The resonant tunneling through the magnetic quantum dot In this section, we present a phenomenological discussion of the tunneling of an electron through the magnetic quantum dot placed in the quantum wire (narrow two-dimensional conductor). More accurate numerical calculations of such structure will be presented in the next section. We consider a narrow two-dimensional conductor with a magnetic quantum dot at the center. Under the applied strong magnetic /elds which give the quantum Hall plateaus, the transport along the boundary of the sample, which is usually promoted by conventional edge states, can be backscattered by the resonant tunneling into the magnetic edge states along the boundary of the dot, because of the impurity e8ect in the narrow region between two boundaries. In usual quantum dots or ring structures, the resonant tunneling e8ect in magnetoresistance measurements gives rise to the Aharonov–Bohm oscillations [27,29], which are periodic with magnetic /eld. In the magnetic quantum dot considered here, we do not see such periodic oscillations. We calculate the two-terminal conductance, which is the inverse of the sum of magnetoresistance and Hall resistance, taking into account the resonant backscattering via the magnetic edge channels as follows; 2e2 +2 G(B) = 1− ; (3.8) h (EF − Enm (B))2 + +2 n; m
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Fig. 3.5. Classical trajectories of electrons and corresponding probability densities for the eigenstates (a) (0; 0), (b) (0; −1), and (c) (0; 1).
where + is the elastic resonance width and a constant value of + = 0:005 is used for simplicity. The calculated conductance is plotted as a function of magnetic /eld in Fig. 3.6, with the Fermi energy of EF = 2 in units of Fig. 3.4. In this case, the magnetic /elds represented by s are in the , = 2 quantum Hall plateau region, where , is the Landau level /lling factor. We /nd that the oscillations are not periodic, in contrast to the Aharonov–Bohm type of oscillations. The /rst dip in the conductance around s = 3:7 is due to the resonant backscattering via the (1; −3) magnetic edge state. The other dips are found to be associated with the (0; 3), (1; 1), (1; −2), and (1; −1) states in the increasing order of s. In the narrow ring structure of Jain [27], the intervals between the dips were shown to be periodic, which indicates the subsequent change of one 6ux quantum passing through the inner boundary. In our magnetic dot structure, the resonances occur via the two di8erent magnetic edge states circulating in di8erent directions, depending on the sign of m. Since there is
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Fig. 3.6. Magnetoconductance as a function of s.
no magnetic /eld inside the magnetic dot, the magnetic edge states may not enclose the magnetic 6ux, resulting in the missing of 6ux quanta, which is absent in the edge states formed by electrostatic con/nements. 4. Quantum wires with magnetic quantum dots 4.1. Edge state transport Transport properties of two-dimensional electron gas (2DEG) in spatially nonuniform magnetic /elds have also attracted much attention recently. As a counterpart of electrostatic structures, various magnetic structures have been realized experimentally [3,30,31,21], patterning of ferromagnetic or superconducting materials on 2DEG or using nonplanar 2DEG. Theoretically, it was shown that nonuniform magnetic /elds can cause electron drifts [25,32], transmission barriers [33], commensurability e8ects [34], and electron con/nements [35,36]. Magnetic edge states, which exist along the boundary between two di8erent magnetic domains, were also proposed [35,36] in the same analogy with the conventional edge states [22,19] in quantum Hall systems. Recently, the e8ects of them on magnetoresistance were reported in transverse magnetic steps [21]. In the edge state transport regime, the conductance of quantum wires with a local electrostatic modulation is quantized except for resonant re6ections [37] and exhibits Aharonov–Bohm oscillations [38]. These interesting features can be modi/ed when such a modulation is replaced by a magnetic one such as a magnetic quantum dot (or magnetic antidot) [35,36] which is formed in 2DEG by nonuniform perpendicular magnetic /elds; ˜B = B∗ zˆ within a circular disk with radius r0 , while ˜B = B0 zˆ outside it. Classical electron trajectories (see Fig. 4.1) scattered by a magnetic dot with -[ = B∗ =B0 ] ¡ 0 are very di8erent from those for - ¿ 0 and those by an electrostatic dot (or antidot). This indicates that the edge-channel scattering by local magnetic modulations can be quite di8erent from that by electrostatic ones. The study of such a scattering mechanism is important to understand electron transport in magnetic structures and to suggest future device application. However, to our knowledge, little attention has been paid to it [33]. In this chapter, we study the ballistic transport of conventional edge channels through quantum wires with a magnetic quantum dot [35]. The magnetic edge states near the dot and two-terminal
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Fig. 4.1. Schematic diagram of a quantum wire with a magnetic quantum dot. The solid (dotted) arrows represent classical electron trajectories for - ¿ 0 (- ¡ 0), where - = B∗ =B0 .
conductance G(EF ) of the wires in the limit of zero bias are found to exhibit distinct features between two cases of - ¿ 0 and - ¡ 0, where EF is the Fermi energy. For - ¿ 0, G(EF ) is quantized except for resonances and shows the behavior of a transmission barrier and a resonator, depending on the value of -, when the magnetic length inside the dot is smaller than r0 . This feature results from the harmonic-potential-like magnetic con/nements and is similar to those of electrostatic dots (or antidots). On the other hand, for - ¡ 0, G(EF ) is not quantized when incident edge channels are scattered by the dot. Moreover, for - ¡ − 1, incident channels can be completely re6ected by the dot in some ranges of EF , resulting in the plateaus of G(EF ) = 0. This interesting feature is due to the double-well and merged-well magnetic con/nements caused by the /eld reversal at the dot boundary. We also propose a calculation method for conductance, based on the Green’s function along with the lattice-Hamiltonian and the symmetric gauge. In the /nal section of this chapter, the method of conductance calculation based on the Green’s function is generalized to include many magnetic quantum dots. 4.2. Magnetic edge states of magnetic quantum dots As shown in Fig. 4.1, the dot is assumed to be located at the center of the wire. For simplicity, the wire potential along the transverse (y) ˆ direction is assumed to be an in/nite square well with width Ly . The magnitudes of the magnetic /eld determine the characteristic length andthe energy scales; the magnetic energy inside (outside) the dot is lB∗ (j) = (2j + 1)˝=(e|B∗ |) length and the Landau (lB0 (j)= (2j + 1)˝=(eB0 )) and E ∗ (j)=(j+1=2)˝e|B∗ |=m∗ (E0 (j)=(j+1=2)˝eB0 =m∗ ), respectively, where j = 0; 1; 2; : : : and m∗ is the e8ective mass. We focus on the edge state transport regime [i.e., Ly lB0 (N − 1)] and ignore the e8ects of spin and disorder, where N is the number of Landau levels below EF far away from the dot. Note that in previous studies [16] for magnetic dots, B∗ is /xed to be zero. We /rst consider the case 1 ¿ lB0 (N − 1), where 1 is the distance between the dot and wire edge. In this case, current-carrying conventional edge channels do not interact with the dot, thus, G(EF ) = NG0 (including the spin degeneracy) except for backward scatterings of channels by the resonant tunneling into the magnetic edge states of the dot, where G0 =2e2 =h. To study these magnetic edge states, one can neglect the wire con/nement.
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For a magnetic dot in an in/nite 2DEG, the SchrLodinger’s equation is given by [(˜ p +e˜A)2 =2m∗ ] = ˜ E where the vector potential A can be chosen in the form of the symmetric gauge: 1 (r ¡ r0 ) ; -B0 r ˜A = ˆ 2 (4.1) 1 r 2 B (- − 1) + 1 B r (r ¿ r ) : 0 0 0 2r 0 2 Here we used the polar coordinates (r; ) and assumed that the dot center is located at r = 0. Then the eigenstates can be written as nm (˜r) = Rnm (r)eim , where m is the angular momentum quantum number and n(=0; 1; 2; : : :) is the number of nodes in R(r). The states can be classi/ed by their radial locations. A (n; m ¡ 0) state located far away from the dot interacts with B0 . From the gauge invariance [39], its radial wave function is found to be the same as that of the (n; me8 ) state in the uniform /eld B0 . Here, me8 = m − s and s[ = (1 − -)r 2 B0 =0 ] is the number of removed magnetic 6ux quanta (or additional ones for s ¡ 0) to form the magnetic dot in 2DEG where uniform B0 is already applied. Then, nm¡0 is located at rp (me8 ; B0 ), encloses |m| 6ux quanta, and its energy Enm is E0 (n), where rp (m; B) = 2|m|h=(eB). On the other hand, nm ’s near the dot interact with both of B0 and B∗ , thus, Enm ’s deviate from E0 (n). They are magnetic edge states, carry nonzero probability current Inm (˙ 9Enm =9m), and result in resonances when they interact with conventional edge channels. When r0 lB∗ (n − 1) and |m| is small, nm ’s are located at rp (m; B∗ ) inside the dot and Enm = E ∗ (n). Interestingly, for - ¡ 0, nm ’s with small m ¡ 0 can be located also at rp (me8 ; B0 ) outside the dot. √ The above features are clearly shown in Fig. 4.2. In dimensionless units of E0 (0) → 1 and 2lB (0) → 1, Enm is calculated from the radial part of the single particle SchrLodinger equation, 2 d 1 d + (4.2) + 2(Enm − Ve8 (r)) Rnm (r) = 0 ; dr 2 r dr where
1 m 2 (r ¡ r0 ) ; + -r 2 r Ve8 (r) = (4.3) 1 me8 + r 2 (r ¿ r ) : 0 2 r The magnetic con/nement can be de/ned as the e8ective potential Ve8 and becomes the harmonic potential in uniform /elds (i.e., Ve8 with - = 1). For - ¿ 0, magnetic con/nements are similar to the harmonic potential. Thus, Enm ’s vary monotonously from E0 (j) at large |m| [i.e., rp (me8 ; B0 )r0 ] to E ∗ (j) at small |m| [i.e., rp (m; B∗ )r0 ], where j = n + (m + |m|)=2. For - ¿ 1, magnetic edge states circulate counterclockwise around the dot, while either clockwise or counterclockwise for 0 ¡ - ¡ 1. These magnetic edge states are similar to edge states around electrostatic dots or antidots. For - ¡ 0, magnetic con/nements are very di8erent from the harmonic potential. For |m| ¡ |-|s0 , Ve8 is a double-well potential, where s0 = r02 B0 =0 . The barrier in this potential is enough high to con/ne nm only in one of the wells, if r0 lB∗ . Then, for small m ¡ 0, the inner well allows energies E ∗ (j1 ) with j1 = |m|; |m| + 1; |m| + 2; : : : ; while the outer well E0 (j2 ) with j2 = 0; 1; 2; : : : Thus, nm with small m ¡ 0 can be located either inside or outside the dot, depending on n, as discussed before. This feature results in abrupt changes of Enm ’s from E0 to E ∗ [see Fig. 4.2(d)].
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Fig. 4.2. (a) – (d) Enm ’s and (e) Ve8 (r; m)’s for s0 (=r02 B0 =0 ) = 5 and some -’s. In (e), m = −1 (solid), 4 (dashed), 15 (dotted) are chosen. The energy unit is E0 (0).
Note that the abrupt change appears only in the n = 0 level in Fig. 4.2(c), since lB∗ (0) ≈ r0 . For |-|s0 6 m 6 s + s0 , the two wells in Ve8 merge into a single well [see dotted line in Fig. 4.2(e)], which minimum occurs at r0 . Magnetic edge states in this merged well circulate counterclockwise along r = r0 with snake-like classical motions. 4.3. Edge-channel scattering by magnetic quantum dots Next, we study the scattering of incident conventional edge channels by the magnetic dot when WlB0 (N − 1). We calculate a transmission probability T ()) [or G()) = T ())G0 ] of incident channels with dimensionless energy )[ = EF =(2E0 (0))] in a quantum mechanical way based on the lattice Green’s function [38], where a continuous 2DEG is approximated by a tight-binding square lattice with lattice constant a. The vector potential is included as the Peierls’ phase factor [exp(−ie=h l ˜A · d˜l)] in hopping matrix element. The symmetric gauge is essential to study the scattering by the magnetic dot [40], however, to our knowledge, it has never been used so far to calculate T in a quantum mechanical way. The behavior in T ()) can be classi/ed by - (see Fig. 4.3). For - ¿ 0, T ()) is quantized when lB∗ ¡ r0 . In this case, magnetic con/nements are similar to the harmonic potential. Thus, when edge channels pass the constriction between the dot and wire edge, they are still well con/ned near edges and do not interact with those in the opposite edge, resulting in the quantization of T ()). For - ¿ 1, T ()) is smaller than that of the uniform-/eld case (-=1). This feature results from that some of incident edge channels are re6ected by the dot due to the large magnetic energy E ∗ (¿ E0 ). Thus, the magnetic dot with - ¿ 1 is similar to an electrostatic antidot [38]. A transition energy Et (j), where T ()) changes from j − 1 to j, is approximately determined by min {E ∗ (j); (˝j)2 =(2m2 12 )}.
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Fig. 4.3. Dependence of T on ) for some -’s and r0 ’s. For all cases, Ly and lB (0) are /xed as 35a and 5a.
As 1 decreases, Et (j) approaches to E ∗ (j) and the number of resonances decreases [see Figs. 4.3(a) and (c)], because magnetic edge states are con/ned in a narrower region. For 0 ¡ - ¡ 1, T ()) is the same as that for - = 1 except for resonant dips. In this case, the small magnetic energy E ∗ (¡ E0 ) does not re6ect any incident edge channels and binds electrons like as electrostatic dots, thus, the magnetic dot behaves as a resonator. The number of resonances increases as - decreases from 1 and r0 increases. The features of the magnetic dot with - ¡ 0 are very di8erent from those for - ¿ 0 and those by electrostatic dot or antidots. For −1 ¡ - ¡ 0, T ()) is not quantized and smaller than that of the uniform-/eld case, although E ∗ ¡ E0 , in contrast to the case of 0 ¡ - ¡ 1. For - ¡ − 1, T ()) is not quantized. Moreover, when 1 ≈ lB (0), incident edge channels are completely re6ected, except for resonances, in some ranges of ), so that G()) oscillates between 0 and G0 with G = 0 plateaus. It contrasts with the barrier with same 1 for - ¿ 1. The features for - ¡ 0 result from the double-well and merged-well magnetic con/nements, which are caused by the /eld reversal. To understand these features, we imitate the region near the dot as a magnetic step [32], which is con/ned by an in/nite square well U (y) with width Ly and is divided into three strips by di8erent magnetic /elds; in the middle strip (|y| ¡ r0 ), B = B∗ , while in the upper (y ¿ r0 ) and lower ones (y ¡ − r0 ), B = B0 . Eigenstates can be written as eikx Yk (y) and Ve8 (y; k) is de/ned in a similar way to that for the dot; Ve8 (y; k) = ˝2 {k + F(y)=l2B0 }2 =(2m∗ ) + U (y), where F(y) is y + r0 (- − 1) for y ¿ r0 , -y for |y| ¡ r0 , and y − r0 (- − 1) for y ¡ − r0 . In Fig. 4.4,
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Fig. 4.4. (a) – (b) Ve8 (y; k)’s and (c) – (d) E(k)’s for magnetic steps. The energy unit is 2E0 (0) while the length unit is arbitrary. For all cases, lB (0) = 2:47 and r0 = 9.
Ve8 (y)’s and the calculated energy levels E(k ¿ 0)’s are shown. Note that E(k ¡ 0) is the same as E(|k|). The states near y = ±Ly =2 correspond to the current-carrying states near the magnetic dot, while those near y = ±r0 to the magnetic edge states circulating around the dot. And, the triple wells [solid and dashed lines in Fig. 4.4(a) – (b)] correspond to the double-well magnetic con/nements of the magnetic dot, while the double wells (dotted lines) to the merged-well magnetic con/nements. For −1 ¡ - ¡ 0, as 1(=Ly =2 − r0 ) decreases, the edge states near y = −Ly =2 are determined by Ve8 with smaller k ¿ 0, which has a smaller barrier at y = −r0 , due to the wire con/nement [see Fig. 4.4(a)]. When 1 ≈ lB0 , the barrier is so small that the states near y = −Ly =2 can be extended to the center or upper strip. Then, the states in the lower strip can easily interact with those in the upper one. The same behavior arises in the case of the magnetic dot: When 1 ≈ lB0 , conventional edge channels can interact with the doublewell magnetic con/nement with small barrier, so that they are extended in the transverse direction. Then, the left-going channels easily interact with the right-going ones, thus, the conductance is not quantized. This behavior well corresponds to the classical trajectories in Fig. 4.1. For - ¡ − 1 and k ¿ 0, when lB0 1 ¡ |-|r0 , states con/ned in the local minimum at y = Ly =2 of triple wells are the conventional edge states. Their energies are much larger than E0 at k = 0 and meet the relation dE=d k ¡ 0. As 1 decreases, the well near y = Ly =2 becomes narrower, so that the conventional edge states have larger energies and begin to be mixed with the magnetic edge states near r = r0 , resulting in the level splitting. The number of the pure conventional edge channels near y = Ly =2 is, ∼ M , where M is the largest number satisfying 2lB (M − 1) ¡ 1. In Fig. 4.4(c), two energy levels of the pure conventional edge channels are shown; note that the levels of channels near y = −Ly =2 do not appear because of their very large energies. Thus, when 1 ≈ lB (0), there exist no pure conventional edge states. In this case, eigenstates are classi/ed into those with dE=d k = 0 inside the middle strip, those with dE=d k ¿ 0 caused by the merged wells at y = r0 , and those with dE=d k ¡ 0 which result from the triple wells with small barrier at r0 [see Fig. 4.4(d)]. The third states are the mixed ones of the magnetic and conventional edge states and their energies are smaller
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than E ∗ ’s. Then, in some energy ranges above E ∗ ’s, no states with dE=d k ¡ 0 are allowed in the upper strip. This feature indicates that all conventional edge channels cannot pass the constriction between the magnetic dot and wire edge, thus, G(EF ) = 0 in some ranges above E ∗ ’s. The plateaus of G(EF ) = 0 appear in longer energy ranges for larger |-|, smaller EF , and smaller 1. The resonant peaks in the ranges of T ()) = 0 in Fig. 4.3(b) result from the snake magnetic edge states in the merged-well magnetic con/nement. Finally, the shapes of magnetic con/nements for - ¡ 0 (- ¿ 0) are still the double wells or merged wells (the harmonic-like potentials) in realistic situations, where magnetic /elds slowly vary near the boundary of a magnetic dot or a step, thus, our /ndings can be observed experimentally. We propose that the constriction between the magnetic dot and wire edge can be considered as a magnetic quantum point contact. The conductance in this geometry with - ¿ 1 is similar to that in electrostatic quantum point contacts [23], while it can be very di8erent for - ¡ 1, showing a switching behavior with the plateaus of G(EF ) = 0. 4.4. Conductance and local density of states of quantum wire with two magnetic quantum dots In this section, we present the theoretical formulation to calculate the conductance of quantum wires in which many magnetic quantum dots are placed in series [41]. The proposed calculation method is applied to calculate the conductance and the local density of states (LDOS) of quantum wire in which two magnetic quantum dots are placed in series. In order to calculate the conductance of such systems, the method of calculation given in the last section (single magnetic dot problem) has to be modi/ed so as to allow us to treat more complicated distribution of the magnetic /elds. To be speci/c, we consider a quantum wire with width Lz (narrow two-dimensional system) de/ned by the con/nement potential Uconf (y) = 0 for |y| 6 Ly =2 and ∞ otherwise. The presence of the magnetic quantum dots is de/ned by the following magnetic /eld pro/le: ˜ r) = B(˜
N
Bi (Ri −
(x − xi )2 + y2 )zˆ ;
(4.4)
i=1
where (x) is the step function, zˆ the unit vector along the z-direction, ˜r i = (xi ; 0) and Ri are the center position and the radius of the each ith magnetic quantum dot, respectively. Here we restrict our attention to the case xi+1 − xi ¿ Ri + Ri+1 (i = 1; 2; : : : ; N − 1), so that those magnetic dots are separated with each other and are arranged in the form of an array. In the presence of an applied perpendicular (static) magnetic /eld ˜B0 = B0 z, ˆ the total magnetic /eld felt by electrons is given by ˜ r) ; ˜B(˜r) = ˜B0 + B(˜
(4.5)
which can be rewritten as ˜B(˜r) = Bi∗ z(≡ ˆ (B0 + Bi )z) ˆ if ˜r is in the ith dot region while ˜B(˜r) = B0 if ˜r is outside dot regions. In order to attack the quantum mechanical scattering problem of electrons in such system, it is convenient to model the considered system by a square lattice with lattice spacing a in the x–y plane. Let us consider M lattice sites along the y-direction, so that the width of the quantum wire is given by Ly = (M + 1)a. The lattice coordinates in the x- and the y-directions are speci/ed by the lattice indices l and m, respectively: (la; (m − (M + 1)=2)a). Assuming that the functional form of the vector potential ˜A(˜r) which satis/es ˜B(˜r) = ∇ × ˜A(˜r) is known, the total Hamiltonian can be
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written as H=
∞ M
[4t|l; ml; m|
l=−∞ m=1
− t{Px (l; m)|l + 1; ml; m| + Py (l; m)|l; m + 1l; m| + h:c:}] 2
(4.6)
2
Here, t ≡ ˝ =2ma is the hopping integral between the nearest-neighbor sites within the 2D plane, with m∗ being the e8ective mass of an electron. In Eq. (4.6), the position dependent perpendicular magnetic /eld ˜B(˜r) has been implemented in the Peierls phase factor ie Ax ((˜r l; m + ˜r l+1; m )=2)a ; (4.7) Px (l; m) = exp h ie Ay ((˜r l; m + ˜r l; m+1 )=2)a : Py (l; m) = exp (4.8) h However, the complicated form of the magnetic /eld pro/le given in Eq. (4.5) makes it diVcult to /nd the functional form of ˜A(˜r). Therefore, we make use of di8erent vector potentials depending on regions, along with appropriate gauge transformations for wavefunctions to take into account the uni/cation of the gauge. As seen in Eq. (4.1), the vector potential in a region including only the ith magnetic quantum dot (i = 1; 2; : : : ; N ) can be chosen in the form of symmetric gauge: - i B0 (r(i) ¡ Ri ) ; 2 (−y; x(i); 0) (i) ˜A (˜r) = (4.9) B0 R2i (-i − 1) + 1 (−y; x(i); 0) (r(i) ¿ Ri ) ; 2 r 2 (i) where -i ≡ (B0 + Bi )=B0 . Here we have introduced a new coordinate system: (x(i); y) ≡ (x − xi ; y) which has the origin at the center of the each ith magnetic quantum dot, and r(i) ≡ (x2 (i) + y2 )1=2 the distance from the ith origin. The each vector potential ˜A(i) (˜r) gives rise to the magnetic /eld ˜B(˜r) = (B0 + Bi )zˆ for r(i) ¡ Ri and ˜B(˜r) = B0 zˆ otherwise, as expected. On the other hands, in the left (i = 0) and the right (i = N + 1) lead regions where magnetic quantum dot does not exist, it is natural to employ the Landau gauge for vector potential: ˜A(0) (˜r) = ˜A(N +1) (˜r) = B0 (−y; 0; 0) :
(4.10)
Here we note that the boundary between the region where we use ˜A(i) and the region where we use ˜A(i+1) is appropriate, as long as the magnetic /eld at that boundary is ˜B0 . Suppose that the magnetic /eld in the region including lattice columns l − 1, l, and l + 1 can be described by the vector potential ˜A(i) . Then the SchrLodinger equation (Hˆ − E Iˆ)|: = 0 can be reduced to the equation ˜ (i) ˜ (i) ˆ (i)∗ ˆ (i) ˜ (i) (E Iˆ − hˆ(i) l )C l + t P x (l − 1)C l−1 + t P x (l)C l+1 = 0 ;
(4.11)
where hˆ(i) l is the M × M -matrix which describes the Hamiltonian for an isolated lth column chain (i)∗ ˆ(i) (slice), and the (m; m ) element of hˆ(i) (l; m − 1) m; m +1 + l is given by {hl }mm = 4t mm − t(P (i) (i) ˆ P (l; m) m ; m+1 ). The M × M -matrix P x (l) couples the nearest neighbor column chains, and the (i) (i) (i) (m; m ) element of which is given by {Pˆ (i) x (l)}mm = Px (l; m) mm . Here Px (l; m) and Py (l; m) are the Peierls phase factors (Eqs. (4.7) and (4.8)) de/ned using the vector potential ˜A(i) . In Eq. (4.11),
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˜ (i) is a M -dimensional vector which describes the wavefunction in the lth column chain, such that C l ˜ (i) }m = l; m| (i) . Let l be a lattice column in the boundary region between the ith region and the {C l i + 1th region (i = 0; 1; : : : ; N ), such that ˜B(xl ; y) = ˜B0 . Then, noting the fact that ∇ × ˜A(i) (x; y)|x=xl = ∇ × ˜A(i+1) (x; y)|x=xl = ˜B0 , the wavefunction vector in the column l expressed using the ˜A(i) is related to that expressed using the ˜A(i+1) by the following gauge transformation: ˜ (i) ; ˜ (i+1) = Pˆ l (i + 1; i)C C l l where Pˆ l (i + 1; i) is the M × M gauge function matrix de/ned by
ie ˆ ;i+1; i (l; m) mm ; {Pl (i + 1; i)}m; m = exp h
(4.12)
(4.13)
and is introduced to account for the required uni/cation of the gauge [42]. The functional form of the gauge function ;i+1; i is de/ned by the relation (;i+1; i (l; m + 1) − ;i+1; i (l; m))=a = A(i) r l; m + ˜r l; m+1 )=2) − A(i+1) ((˜r l; m + ˜r l; m+1 )=2) ; y ((˜ y
(4.14)
with arbitrary boundary condition (e.g., ;i; i+1 (xl ; ym=0 )=0). Eq. (4.14) is the /nite di8erence version of the following di8erence equation: ∇;i; i+1 (˜r)|x=xl = (˜A(i) (˜r) − ˜A(i+1) (˜r))|x=xl . Combining a set of Eqs. (4.9)–(4.14) with the recursive Green’s function method [43] generalized to the strong magnetic /eld case [44], one can calculate the transmission/re6ection probability in our system. Then the Landauer–BLuttiker formalism gives us the conductance G [18,19]. Here we note that the calculation method presented in this section has an advantage over the previous method given in the last section even in the case of single magnetic dot; the method given in this section is eVcient because it does not require to introduce the /ctitious gradation of the magnetic /eld, which is necessary in the previous method. First in order to con/rm the validity of the calculation method presented here, we carried out the numerical calculation for the case of single magnetic dot. Fig. 4.5 shows the calculated conductance (single dot case) as a function of the Fermi energy for the quantum limit [EF =(˝!0 ) ¡ 1:5] with !0 = eB0 =m∗ . Here the total magnetic /eld in the dot region (r ¡ R1 ) is chosen to be zero such that $1 = −B0 (i.e., B1∗ ≡ $1 + B0 = 0, -1 = B1∗ =B0 = 0:0). The ratio between the dot radius and the wire width is chosen asR1 = 0:28Ly , and the ratio between the wire width and the magnetic length is chosen as Ly = 7:2 ˝=(eB0 ). Therefore the number of additional magnetic 6ux quanta threading the dot region is given by s1 = R21 (B1∗ − B0 )=0 = −2:0 (the number of missing 6ux quanta is 2.0), with 0 = h=e being the magnetic 6ux quantum. As seen in Fig. 4.5, the calculated conductance is almost quantized except for the appearance of aperiodic dips. By comparing the positions of those dips with the energy spectrum of magnetic quantum dot given in Fig. 3.4 (at s = 2), one can recognize that four dips seen in Fig. 4.5 correspond to the magnetic dot eigenstates (a) (n; m) = (0; 1), (b) (1; 0), (c) (0; 2), and (d) (1; −1), respectively. Note that the energy axis in Fig. 4.5 is scaled by eB=(2m∗ ) with B satisfying R21 B=0 = 5:0. In order to understand the formation of the magnetic quantum dot more clearly, in Fig. 4.6 we show the two-dimensional distribution of the local density of states (LDOS) at those four dip positions [(a) – (d)] and at an o8-resonance point EF = 0:8˝!c [(e)]. Here the LDOS at a particular position ˜r is calculated by using the diagonal element of the Green’s function as <(˜r; E)=−(1=)G R (˜r;˜r; E). As seen in Fig. 4.6, the LDOS at the each dip position clearly
S.J. Lee et al. / Physics Reports 394 (2004) 1 – 40
21
1
2
G[2e /h]
(e)
0.5
0 0.4
0.6 (a)
0.8
1
EF / hω 0
1.2 (b)
(c)
1.4 (d)
Fig. 4.5. The dependence of the conductance G on the Fermi energy EF =(˝!0 ) for the case of single magnetic dot.
Fig. 4.6. The spatial distribution of the local density of states (LDOS) for given values of EF =˝!0 [(a) – (e)] denoted in Fig. 4.5.
shows the concentrated signature of electron’s wavefunction (magnetic edge state) corresponding to the each isolated magnetic quantum dot eigenstate (n; m) discussed in Section 3. It should be noted that the large negative value of me8 (i.e., me8 0) gives the broader dip (anti-peak). This feature can be understood by Eq. (3.7) of Section 3. That is, the large negative value of me8 (0) gives the magnetic edge state extended over a wider region, allowing the strong coupling between the (circular) magnetic edge state and the straight conventional edge states along the wire boundaries. At the o8-resonance position (EF = 0:8˝!c ), on the other hand, the LDOS is not concentrated over the dot region, but shows the clear “conventional” edge state along the wire boundaries.
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2
G[2e /h]
1
(a)
0.5
0 0.4
0.6
0.8
1
1.2
1.4
1
1.2
1.4
1 G[2e 2/h]
2
G[2e /h]
1
0.5
0
0 0.4
0.62 0.64 0.66
EF / hω 0
0.6
0.8
E F / hω 0
(b) (a)
(b)
(c) (d)
(e) (f ) (g)
(h)
Fig. 4.7. The dependence of the conductance G on the Fermi energy EF =(˝!0 ) for the case of two magnetic dots, (a) x2 − x1 = 10 ˝=(eB0 ); (b) x2 − x1 = 5:2 ˝=(eB0 ).
Fig. 4.8. The spatial distribution of the local density of states (LDOS) for given values of EF =˝!0 [(a) – (h)] denoted in Fig. 4.7.
Now let us move onto the discussion of two magnetic dots case. Fig. 4.7 shows the calculated EF =(˝!0 )-G curves for two di8erent values of d ≡ x2 − x1 : the separation between two magnetic quantum dots. Other parameters are the same as those in the single dot case of Fig. 4.5. That is, B1∗ = B2∗ = 0:0 (-1 = -2 = 0:0), Ly = 7:2 ˝=(eB0 ) and R1 = R2 = 0:28Ly = 2:0 ˝=(eB0 ). As understood
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from Fig. 4.7, when d is much larger than the magnetic length ˝=(eB0 ), we obtain the conductance pro/le which is almost the same as the single dot case.As decreasing d, the EF =(˝!0 )-G curve starts to deviate from that of singe dot case. When d = 5:2 ˝=(eB0 ), interestingly, the each conductance dip splits into two distinct dips. We obtained the larger splitting width when the corresponding me8 ≡ m + s0 has large negative value. In Fig. 4.8 we show the calculated LDOS distributions at those “splitted” conductance dips [(a) – (h)]. One can observe the “coupled” magnetic edge states which are extended over two dots regions. Therefore, when the separation between two dots is very small, the degenerated energy level of two magnetic edge states splits into two distinct levels due to the /nite coupling between them, giving rise to the appearance of the splitted conductance dips.
5. Modi ed magnetic quantum dot 5.1. Formulation of modi>cation The electronic properties of a modi/ed magnetic quantum dot are also studied. The modi/ed magnetic quantum dot is a quantum structure that is formed by spatially inhomogeneous distributions of magnetic /elds. Electrons are magnetically con/ned to the plane where the magnetic /elds inside and outside the dot are di8erent from each other. The energy spectrum exhibits quite di8erent features depending on the directions of the magnetic /elds inside and outside the dot. In particular, the case of opposite directions of the /elds is more interesting than that of the same direction. An electrostatic potential is introduced to the system to study the e8ects of an electric con/ning potential on the eigenenergy of a single electron in the modi/ed magnetic quantum dot. The additional potential raises the whole energy spectrum and changes its shape. The ground-state angular momentum transitions occurring in a bare modi/ed magnetic quantum dot disappear on introduction of the additional parabolic potential. The modi/ed magnetic quantum dot is formed by spatially inhomogeneous distributions of magnetic /elds [45]. Electrons are magnetically con/ned to the plane where the magnetic /elds inside and outside the dot are di8erent, i.e., (0; 0; B1 ) for r ¡ r0 and (0; 0; B2 ) for r ¿ r0 . This is a more complicated system than a magnetic quantum dot (B1 = 0 for r ¡ r0 ). Here, r0 is the radius of the dot. This kind of magnetic /eld pro/le may be obtained by the combination of a ferromagnetic disk with a homogeneous magnetic /eld as shown in Ref. [46]. An electrostatic potential is introduced to the system to study the e8ects of an electric con/ning potential on the eigenenergy of a single electron in the modi/ed magnetic quantum dot. The electrostatic potentials that we have considered are V (r) = ar 2 + b=r 2 and describe a quantum dot, an antidot, and a quantum ring, depending on the values of a and b. The exact single-electron eigenstates and energies of a modi/ed magnetic quantum dot are calculated by using a general form of the single-particle SchrLodinger equation without electron–electron interactions, i.e.,
2 1 ˜ p ˜ + eA + V (r) (˜r) = E (˜r) ; 2m∗
(5.1)
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where e is the absolute value of the electron charge. In plane polar coordinates (r; ), the vector potential ˜A can be chosen in the symmetric gauge as 1 (r ¡ r0 ) ; - 1 B0 r ˜A = ˆ 2 (5.2) 1 r 2 B (- − - ) + 1 - B r (r ¿ r ) ; 0 1 2 2 0 0 2r 0 2 where -1 = B1 =B0 and -2 = B2 =B0 are the ratios of the applied external magnetic /eld to the standard magnetic /eld B0 . The additional electrostatic potential V (r) is expressed as V (r) =
∗ 2 4 1 ∗ 2 2 2 m !0 r0 + m ) !0 r : 2r 2 2
(5.3)
Here m∗ is the e8ective mass of the electron and d and a are parameters that decide the strength of the antidot and the parabolic potential, respectively. The standard cyclotron frequency is !0 = eB0 =2m∗ . The wave functions are separable, i.e., nm (˜r) = Rnm (r)eim , where m is the angular momentum quantum number and n (=0; 1; 2; : : :) is a radial quantum number which gives the number of nodes in the radial wave function. All quantities are expressed in dimensionless units by setting ˝!0 (=˝eB0 =2m∗ ) and the inverse length $ = m∗ !0 =˝ equal to 1. In these units, ˝2 =m∗ = ˝!0 =$2 → 1, the radius of √ the magnetic quantum dot r0 → s0 where s0 = B0 r02 =0 is the number of magnetic 6ux quanta enclosed by a circle of radius r0 with magnetic /eld B0 and magnetic 6ux quantum 0 (=h=e). The SchrLodinger equation of the radial part is written as 2 d 1 d + (5.4) + 2(E − Ve8 ) Rnm (r) = 0 : dr 2 r dr Here the e8ective potential Ve8 is expressed as 2 m + s02 1 2 + (-1 + )2 )r 2 + m-1 2r 2 2 Ve8 = 2 2 me8 + s0 + 1 (-2 + )2 )r 2 + me8 -2 2r 2 2 2
(r ¡ r0 ) ;
(5.5)
(r ¿ r0 ) ;
where me8 = m + (-1 − -2 )s0 . The wave functions can be solved using Eq. (5.4) and the energy eigenvalues are determined by the continuity of the wave functions and their derivatives at the dot boundary r = r0 . The wave functions are expressed in con6uent hypergeometric functions M and U as
√m2 + s02 √ 4 − -21 +)2 r 2 =2 2 2 2 2 2 (r ¡ r0 ) ; Rnm (r) = C1 -1 + ) r e M a; b; -1 + ) r E m-1 1 − 2 − m2 + s02 − 1 ; a=− 2 -21 + )2 -1 + ) 2 b = m2 + s02 + 1 ;
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Fig. 5.1. Eigenenergies for √ the states (n; m) (n = 0; 1 and m = −2; −1; 0; 1) of a modi/ed magnetic quantum dot as a function of -1 with r0 = 5 and -2 = 1.
4
√m2e8 + s02
−
√
-22 +)2 r 2 =2
Rnm (r) = C2 + e M (c; d; -22 + )2 r 2 ) (r ¿ r0 ) ; E 1 me8 -2 2 2 c=− − 2 − me8 + s0 − 1 ; 2 -22 + )2 -2 + ) 2 (5.6) d = m2e8 + s02 + 1 : √ U for magnetic The radius of the modi/ed magnetic quantum dot is r0 = 5, which is about 1500 A /elds of 0:1 T. For calculations at the special point = 0, the radial wave function Rnm (r) = 1 √ C1 J|m| ( 2Er) is used in the region of r ¡ r0 just as in a regular magnetic quantum dot [16]. Here the function J|m| is the Bessel function of order m. -22
)2 r
5.2. Angular momentum transitions The low lying energy levels for the states (n; m) (n = 0; 1 and m = −2; −1; 0; 1) of a modi/ed magnetic quantum dot () = 0; = 0) as a function of -1 are shown in Fig. 5.1. Here, the magnetic /eld outside the dot is /xed (-2 = 1) and the magnetic /eld inside the dot is varied. The energy spectrum exhibits quite di8erent features depending on the directions of the magnetic /elds inside and outside the dot. In particular, when the /elds inside and outside the dot are opposite to each other in direction the spectrum shows more interesting features. These peculiar behaviors of eigenstates (n; m) can be understood from Ve8 in Eq. (5.5) without detailed calculations. Ve8 (r) for several cases are shown in Fig. 5.2. When r1 (= |m=-1 |) ¡ r0 and r2 (= |me8 =-2 ) ¿ r0 , Ve8 has a double well structure with two local minima, Ve8 (r1 ) = |-1 m| + -1 m and Ve8 (r2 ) = |-2 me8 | + -2 me8 . However, when r1 ¿ r0 and r2 ¡ r0 , Ve8 has one minimum at r = r0 . For states with m ¡ 0, there are |m| 6ux quanta inside a circle of radius rmin , where Ve8 (rmin ) = 0. In fact, this is manifest in Eq. (5.5) and
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Fig. 5.2. E8ective potential Ve8 (r). (a) The cases of a double well (m = −1, -1 = −1, and -2 = 1) and a single well (m = −3, -1 = 0:5, and -2 = 1). (b) The cases having a minimum inside the dot (m = −5, -1 = 3, and -2 = 1) and a minimum outside the dot (m = −5, -1 = −3, and -2 = 1).
is explained in detail in Ref. [16]. These behaviors are clearly shown in Fig. 5.2. Thus, as -1 (¿ 0) increases, (n; m ¡ 0) states are located in the deeper region of the dot, resulting in the Landau levels (2n + 1)|-1 |˝!0 , as shown in the region of -1 ¿ 1 in Fig. 5.1. For -1 ¡ 0, when |-1 | is increased, (n; m ¡ 0) states are located farther away from the dot to enclose |m| 6ux quanta and approach the Landau levels (2n + 1)|-2 |˝!0 . [See the dotted line in Fig. 5.2(b).] States for m ¿ 0 show distinctive variations as also predicted from the shape of Ve8 . Besides convergence to the Landau levels, there is also breaking of Landau level degeneracy except at -1 = 1 where the magnetic /eld is homogeneous. There are also ground-state angular momentum transitions as |-1 | increases in contrast to the regular magnetic quantum dot [16]. These are mainly caused by change of magnetic 6ux quanta due to the inhomogeneous magnetic /elds in the dot. This type of angular momentum transition can occur in conventional quantum rings with /nite width, or in magnetic quantum rings, or in conventional quantum dots including electron–electron interactions. When -1 = 0, the ground state occurs at m = 0 [16]. As |-1 | increases, the ground state occurs at the state m(¿ 0) for negative -1 as in Fig. 5.1 and at m(¡ 0) for positive -1 , and the values m are determined by the given parameters. The main reason is that for -1 6 0 the ground state corresponds to nearby states from the dot while for -1 ¿ 1 it corresponds to states away from the dot as we see from Ve8 in Eq. (5.5).
S.J. Lee et al. / Physics Reports 394 (2004) 1 – 40
Fig. 5.3. Probability densities |Rnm (r)|2 of states (a) (0; 2) and (b) (0; −10) for di8erent -1 ’s with r0 =
27
√
5 and -2 = 1.
The probability densities |Rnm (r)|2 for (0; 2) and (0; −10) states are plotted in Fig. 5.3 for several di8erent -1 ’s. In Fig. 5.3(a), the (0; 2) state can be localized inside or outside the dot but not on the boundary since the condition of r1 ¿ r0 and r2 ¡ r0 is never satis/ed. These values of r1 and r2 are the positions of the minima in Ve8 inside and outside the dot, respectively. Otherwise, states are distributed away from the boundary. In Fig. 5.3(b), states with m = −10 are located farther away from the dot to enclose 10 magnetic 6ux quanta as -1 goes to negative values. When -1 = 2, the probability density has a maximum peak at r0 because the modi/ed magnetic quantum dot includes 10 magnetic 6ux quanta inside the dot exactly at -1 = 2 and s0 = 5. The energy levels for the states (n; m) (n = 0; 1 and m = −2; −1; 0; 1) of the modi/ed magnetic quantum dot with an additional electrostatic potential ()=1, =0) as a function of -1 are shown in Fig. 5.4. The additional potential raises the whole energy spectrum and changes its shape, and breaks the Landau level degeneracy at -1 = 1. The ground state angular momentum transitions that occur in a bare modi/ed magnetic quantum dot disappear due to the additional parabolic potential. When -1 ¿ 0, states gain more energy by virtue of the parabolic potential. These states show similar energy dispersion to that of a conventional quantum dot with a homogeneous external magnetic /eld (see the inset in Fig. 5.4). However, when -1 ¡ 0, the energy spectrum becomes more complicated. Most of the low lying energy states that are localized outside the dot without the parabolic potential are con/ned in the dot by the parabolic potential. As |-1 | increases, i.e., the e8ect of magnetic nonuniformity gets stronger,
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S.J. Lee et al. / Physics Reports 394 (2004) 1 – 40
Fig. 5.4. Eigenenergies for the states (n; m) (n = 0; 1 and m = −2; −1; 0; 1) of a modi/ed magnetic quantum dot as a √ function of -1 with additional parabolic potential () = 1 and = 0), -2 = 1, and r0 = 5.
the higher energy states are less a8ected by the parabolic potential and tend to stay where they were despite the electric con/nement. But those states are no longer Landau levels because of gaining additional energy due to the electric con/nement. If we choose a big enough ) to neglect magnetic /eld nonuniformity, we can get a similar energy dispersion to that of the inset. Figs. 5.5(a) and (b) show the energy dependence Enm on di8erent angular momentum quantum number m for ) = 0 and ) = 1, respectively. These show that all states for ) = 1 are increased in energy and more states (m ¿ 0 → m ¿ − 3) are localized inside the dot compared to the case of ) = 0. In Fig. 5.5(a), the states m ¡ 0 are located outside the dot and are just Landau levels. The state (1; 0) represents the localized state inside the dot having doubled energy E1; 0 compared to E0; 0 . This simply re6ects the /eld ratio of -1 =-2 = 2. The ground state of the system occurs at m ¿ 0 (in fact m = 7), which is associated with a double well situation of Ve8 as already discussed. The states m ¿ 10 represent the edge states, which are localized at the dot boundary and show rapid increase with increase of m. 6. Magnetic quantum ring 6.1. Electronic structures of magnetic quantum ring Now we consider the electronic structure of a magnetic quantum ring formed by inhomogeneous magnetic /elds [47], where electrons are con/ned to a plane, and the magnetic /elds are zero inside the ring and constant elsewhere. The energy states that deviate from the Landau levels are found to form the magnetic edge states along the boundary regions of the magnetic quantum ring. The probability densities of these magnetic edge states are found to be well corresponded to the circulating classical trajectories. In contrast to magnetic or conventional quantum dots, the eigenstates of the magnetic quantum ring show angular momentum transitions in the ground state as the magnetic /eld increases, even without including electron–electron interactions. For a modi/ed magnetic quantum ring with the distribution of nonzero magnetic /elds inside the ring and di8erent /elds outside it,
S.J. Lee et al. / Physics Reports 394 (2004) 1 – 40
29
Fig. 5.5. Dependence of the energy eigenvalues Enm of a modi/ed magnetic quantum dot on the angular momentum quantum number m for (a) ) = 0 and (b) ) = 1 at -1 = −2, -2 = 1, and s0 = 5.
we also /nd similar behaviors such as the angular momentum transitions in the ground state with increasing the magnetic /eld. In the discussions of the magnetic quantum dots given in Sections 3–5, the magnetic edge state was shown to have quite di8erent properties from the conventional electrostatic edge state; a notable feature is that for a small conductor with a magnetic quantum dot at the center, magnetoconductances have aperiodic oscillations instead of the well-known Aharonov–Bohm-type periodic oscillations. Therefore it is interesting to see the formation of magnetic edge states in other magnetic quantum structures like magnetic quantum rings, which have di8erent magnetic /elds in the ring region and outside it. Since the magnetic edge states are related to the characteristics of the electronic-structure, the related physical properties are expected to be di8erent from those of the magnetic quantum dot. In this chapter, we investigate the electronic structure and the magnetic edge states of magnetic quantum rings. We calculate exactly single electron energies by neglecting electron–electron interactions, and /nd that the energy spectra critically depend on the number of missing magnetic 6ux quanta rather than the geometry of the structure or the /eld abruptness. In contrast to the magnetic quantum dot [16], the angular momentum transitions in the ground state are found to occur as the magnetic /eld varies. The classical trajectories of the quantum states are obtained by using the general rules, which are derived from the energy and angular momentum conservation laws.
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We /rst consider the 2DEG con/ned in a magnetic quantum ring formed by inhomogeneous magnetic /elds; the magnetic /eld perpendicular to the plane is zero within a circular ring and constant B outside it. When electron–electron interactions are neglected, the single-particle SchrLodinger equation for the magnetic quantum ring is 1 (˜ p + e˜A)2 (˜r) = E (˜r) ; (6.1) 2m∗ where m∗ is the e8ective mass of an electron and e is the absolute value of the electron charge. In polar coordinates (r; ) on the plane, the vector potential ˜A is chosen in a symmetric gauge such as 1 Br (r ¡ r1 ) ; 2 1 ˜A = ˆ (6.2) Br12 (r1 ¡ r ¡ r2 ) ; 2r 1 Br − 1 (r22 + r12 ) (r2 ¡ r) : 2 2r Then, the wave functions are separable, i.e., Rnm (˜r) = Rnm (r)eim , where m is the angular momentum quantum number and n(=0; 1; 2; : : :) is the number of nodes in Rnm (r), and the equation for the radial part is written as 2 d 1 d + (6.3) + 2(E − Ve8 ) Rnm (r) = 0 ; dr 2 r dr 1 2 m2e8 ; 1 + me8 ; 1 (r ¡ r1 ) ; r + 2 2r 2 2 me8 ; 2 Ve8 = (6.4) (r1 ¡ r ¡ r2 ) ; 2 2r 2 1 r 2 + me8 ; 3 + me8 ; 3 (r ¿ r2 ) ; 2 2r 2 where me8 ; 1 , me8 ; 2 , and me8 ; 3 are de/ned as m, m + s1 , and m − S, respectively. All quantities are ∗ ∗ expressed in dimensionless units by letting ˝!L [=˝eB=(2m )] and the inverse length $= m !L =˝ to be 1. The dimensionless parameter S(=s2 − s1 ) represents the number of missing 6ux quanta, where s1 = r12 B=0 , s2 = r22 B=0 , and 0 (=h=e) is the 6ux quantum. In these units ˝2 =m∗ = ˝!L =$2 → 1, √ √ r1 → s1 , and r2 → s2 , where r1 and r2 are the inner and outer radii of the ring, respectively. The solutions for Rnm (r) are found to be 2 (r ¡ r1 ) ; C1 r |me8 ; 1 | e−r =2 M (a1 ; b1 ; r 2 ) √ √ 2Er + C3 N|me8 ; 2 | 2Er (r1 ¡ r ¡ r2 ) ; Rnm (r) = C2 J|me8 ; 2 | (6.5) 2 C4 r |me8 ; 3 | e−r =2 U (a2 ; b2 ; r 2 ) (r ¿ r2 ) ; where a1 = (|me8 ; 1 | + 1 − E + me8 ; 1 )=2, b1 = |me8 ; 1 | + 1, a2 = (|me8 ; 3 | + 1 − E + me8 ; 3 )=2, b2 = |me8 ; 3 | + 1. Here, J and N denote the Bessel functions, and M and U are the con6uent hypergeometric functions. The eigenenergies are determined by the continuity of the wave functions and their derivatives at the boundaries of the inner and outer circles. From the e8ective potential Ve8 in Eq. (6.4), we can obtain
S.J. Lee et al. / Physics Reports 394 (2004) 1 – 40
31
Fig. 6.1. E8ective potentials for the states (m = 0; −1; 1) in the magnetic quantum ring with s1 = 2 and s2 = 6.
very useful information on the properties of the (n; m) eigenstates without detailed calculations. For the states with m 6 0, the minimum value of Ve8 is always 0, and |m| magnetic 6ux quanta are enclosedat the minimum. For m ¿ 0 and |me8 ; 3 | ¿ s2 , Ve8 has the minimum value of |me8 ; 3 | + me8 ; 3 at r = |me8 ; 3 |, which is located outside the outer circle of the magnetic ring. For m ¿ 0 and |me8 ; 3 | 6 s2 , the minimum of Ve8 is always located at r = r2 , and the corresponding states are localized near the ring boundaries. The e8ective potentials for m=0; −1, and 1 are drawn√in Fig. 6.1. √ √ In this case, we choose the parameters, s1 = 2 and s2 = 6, i.e., r1 = s1 and r2 = s2 = 3r1 , which U for magnetic /elds of few teslas. As the magnetic give the ring size of about several hundreds A /eld S increases, more states are populated in the ring region, with the energies deviated from the Landau levels. This deviation results in the formation of magnetic edge states, similar to the case of magnetic quantum dots [16].
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S.J. Lee et al. / Physics Reports 394 (2004) 1 – 40
Fig. 6.2. Energy spectra of the magnetic quantum ring as a function of S. Dotted lines represent the Landau levels, and Enm represents the energies of the (n; m) states, which are normalized by that for the lowest Landau level at S = 4, where n(=0; 1; 2; : : :) and m denote the radial and angular momentum quantum numbers, respectively.
For the magnetic ring considered here, the calculated energies Enm are plotted as a function of the magnetic /eld S in Fig. 6.2. For weak-magnetic /elds, since the density of magnetic 6ux is low over the ring, the (0; m ¡ 0) states must be localized in the region very far from the ring to enclose |m| 6ux quanta, if the magnitude of m is very large. In this case, the states resemble the Landau levels, which are normally formed by the uniform distribution of magnetic /elds over the whole region. As B increases, the localized region of the states get closer to the ring, and these states feel the absence of magnetic /elds inside the ring. Then, these states start to deviate from the Landau levels, with lower energies. When B is strong enough for the (0; m ¡ 0) states to be prominent inside the inner circle, where uniform magnetic /elds are present, the energies return to the Landau levels. The larger the magnitude of m, the faster the recovery of the Landau levels takes place with increasing the magnetic /eld. Thus, the magnitude of m in the ground state continuously increases as B increases, i.e., the high-energy states with large values of m (m ¡ 0) turn into the ground state. Such angular momentum transitions are mainly caused by the missing of 6ux quanta in the ring area, while these transitions were not seen in the magnetic quantum dot [16]. In conventional quantum dots con/ned by electrostatic potentials, this type of angular momentum transitions in the ground state can occur only if electron–electron interactions are included [48]. For a conventional quantum ring with a /nite width con/ned by electrostatic potentials, the angular momentum transition was also observed with increasing magnetic /eld [49]. When the (0; m ¡ 0) states have the quantum number of m = −s1 , Ve8 is zero over the ring region. Thus, for S = 4, the ground state is the (0; −2) state instead of the (0; 0) state, as clearly shown in Fig. 6.2. In the limit of B → ∞, the whole energy spectra of the magnetic quantum ring become identical to those of the conventional quantum ring with the angular momentum quantum number shifted from m to me8 ; 2 . This is because very high-magnetic /elds outside the ring area act as an in/nite barrier for electrons in the ring. In the case of m ¿ 0,
S.J. Lee et al. / Physics Reports 394 (2004) 1 – 40
33
Fig. 6.3. Dependence of Enm on m in the magnetic quantum ring for S = 4, i.e., s1 = 2 and s2 = 6.
as expected from the behavior of Ve8 , the states with |me8 ; 3 | 6 s2 are mainly localized near the outer circle, while for |me8 ; 3 | ¿ s2 they are distributed outside the ring. Then, once the (0; m ¿ 0) states deviate from the Landau levels, they never turn to the Landau levels again even for very high-magnetic /elds, as shown in Fig. 6.2. For a magnetic /eld given by s1 = 2 and s2 = 6, the dependence of Enm on m is drawn in Fig. 6.3. If me8 ; 3 in Ve8 satis/es the condition |me8 ; 3 |s2 , which gives m10 or m − 2, the (0; m ¡ 0) states are distributed very far from the ring, and their energies becomes the Landau levels. As mentioned earlier, the (0; m ¡ 0) states near m = −2, where the ground state occurs, are perturbed by the missing of S 6ux quanta in the ring region. Since these states have lower energies than the Landau level, they have nonzero probability currents Inm [22], where Inm = 1=h9Enm =9m, forming magnetic edge states. Depending on the sign of 9Enm =9m, the magnetic edge states carry currents circulating either clockwise or counterclockwise, while the degenerated Landau levels of m ¡ 0 carry no probability currents. In Fig. 6.4, the classical trajectories of electrons and their corresponding probability densities |Rnm (r)|2 are drawn for the (0; 0), (0; −1), and (0; 1) states, which exhibit clearly the classical behavior of the magnetic edge states formed near the ring boundaries. In fact, these states represent the ensemble average of trajectories, which consist of straight line paths in the ring region and cyclotron orbits with the radius ri = Enm =2 and the center located at rj = ri2 − me8 outside the ring. Here, the value of me8 depends on the region, as shown in Eq. (6.4). Thus, for given n and m, the value of ri is /xed whereas that for rj varies with region. The general rules for ri and rj are derived from the conservation of both energy and angular momentum [16]. Since the (0; −2) state is the ground state, the (0; 0), (0; −1), and the (0; 1) states have the probability currents drifting along the counterclockwise direction, i.e., Inm =1=h9Enm =9m is positive (see Fig. 6.3). Besides the direction of classical motions, we /nd other interesting feature that although no tunneling is allowed between the classical trajectories which consist of separate sets of motions in general, the corresponding probability densities between the trajectories are connected smoothly due to quantum mechanical tunneling.
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S.J. Lee et al. / Physics Reports 394 (2004) 1 – 40
Fig. 6.4. Classical trajectories of electrons and corresponding probability densities for the (a) (0; 0), (b) (0; −1), and (c) (0; 1) states in the magnetic quantum ring.
6.2. Modi>ed magnetic quantum ring We extend our study to the modi/ed structure of the magnetic quantum ring, which has nonzero magnetic /eld B∗ in the ring area (r1 ¡ r ¡ r2 ) and B(= B∗ ) elsewhere. The vector potential for this new structure is chosen to be 1 Br (r ¡ r1 ) ; 2 ˜A = ˆ 1 B∗ r + 1 (B − B∗ )r 2 (6.6) (r1 ¡ r ¡ r2 ) ; 1 2 2r 1 Br + 1 B∗ (r22 − r12 ) − 1 B(r22 − r12 ) (r2 ¡ r) : 2 2r 2r
S.J. Lee et al. / Physics Reports 394 (2004) 1 – 40
35
Fig. 6.5. Dependence of Enm on m in the modi/ed magnetic quantum ring with s1 = 2 and s2 = 6.
Then, the e8ective angular momentum quantum numbers me8 are modi/ed such as me8 ; 1 = m
(r ¡ r1 ) ; ∗
me8 ; 2 = m + s1 − s1
(r1 ¡ r ¡ r2 ) ;
me8 ; 3 = m + s2∗ − s1∗ − (s2 − s1 )
(r2 ¡ r) ;
(6.7)
where s1 = r12 B=0 , s2 = r22 B=0 , s1∗ = r12 B∗ =0 , and s2∗ = r22 B∗ =0 . Using the same dimensionless √ √ units as those in the last section, we can express r1 = s1 and r2 = s2 . The radial function in the ring region is written as the combination of the con6uent hypergeometric functions M and U , 2 2
R(r) = -(-r)|me8 ; 2 | e−1=2- r [C2 M (a3 ; b3 ; -2 r 2 ) + C3 U (a3 ; b3 ; -2 r 2 )]
(r1 ¡ r ¡ r2 ) ;
(6.8)
where a3 =[|me8 ; 2 |+1−-−2 E +sign(B∗ )me8 ; 2 ]=2, b3 =|me8 ; 2 |+1, and -2 =|B∗ =B|. The wave functions outside the ring have the same forms as those of the magnetic quantum ring with the modi/ed me8 ’s in Eq. (6.7). Here, we only consider a special case of B∗ = −B, which gives the e8ective potential Ve8 = m2e8 =(2r 2 ) + r 2 =2 ± me8 . The positive and negative signs of the last term corresponds to the regionsof r ¿ r2 and r ¡ r1 , respectively. In each region, Ve8 has a minimum value of |me8 | ± me8 at r = |me8 |. The calculated energies of the modi/ed magnetic quantum ring are plotted as function of m for s1 = 2 and s2 = 6 in Fig. 6.5. Both the ground and excited states are found to deviate more severely from the Landau levels, as compared to the regular magnetic quantum ring considered in the last section. The results indicate that the energy levels and the angular momentum transitions can be modulated for both the ground and excited states by varying the ratio B∗ =B, i.e., the number of missing 6ux quanta in the ring region. For the states localized far outside the ring, which satisfy the condition |me8 ; 3 |s2 , i.e., m2 or m14, their energies approach to the Landau levels under uniform magnetic /elds with the quantum number shifted from m to me8 ; 3 . For the (n; m) eigenstates, the corresponding
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S.J. Lee et al. / Physics Reports 394 (2004) 1 – 40
Fig. 6.6. Classical trajectories of electrons and corresponding probability densities for the (a) (0; 0), (b) (0; −1), and (c) (0; 1) states in the modi/ed magnetic quantum ring.
classical trajectories can also be constructed usingthe energy and angular momentum conservation ∗ | E =2 and r = |B=B∗ | E =2 − B∗ =Bm laws such as r = |B=B i nm j nm e8 inside the ring whereas ri = Enm =2 and rj = Enm =2 − me8 elsewhere, where me8 ’s are given in Eq. (6.7). In each region, the trajectories consist of circular orbits centered at rj with the radius ri , and the resulting magnetic edge states circulate clockwise for r1 ¡ r ¡ r2 while counterclockwise elsewhere. For s1 = 2, s2 = 6, and B∗ =B = −1, the classical trajectories and the corresponding probability densities |Rnm (r)|2 are drawn for the (0; 0), (0; −1), and (0; 1) states in Fig. 6.6. These states exhibit wavelike trajectories unlike the magnetic quantum dot in the last section. Since the tunnelings between electrons are prohibited classically, some trajectories may consist of separated orbit sets. The radial distribution of the eigenstates obtained quantum mechanically shows rather smooth variations and agrees excellently with the corresponding trajectories. Although the (0; 0) and (0; −1) states are almost degenerate as shown in Fig. 6.5, they have di8erent probability currents, opposite to each other. The (0; 0) state circulates counter-clockwise along the outer boundary of the ring, while the (0; −1) state does
S.J. Lee et al. / Physics Reports 394 (2004) 1 – 40
37
clockwise along the inner boundary (see Fig. 6.6). For the m = 0 states, the e8ective potential has three local minima of Ve8 = 0, in contrast to the magnetic quantum ring in Section 2, where a minimum of Ve8 occurs only at r = 0. This behavior illustrates why the probability density of the (0; 0) state in Fig. 6.6(a) is more reduced for r ¡ r1 than that for the regular magnetic ring in Fig. 6.4(a). The (0; 1) state is found to have no trajectories for r ¡ r1 and show a wavelike circulation along the outer boundary of r = r2 , which is also manifested in the radial distribution in Fig. 6.6(c). 7. Summary We have investigated the electronic structure and the electronic transport in various magnetic nano-structures which are de/ned by various non-uniform distributions of magnetic /elds; the magnetic quantum dots, the modi/ed magnetic quantum dots, and the magnetic quantum ring. Through the theoretical study of such magnetic nano-structures, we found that the magnetic edge states are formed along the boundaries of those magnetic quantum dots or rings. Such magnetic edge states are essentially di8erent from conventional (electrostatic) edge states formed due to the electrostatic modulations, and are interpreted to correspond to the complicated classical electron trajectories formed due to the non-uniform magnetic /elds. The formation of magnetic edge states are essentially determined by the number of additional (or missing) magnetic 6ux quanta threading the systems, and are directly related to the discrete energy eigenvalues of such systems. As a simplest case, if the distribution of magnetic /elds is given by ˜Bin = 0 and ˜Bout = B0 zˆ inside and outside the circular region with radius r0 , respectively, the number of missing magnetic 6ux quanta s threading the circular region importantly determines the energy spectrum, and the e8ective angular momentum quantum number me8 = m − s serves as an important factor which determines the e8ective potential, the e8ective radius of the electron’s wavefunction, and so on. When such a magnetic quantum dot (˜Bin = 0 and ˜Bout = B0 z) ˆ is placed at the center of quantum wire (narrow 2DEG), the formation of the magnetic edge states is found to give rise to quite distinctive aperiodic oscillations in the magnetoconductance. When a magnetic quantum dot, de/ned by ˜Bin = B∗ zˆ and ˜Bout = B0 z, ˆ is located at the center of quantum wire (narrow 2DEG), more varieties of phenomena can be found. The magnetic quantum dot is found to be a characteristic scattering center which results in a transmission barrier and a resonator. The /eld reversal at the dot boundary gives rise to distinct magnetic edge states and transport properties, such as non-quantized conductance and the conductance plateaus. The scattering mechanism based on the magnetic con/nement can be useful to understand electron transport in other magnetic structures combined with electrostatic con/nements. Moreover, the electronic transport in quantum wires with two magnetic quantum dots were also investigated, by using the calculation method based on the gauge transformations. The calculations of the local density of states have shown the presence of magnetic edge states extended over the two magnetic dots. Studies on the electronic structures of the magnetic quantum dot is generalized by considering a modi/ed magnetic quantum dot [˜Bin = -1 B0 zˆ and ˜Bout = -2 B0 z] ˆ and the e8ects of an electric con/ning potential on the eigenenergy of a single electron in a modi/ed magnetic quantum dot have been studied. The energy spectrum exhibits quite interesting features depending on the direction of the magnetic /eld inside the dot when the direction of the magnetic /eld outside the dot is
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S.J. Lee et al. / Physics Reports 394 (2004) 1 – 40
/xed. As -1 (¿ 0) increases, (n; m ¡ 0) states are located in the deeper region of the dot, resulting in the Landau levels (2n + 1)|-1 |˝!0 . For -1 ¡ 0, when |-1 | is increased, (n; m ¡ 0) states are located farther away from the dot to enclose |m| 6ux quanta and approach to the Landau levels (2n + 1)|-2 |˝!0 . Additional electrostatic potentials lift up the whole energy spectrum, break the Landau level degeneracy at -1 = 1, and change the shape of the electronic structure of the modi/ed magnetic quantum dot. Low lying energy levels, which are a8ected strongly by added electrostatic potentials, no longer have ground state angular momentum transitions. Finally, the electronic structure of the magnetic quantum ring have been investigate. The eigenstates are found to deviate from the Landau levels due to the missing of magnetic 6ux quanta and form the magnetic edge states. These edge states carry nonzero probability currents and depend sensitively on the number of the enclosed magnetic 6ux quanta. We /nd that the magnetic quantum ring exhibits the angular momentum transitions in the ground state as the magnetic /eld increases. For extremely high-magnetic /elds, the energy spectra resemble those for a conventional quantum ring without magnetic /elds. For the modi/ed magnetic quantum ring with nonzero but di8erent magnetic /elds inside the ring, the angular momentum transitions are also found in the ground state, which are enhanced by modifying the geometry of the quantum structure and varying the strength of magnetic /eld. Acknowledgements This work was supported by the Korea Science and Engineering Foundation through the QuantumFunctional Semiconductor Research Center at Dongguk University and the authors are grateful for the discussions with Dr. H.S. Sim, Dr. N. Kim, and Dr. J.W. Kim. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15]
S.J. Bending, K. von Klitzing, K. Ploog, Phys. Rev. Lett. 65 (1990) 1060. S.J. Bending, K. von Klitzing, K. Ploog, Phys. Rev. B 42 (1990) 9859. M.A. McCord, D.D. Awschalom, Appl. Phys. Lett. 57 (1990) 2153. M.L. Leadbeater, S.J. Allen Jr., F. DeRosa, J.P. Harbison, T. Sands, R. Ramesh, L.T. Florez, V.G. Keramidas, J. Appl. Phys. 69 (1991) 4689. K.M. Krishnan, Appl. Phys. Lett. 61 (1992) 2365. R. Yagi, Y. Iye, J. Phys. Soc. Japan 62 (1993) 1279. H.A. Carmona, A.K. Geim, A. Nogaret, P.C. Main, T.J. Foster, M. Henini, S.P. Beaumont, M.G. Blamire, Phys. Rev. Lett. 74 (1995) 3009. P.D. Ye, D. Weiss, R.R. Gerhardts, M. Seeger, K. von Klitzing, K. Eberl, H. Nickel, Phys. Rev. Lett. 74 (1995) 3013. F.M. Peeters, A. Matulis, Phys. Rev. B 48 (1993) 15166. A. Matulis, F.M. Peeters, P. Vasilopoulos, Phys. Rev. Lett. 72 (1994) 1518. M.C. Chang, Q. Niu, Phys. Rev. B 50 (1994) 10843. J.Q. You, L. Zhang. P.K. Ghosh, Phys. Rev. B 52 (1995) 17243. I.S. Ibrahim, F.M. Peeters, Phys. Rev. B 52 (1995) 17321. J.Q. You, L. Zhang, Phys. Rev. B 54 (1996) 1526. A. Nogaret, S. Carlton, B.L. Gallagher, P.C. Main, M. Henini, R. Wirtz, R. Newbury, M.A. Howson, S.P. Beaumont, Phys. Rev. B 55 (1997) R16037.
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39
H.-S. Sim, K.-H. Ahn, K.J. Chang, G. Ihm, N. Kim, S.J. Lee, Phys. Rev. Lett. 80 (1998) 1501. L.L. Sohn (Ed.), Mesoscopic Electron Transport, Kluwer Academic Publishers, Dordrecht, 1997. R. Landauer, IBM J. Res. Dev. 32 (1998) 306. M. BLuttiker, Phys. Rev. B 38 (1988) 9375. M.L. Leadbeater, C.L. Foden, T.M. Burke, J.H. Burroughes, M.P. Grimshaw, D.A. Ritchie, L.L. Wang, M. Pepper, J. Phys.: Condens. Matter 7 (1995) L307. A. Nogaret, S.J. Bending, M. Henini, Phys. Rev. Lett. 84 (2000) 2231. B.I. Halperin, Phys. Rev. B 25 (1982) 2185. B.J. van Wees, H. van Houten, C.W.J. Beenakker, J.G. Williamson, L.P. Kouwenhoven, D. van der Marel, C.T. Foxon, Phys. Rev. Lett. 60 (1988) 848; D.A. Wharam, T.J. Thornton, R. Newbury, M. Pepper, H. Ahmed, J.E.F. Frost, D.G. Hasko, D.C. Peacock, D.A. Ritchie, G.A.C. Jones, J. Phys. C 21 (1988) L209. These conventional channels are often referred to the magnetic edge channels to distinguish from the narrow channels in the absence of magnetic /elds. In our work this is not the case. J.E. MLuller, Phys. Rev. Lett. 68 (1992) 385. L. Solimany, B. Kramer, Solid State Commun. 96 (1995) 471. J.K. Jain, Phys. Rev. Lett. 60 (1988) 2074. C.S. Lent, Phys. Rev. B 43 (1991) 4179. B.J. van Wees, L.P. Kouwenhoven, C.J.P.M. Harmans, J.G. Williamson, C.E. Timmering, M.E.I. Broekaart, C.T. Foxon, J.J. Harris, Phys. Rev. Lett. 62 (1989) 2523; W.-C. Tan, J.C. Inkson, Phys. Rev. B 53 (1996) 6947. H.A. Carmona, A.K. Geim, A. Nogaret, P.C. Main, T.J. Foster, M. Henini, S.P. Beaumont, M.G. Blamire, Phys. Rev. Lett. 74 (1995) 3009; P.D. Ye, D. Weiss, R.R. Gerhardts, M. Seeger, K. von Klitzing, K. Eberl, H. Nickel, Phys. Rev. Lett. 74 (1995) 3013; A. Nogaret, S. Carlton, B.L. Gallagher, P.C. Main, M. Henini, R. Wirtz, R. Newbury, M.A. Howson, S.P. Beaumont, Phys. Rev. B 55 (1999) R16037. M.L. Leadbeater, C.L. Foden, J.H. Burroughes, M. Pepper, T.M. Burke, L.L. Wang, M.P. Grimshaw, D.A. Ritchie, Phys. Rev. B 52 (1995) R8629. B.-Y. Gu, W.-D. Sheng, X.-H. Wang, J. Wang, Phys. Rev. B 56 (1997) 13434; J. Reijniers, F.M. Peeters, arXiv:cond-mat/0009303 (2000). A. Matulis, F.M. Peeters, P. Vasilopoulos, Phys. Rev. Lett. 72 (1994) 1518; J. Reijniers, F.M. Peeters, A. Matulis, Physica E 6 (2000) 759. F.M. Peeters, P. Vasilopoulos, Phys. Rev. B 47 (1994) 1466. H.-S. Sim, G. Ihm, N. Kim, K.J. Chang, Phys. Rev. Lett. 87 (2001) 146601. J. Reijniers, F.M. Peeters, A. Matulis, Phys. Rev. B 59 (1999) 2817. J.K. Jain, S.A. Kivelson, Phys. Rev. Lett. 60 (1988) 1542. Y. Takagaki, D.K. Ferry, Phys. Rev. B 48 (1993) 8152. Let us consider a state with m ¡ 0 [ m = eim Rm (r)] in the uniform /eld B0 . This state is located at rp (m ; B0 ) and encloses |m | 6ux quanta. If s 6ux quanta are removed inside r0 (rp ), m can be written as ei(m +s) Rm (r) due to the gauge invariance and then rewritten as eim Rme8 (r). In the calculations given in this section, to make ˜A = 0 in leads (i.e., wire regions of |x| ¿ Lx =2), we choose that B = 0 for r ¿ r1 (r0 ) and ˜A = 0 for r ¿ r2 (∼ Lx =2) (see Ref. [38]). Here, r1 and Lx are chosen as 38a and 4000a, respectively. This arti/cial choice for ˜A does not a8ect our results only when Ly =Lx 1; otherwise, the Peierls’ phase is counted incorrectly. Also, to imitate the continuum limit by the lattice Hamiltonian, the condition |2Ba2 =0 1| should be satis/ed. S. Souma, S.J. Lee, N. Kim, T.W. Kang, G. Ihm, J.C. Woo, A. Suzuki, in: J.H. Davies, A.R. Long (Eds.), Proc. 26th Int. Conf. Physics of Semiconductors, Edinburgh, 2002, IOP Conf. Ser. 171, H187, IOP, Bristol, 2003; S. Souma, S.J. Lee, N. Kim, T.W. Kang, G. Ihm, K.S. Yi, A. Suzuki, J. Superconductivity: Incorporating Novel Magnetism 16 (2003) 339. J.J. Palacios, C. Tejedor, Phys. Rev. B 48 (1993) 5386. D.S. Fisher, P.A. Lee, Phys. Rev. B 23 (1991) 6851.
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[44] T. Ando, Phys. Rev. B 44 (1991) 8017; S. Souma, A. Suzuki, J. Korean Phys. Soc. 39 (2001) 553. [45] N. Kim, G. Ihm, H.-S. Sim, T.W. Kang, Phys. Rev. B 63 (2001) 235317. [46] J.A.K. Freire, A. Matulis, F.M. Peeters, V.N. Freire, G.A. Farias, Phys. Rev. B 61 (2000) 2895. [47] N. Kim, G. Ihm, H.-S. Sim, K.J. Chang, Phys. Rev. B 60 (1999) 8767. [48] U. Merkt, J. Huser, M. Wagner, Phys. Rev. B 43 (1991) 7320; M. Wagner, U. Merkt, A.V. Chaplik, ibid. 45 (1992) 1951; J.H. Oh, K.J. Chang, G. Ihm, S.J. Lee, ibid. 50 (1994) 15397. [49] G. Kirczenow, Superlatt. Microstruct. 14 (1994) 237; W.-C. Tan, J.C. Inkson, Phys. Rev. B 53 (1996) 6947.
Available online at www.sciencedirect.com
Physics Reports 394 (2004) 41 – 156 www.elsevier.com/locate/physrep
Random matrix theory and symmetric spaces M. Casellea;∗ , U. Magneab a
Department of Theoretical Physics, University of Torino and INFN, Sez. di Torino Via P. Giuria 1, I-10125 Torino, Italy b Department of Mathematics, University of Torino, Via Carlo Alberto 10, I-10123 Torino, Italy Accepted 30 December 2003 editor: C.W.J. Beenakker
Abstract In this review we discuss the relationship between random matrix theories and symmetric spaces. We show that the integration manifolds of random matrix theories, the eigenvalue distribution, and the Dyson and boundary indices characterizing the ensembles are in strict correspondence with symmetric spaces and the intrinsic characteristics of their restricted root lattices. Several important results can be obtained from this identi4cation. In particular the Cartan classi4cation of triplets of symmetric spaces with positive, zero and negative curvature gives rise to a new classi4cation of random matrix ensembles. The review is organized into two main parts. In Part I the theory of symmetric spaces is reviewed with particular emphasis on the ideas relevant for appreciating the correspondence with random matrix theories. In Part II we discuss various applications of symmetric spaces to random matrix theories and in particular the new classi4cation of disordered systems derived from the classi4cation of symmetric spaces. We also review how the mapping from integrable Calogero–Sutherland models to symmetric spaces can be used in the theory of random matrices, with particular consequences for quantum transport problems. We conclude indicating some interesting new directions of research based on these identi4cations. c 2003 Elsevier B.V. All rights reserved. PACS: 02.10.−v
Contents 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 Part I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 2. Lie groups and root spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 2.1. Lie groups and manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 ∗
Corresponding author. Tel.: +39-011-6707205; fax: +39-011-6707214. E-mail addresses:
[email protected] (M. Caselle),
[email protected] (U. Magnea).
c 2003 Elsevier B.V. All rights reserved. 0370-1573/$ - see front matter doi:10.1016/j.physrep.2003.12.004
42
M. Caselle, U. Magnea / Physics Reports 394 (2004) 41 – 156 2.2. The tangent space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3. Coset spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4. The Lie algebra and the adjoint representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5. Semisimple algebras and root spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6. The Weyl chambers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7. The simple root systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Symmetric spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1. Involutive automorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2. The action of the group on the symmetric space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3. Radial coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4. The metric on a Lie algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5. The algebraic structure of symmetric spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Real forms of semisimple algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1. The real forms of a complex algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2. The classi4cation machinery . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The classi4cation of symmetric spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1. The curvature tensor and triplicity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2. Restricted root systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3. Real forms of symmetric spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Operators on symmetric spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1. Casimir operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2. Laplace operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3. Zonal spherical functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4. The analog of Fourier transforms on symmetric spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Integrable models related to root systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1. The root lattice structure of the CS models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2. Mapping to symmetric spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
47 48 50 51 55 56 57 57 59 59 62 64 64 65 68 71 72 75 79 81 81 83 88 91 94 95 96
Part II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8. Random matrix theories and symmetric spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1. Introduction to the theory of random matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1.1. What is random matrix theory? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1.2. Some of the applications of random matrix theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1.3. Why are random matrix models successful? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2. The basics of matrix models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3. Identi4cation of the random matrix integration manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3.1. Circular ensembles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3.2. Gaussian ensembles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3.3. Chiral ensembles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3.4. Transfer matrix ensembles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3.5. The DMPK equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3.6. BdG and p-wave ensembles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3.7. S-matrix ensembles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4. Identi4cation of the random matrix eigenvalues and universality indices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4.1. Discussion of the Jacobians of various types of matrix ensembles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.5. Fokker–Planck equation and the Coulomb gas analogy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.5.1. The Coulomb gas analogy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.5.2. Connection with the Laplace–Beltrami operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.5.3. Random matrix theory description of parametric correlations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.6. A dictionary between random matrix ensembles and symmetric spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9. On the use of symmetric spaces in random matrix theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
97 98 98 98 99 100 101 104 104 106 111 114 117 119 120 121 122 124 125 126 127 127 128
3.
4. 5.
6.
7.
M. Caselle, U. Magnea / Physics Reports 394 (2004) 41 – 156 9.1. 9.2. 9.3. 9.4.
Towards a classi4cation of random matrix ensembles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Symmetries of random matrix ensembles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Orthogonal polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Use of symmetric spaces in quantum transport . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4.1. Exact solvability of the DMPK equation in the = 2 case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4.2. Asymptotic solutions in the = 1; 4 cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4.3. Magnetic dependence of the conductance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4.4. Density of states in disordered quantum wires . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10. Beyond symmetric spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.1. NonCartan parametrization of symmetric spaces and S-matrix ensembles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.1.1. NonCartan parametrization of SU (N )=SO(N ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2. Clustered solutions of the DMPK equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.3. Triplicity of the Weierstrass potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11. Summary and conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix A. Zonal spherical functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.1. The Itzykson–Zuber–Harish–Chandra integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.2. The Duistermaat–Heckman theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
43 128 131 131 133 134 137 137 138 139 140 141 143 145 147 149 150 151 152 152
1. Introduction The study of symmetric spaces has recently attracted interest in various branches of physics, ranging from condensed matter physics to lattice QCD. This is mainly due to the gradual understanding during the past few years of the deep connection between random matrix theories and symmetric spaces. Indeed, this connection is a rather old intuition, which traces back to Dyson [1] and has subsequently been pursued by several authors, notably by HNuOmann [2]. Recently it has led to several interesting results, like for instance a tentative classi4cation of the universality classes of disordered systems. The latter topic is the main subject of this review. The connection between random matrix theories and symmetric spaces is obtained simply through the coset spaces de4ning the symmetry classes of the random matrix ensembles. Although Dyson was the 4rst to recognize that these coset spaces are symmetric spaces, the subsequent emergence of new random matrix symmetry classes and their classi4cation in terms of Cartan’s symmetric spaces is relatively recent [3–7]. Since symmetric spaces are rather well understood mathematical objects, the main outcome of such an identi4cation is that several non-trivial results concerning the behavior of the random matrix models, as well as the physical systems that these models are expected to describe, can be obtained. In this context an important tool, that will be discussed in the following, is a class of integrable models named Calogero–Sutherland models [8]. In the early 1980s, Olshanetsky and Perelomov showed that also these models are in one-to-one correspondence with symmetric spaces through the reduced root systems of the latter [9]. Thanks to this chain of identi4cations (random matrix ensemble–symmetric space–Calogero–Sutherland model) several of the results obtained in the last 20 years within the framework of Calogero–Sutherland models can also be applied to random matrix theories. The aim of this review is to allow the reader to follow this chain of correspondences. To this end we will devote the 4rst half of the paper (Sections 2–7) to the necessary mathematical background
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and the second part (Sections 8–10) to the applications in random matrix theory. In particular, in the last section we discuss some open directions of research. The reader who is not interested in the mathematical background could skip the 4rst part and go directly to the later sections where we list and discuss the main results. This review is organized as follows: The 4rst 4ve sections of Part I (Sections 2–6) are devoted to an elementary introduction to symmetric spaces. As mentioned in the Abstract, these sections consist of the material presented in [10], which is a self-contained introductory review of symmetric spaces from a mathematical point of view. The material on symmetric spaces should be accessible to physicists with only elementary background in the theory of Lie groups. We have included quite a few examples to illustrate all aspects of the material. In the last section of Part I, Section 7, we brieQy introduce the Calogero– Sutherland models with particular emphasis on their connection with symmetric spaces. After this introductory material we then move on in Part II to random matrix theories and their connection with symmetric spaces (Section 8). Let us stress that this paper is not intended as an introduction to random matrix theory, for which very good and thorough references already exist [11,26–29]. In this review we will assume that the reader is already acquainted with the topic, and we will only recall some basic information (de4nitions of the various ensembles, main properties, and main physical applications). The main goal of this section is instead to discuss the identi4cations that give rise to the close relationship between random matrix ensembles and symmetric spaces. Section 9 is devoted to a discussion of some of the consequences of the above mentioned identi4cations. In particular we will deduce, starting from the Cartan classi4cation of symmetric spaces, the analogous classi4cation of random matrix ensembles. We discuss the symmetries of the ensembles in terms of the underlying restricted root system, and see how the orthogonal polynomials belonging to a certain ensemble are determined by the root multiplicities. In this section we also give some examples of how the connection between random matrix ensembles on the one hand, and symmetric spaces and Calogero–Sutherland models on the other hand, can be used to obtain new results in the theoretical description of physical systems, more precisely in the theory of quantum transport. The last section of the paper is devoted to some new results that show that the mathematical tools discussed in this paper (or suitable generalizations of these) can be useful for going beyond the symmetric space paradigm, and to explore some new connections between random matrix theory, group theory, and diOerential geometry. Here we discuss clustered solutions of the Dorokhov– Mello–Pereyra–Kumar equation, and then we go on to discuss the most general Calogero–Sutherland potential, given by the Weierstrass P-function, and show that it covers the three cases of symmetric spaces of positive, zero and negative curvature. Finally, in the appendix we discuss some intriguing exact results for the so called zonal spherical functions, which not only play an important role in our discussion, but are also of great relevance in several other branches of physics. There are some important and interesting topics that we will not review because of lack of space and competence. For these we refer the reader to the existing literature. In particular we shall not discuss: • the supersymmetric approach to random matrix theories and in particular their classi4cation in terms of supersymmetric spaces. Here we refer the reader to the original paper by Zirnbauer [4], while a good introduction to the use of supersymmetry in random matrix theory and a complete set of the relevant references can be found in [12];
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• the very interesting topic of phase transitions. For this we refer to the recent and thorough review by Cicuta [13]; • the extension to two-dimensional models of the classi4cation of symmetric spaces, and more generally the methods of symmetric space analysis [14]; • the generalization of the classi4cation of symmetric spaces to non-hermitean random matrices [15] (see however a discussion in Section 11); • the so-called q-ensembles [16]; • the two-matrix models [17] and multi-matrix models [18] and their continuum limit generalization. The last item in the list given above is a very interesting topic, which has several physical applications and would indeed deserve a separate review. The common feature of these two- and multi-matrix models which is of relevance for the present review, is that they all can be mapped onto suitably chosen Calogero–Sutherland systems. These models represent a natural link to two classes of matrix theories which are of great importance in high energy physics: on the one hand, the matrix models describing two-dimensional quantum gravity (possibly coupled to matter) [19], and on the other hand, the matrix models pertaining to large N QCD, which trace back to the original seminal works of ’t Hooft [20]. In particular, a direct and explicit connection exists between multi-matrix models (the so called Kazakov–Migdal models) for large N QCD [21] and the exactly solvable models of two-dimensional QCD on the lattice [22]. The mapping of these models to Calogero–Sutherland systems of the type discussed in this review can be found for instance in [23]. The relevance of these models, and in particular of their Calogero– Sutherland mappings, for the condensed matter systems like those discussed in the second part of this review, was 4rst discussed in [24]. A recent review on this aspect, and more generally on the use of Calogero–Sutherland models for low-dimensional models, can be found in [25]. We will necessarily be rather sketchy in discussing the many important physical applications of the random matrix ensembles to be described in Section 8. We refer the reader to some excellent reviews that have appeared in the literature during the last few years: the review by Beenakker [26] for the solid state physics applications, the review by Verbaarschot [27] for QCD-related applications, and [28,29] for extensive reviews including a historical outline. Part I The theory of symmetric spaces has a long history in mathematics. In this 4rst part of the paper we will introduce the reader to some of the most fundamental concepts in the theory of symmetric spaces. We have tried to keep the discussion as simple as possible without assuming any previous familiarity of the reader with symmetric spaces. The review should be particularly accessible to physicists. In the hope of addressing a wider audience, we have almost completely avoided using concepts from diOerential geometry, and we have presented the subject mostly from an algebraic point of view. In addition we have inserted a large number of simple examples in the text, that will hopefully help the reader visualize the ideas. Since our aim in Part II will be to introduce the reader to the application of symmetric spaces in physical integrable systems and random matrix models, we have chosen the background material presented here with this in mind. Therefore we have put emphasis not only on fundamental issues but on subjects that will be relevant in these applications as well. Our treatment
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will be somewhat rigorous; however, we skip proofs that can be found in the mathematical literature and concentrate on simple examples that illustrate the concepts presented. The reader is referred to Helgason’s book [30] for a rigorous treatment; however, this book may not be immediately accessible to physicists. For the reader with little background in diOerential geometry we recommend the book by Gilmore [31] (especially Chapter 9) for an introduction to symmetric spaces of exceptional clarity. In Section 2, after reviewing the basics about Lie groups, we will present some of the most important properties of root systems. In Section 3 we de4ne symmetric spaces and discuss their main characteristics, de4ning involutive automorphisms, spherical decomposition of the group elements, and the metric on the Lie algebra. We also discuss the algebraic structure of the coset space. In Section 4 we show how to obtain all the real forms of a complex semisimple Lie algebra. The same techniques will then be used to classify the real forms of symmetric spaces in Section 5. In this section we also de4ne the curvature of a symmetric space, and discuss triplets of symmetric spaces with positive, zero and negative curvature, all corresponding to the same symmetric subgroup. We will see why curved symmetric spaces arise from semisimple groups, whereas the Qat spaces are associated to nonsemisimple groups. In addition, in Section 5 we will de4ne restricted root systems. The restricted root systems are associated to symmetric spaces, just like ordinary root systems are associated to groups. As we will discuss in detail in Part II of this paper, they are key objects when considering the integrability of Calogero–Sutherland models. In Section 6 we discuss Casimir and Laplace operators on symmetric spaces and mention some known properties of the eigenfunctions of the latter, so called zonal spherical functions. These functions play a prominent role in many physical applications. The introduction to symmetric spaces we present contains the basis for understanding the developments to be discussed in more detail in Part II. The reader already familiar with symmetric spaces is invited to start reading in the last section of Part I, Section 7, where we give a brief introduction to Calogero–Sutherland models. 2. Lie groups and root spaces In this introductory section we de4ne the basic concepts relating to Lie groups. We will build on the material presented here when we discuss symmetric spaces in the next section. The reader with a solid background in group theory may want to skip most or all of this section. 2.1. Lie groups and manifolds A manifold can be thought of as the generalization of a surface, but we do not in general consider it as embedded in a higher-dimensional euclidean space. A short introduction to diOerentiable manifolds can be found in Ref. [32], and a more elaborate one in Refs. [33,34, Chapter III]. The points of an N -dimensional manifold can be labelled by real coordinates (x1 ; : : : ; xN ). Suppose that we take an open set U of this manifold, and we introduce local real coordinates on it. Let be the function that attaches N real coordinates to each point in the open set U . Suppose now that the manifold is covered by overlapping open sets, with local coordinates attached to each of them. If for each pair of open sets U , U , the function ◦ −1 is diOerentiable in the overlap region
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U ∩ U , it means that we can go smoothly from one coordinate system to another in this region. Then the manifold is diOerentiable. Consider a group G acting on a space V . We can think of G as being represented by matrices, and of V as a space of vectors on which these matrices act. A group element g ∈ G transforms the vector v ∈ V into gv = v . If G is a Lie group, it is also a diOerentiable manifold. The fact that a Lie group is a diOerentiable manifold means that for two group elements g, g ∈ G, the product (g; g ) ∈ G × G → gg ∈ G and the inverse g → g−1 are smooth (C ∞ ) mappings, that is, these mappings have continuous derivatives of all orders. Example. The space Rn is a smooth manifold and at the same time an abelian group. The “product” of elements is addition (x; x ) → x + x and the inverse of x is −x. These operations are smooth. Example. The set GL(n; R) of nonsingular real n × n matrices M , det M = 0, with matrix multiplication (M; N ) → MN and multiplicative matrix M → M −1 is a non-abelian group manifold. inverse t i Xi i Any such matrix can be represented as M = e where Xi are generators of the GL(n; R) algebra and t i are real parameters. 2.2. The tangent space In each point of a diOerentiable manifold, we can de4ne the tangent space. If a curve through a point P in the manifold is parametrized by t ∈ R xa (t) = xa (0) + a t;
a = 1; : : : ; N;
(2.1)
where P = (x1 (0); : : : ; xN (0)), then = ( 1 ; : : : ; N ) = (x˙1 (0); : : : ; x˙N (0)) is a tangent vector at P. Here x˙a (0)=d=dt xa (t)|t=0 . The space spanned by all tangent vectors at P is the tangent space. In particular, the tangent vectors to the coordinate curves (the curves obtained by keeping all the coordinates 4xed except one) through P are called the natural basis for the tangent space. Example. In euclidean 3-space the natural basis is {eˆ x ; eˆ y ; eˆ z }. On a patch of the unit 2-sphere parametrized by polar coordinates it is {eˆ ; eˆ ! }. For a Lie group, the tangent space at the origin is spanned by the generators, that play the role of (contravariant) vector 4elds (also called derivations), expressed in local coordinates on the group manifold as X = X a (x)9a (for an introduction to diOerential geometry see Ref. [35, Chapter 5] or [34]). Here the partial derivatives 9a = 9=9xa form a basis for the vector 4eld. That the generators span the tangent space at the origin can easily be seen from the exponential map. Suppose X is a generator of a Lie group. The exponential map then maps X onto etX , where t is a parameter. This mapping is a one-parameter subgroup, and it de4nes a curve x(t) in the group manifold. The tangent vector of this curve at the origin is then d tX e |t=0 = X : (2.2) dt All the generators together span the tangent space at the origin (also called the identity element).
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2.3. Coset spaces The isotropy subgroup Gv0 of a group G at the point v0 ∈ V is the subset of group elements that leave v0 4xed. The set of points that can be reached by applying elements g ∈ G to v0 is the orbit of G at v0 , denoted Gv0 . If Gv0 = V for one point v0 , then this is true for every v ∈ V . We then say that G acts transitively on V . In general, a symmetric space can be represented as a coset space. Suppose H is a subgroup of a Lie group G. The coset space G=H is the set of subsets of G of the form gH , for g ∈ G. G acts on this coset space: g1 (gH ) is the coset (g1 g)H . We will refer to the elements of the coset space by g instead of by gH , when the subgroup H is understood from the context, because of the natural mapping described in the next paragraph. If g ∈ H , gH corresponds to a point on the manifold G=H away from the origin, whereas hH = H (h ∈ H ) is the identity element identi4ed with the origin of the symmetric space. This point is the north pole in the example below. If G acts transitively on V , then V = Gv for any v ∈ V . Since the isotropy subgroup Gv0 leaves a 4xed point v0 invariant, gGv0 v0 = gv0 = v ∈ V , we see that the action of the group G on V de4nes a bijective action of elements of G=Gv0 on V . Therefore the space V on which G acts transitively, can be identi4ed with G=Gv0 , since there is one-to-one correspondence between the elements of V and the elements of G=Gv0 . There is a natural mapping from the group element g onto the point gv0 on the manifold. Example. The SO(2) subgroup of SO(3) is the isotropy subgroup at the north pole of a unit 2-sphere imbedded in three-dimensional space, since it keeps the north pole 4xed. On the other hand, the north pole is mapped onto any point on the surface of the sphere by elements of the coset SO(3)=SO(2). This can be seen from the explicit form of the coset representatives. As we will see in Eq. (3.20) in Section 3.5, the general form of the elements of the coset is 0 C I2 − XX T X ; = M = exp (2.3) T T −C T 0 −X 1−X X where C is the matrix 2 t C= t1
(2.4)
and t 1 , t 2 are real coordinates. I2 in Eq. (2.3) is the 2 × 2 unit matrix. For the coset space SO(3)=SO(2), M is equal to 0 0 0 0 0 1 2 1 1 0 0 1 0 0 0 t i Li ; L1 = M = exp ; L2 = (2.5) : 2 2 i=1 0 −1 0 −1 0 0 The third SO(3) generator 0 1 0 1 −1 0 0 L3 = 2 0 0 0
(2.6)
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spans the algebra of the stability subgroup SO(2), that keeps the north pole 4xed: 0 0 3 exp(t L3 ) 0=0 : 1 1
49
(2.7)
The generators Li (i = 1; 2; 3) satisfy the SO(3) commutation relations [Li ; Lj ] = 12 jijk Lk . Note that since the Li and the t i are real, C † = C T . In (2.3), M is a general representative of the coset SO(3)=SO(2). By expanding the exponential we see that the explicit form of M is 1 2 2 2 1 2 2 2 1 2 2 2 2 2 cos (t ) + (t ) − 1 1 2 cos (t ) + (t ) − 1 2 sin (t ) + (t ) t t t 1 + (t ) (t 1 )2 + (t 2 )2 (t 1 )2 + (t 2 )2 (t 1 )2 + (t 2 )2 1 2 2 2 1 2 2 2 1 2 2 2 1 2 cos (t ) + (t ) − 1 1 2 cos (t ) + (t ) − 1 1 sin (t ) + (t ) M = t t 1 + (t ) t : 1 )2 + (t 2 )2 (t 1 )2 + (t 2 )2 (t 1 )2 + (t 2 )2 (t 1 )2 + (t 2 )2 1 )2 + (t 2 )2 sin (t sin (t 2 1 −t −t cos (t 1 )2 + (t 2 )2 (t 1 )2 + (t 2 )2 (t 1 )2 + (t 2 )2 (2.8) Thus the matrix X = yx is given in terms of the components of C by (cf. Eq. (3.21)): 1 2 2 2 2 sin (t ) + (t ) t x (t 1 )2 + (t 2 )2 : X= (2.9) = 1 2 2 2 y 1 sin (t ) + (t ) t (t 1 )2 + (t 2 )2 De4ning now z = cos (t 1 )2 + (t 2 )2 , we see that the variables x, y, z satisfy the equation of the 2-sphere: x2 + y2 + z 2 = 1 : When the coset space of the 2-sphere: 0 : M 0=: 1 :
(2.10)
representative M acts on the north pole it is easily seen that the orbit is all : : :
0 x y 0 = y : z 1 z x
(2.11)
This shows that there is one-to-one correspondence between the elements of the coset and the points of the 2-sphere. The coset SO(3)=SO(2) can therefore be identi4ed with a unit 2-sphere imbedded in three-dimensional space.
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2.4. The Lie algebra and the adjoint representation A Lie algebra G is a vector space over a 4eld F. Multiplication in the Lie algebra is given by the bracket [X; Y ]. It has the following properties: (1) (2) (3) (4)
If X; Y ∈ G, then [X; Y ] ∈ G, [X; Y + Z] = [X; Y ] + [X; Z] for , ∈ F, [X; Y ] = −[Y; X ], [X; [Y; Z]] + [Y; [Z; X ]] + [Z; [X; Y ]] = 0 (the Jacobi identity).
The algebra G generates a group through the exponential mapping. A general group element is t i Xi ; t i ∈ F; Xi ∈ G : (2.12) M = exp i
We de4ne a mapping ad X from the Lie algebra to itself by ad X : Y → [X; Y ]. The mapping X → ad X is a representation of the Lie algebra called the adjoint representation. It is easy to check that it is an automorphism: it follows from the Jacobi identity that [ad Xi ; ad Xj ]=ad [Xi ; Xj ]. Suppose we choose a basis {Xi } for G. Then ad Xi (Xj ) = [Xi ; Xj ] = Cijk Xk ;
(2.13)
where we sum over k. The Cijk are called structure constants. Under a change of basis, they transform as mixed tensor components. They de4ne the matrix (Mi )jk = Cikj associated with the adjoint representation of Xi . One can show that there exists a basis for any complex semisimple algebra in which the structure constants are real. This means the adjoint representation is real. Note that the dimension of the adjoint representation is equal to the dimension of the group. Example. Let us construct the adjoint representation of SU (2). The generators in the de4ning representation are 0 1 0 −i 1 1 0 1 J3 = ; J± = ±i (2.14) 2 0 −1 2 1 0 i 0 and the commutation relations are [J3 ; J± ] = ±J± ;
[J+ ; J− ] = 2J3 :
(2.15)
− + + 3 3 The structure constants are therefore C3+ = −C+3 = −C3−− = C− 3 = 1, C+− = −C−+ = 2 and the adjoint representation is given by (M3 )++ = 1, (M3 )− − = −1, (M+ )+3 = −1, (M+ )3− = 2, (M− )−3 = 1, (M− )3+ = −2, and all other matrix elements equal to 0: 0 0 0 0 0 2 0 −2 0 M3 = (2.16) 0 1 0 ; M+ = −1 0 0 ; M− = 0 0 0 : 0 0 −1 0 0 0 1 0 0
These representation matrices are real, have the same dimension as the group, and satisfy the SU (2) commutation relations [M3 ; M± ] = ±M± , [M+ ; M− ] = 2M3 .
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2.5. Semisimple algebras and root spaces In this paragraph we will brieQy recall the basic facts about root spaces and the classi4cation of complex simple Lie algebras, to set the stage for our discussion of real forms of Lie algebras and 4nally symmetric spaces. An ideal, or invariant subalgebra I is a subalgebra such that [G; I] ⊂ I. An abelian ideal also satis4es [I; I] = 0. A simple Lie algebra has no proper ideal. A semisimple Lie algebra is the direct sum of simple algebras, and has no proper abelian ideal (by proper we mean diOerent from {0}). A Lie algebra is a linear vector space over a 4eld F, with an antisymmetric product de4ned by the Lie bracket (cf. Section 2.4). If F is the 4eld of real, complex or quaternion numbers, the Lie algebra is called a real, complex or quaternion algebra. A complexi4cation of a real Lie algebra is obtained by taking linear combinations of its elements with complex coeVcients. A real Lie algebra H is a real form of the complex algebra G if G is the complexi4cation of H. In any simple algebra there are two kinds of generators: there is a maximal abelian subalgebra, called the Cartan subalgebra H0 = {H1 ; : : : ; Hr }; [Hi ; Hj ] = 0 for any two elements of the Cartan subalgebra. There are also raising and lowering operators denoted E . is an r-dimensional vector = ( 1 ; : : : ; r ) and r is the rank of the algebra. 1 The latter are eigenoperators of the Hi in the adjoint representation belonging to eigenvalue i : [Hi ; E ] = i E . For each eigenvalue, or root i , there is another eigenvalue − i and a corresponding eigenoperator E− under the action of Hi . Suppose we represent each element of the Lie algebra by an n × n matrix. Then [Hi ; Hj ] = 0 means the matrices Hi can all be diagonalized simultaneously. Their eigenvalues -i are given by Hi |- = -i |-, where the eigenvectors are labelled by the weight vectors - = (-1 ; : : : ; -r ) [36]. A weight whose 4rst non-zero component is positive is called a positive weight. Also, a weight - is greater than another weight - if - − - is positive. Thus we can de4ne the highest weight as the one which is greater than all the others. The highest weight is unique in any representation. The roots i ≡ (Hi ) of the algebra G are the weights of the adjoint representation. Recall that in the adjoint representation, the states on which the generators act are de4ned by the generators themselves, and the action is de4ned by Xa |Xb ≡ ad Xa (Xb ) ≡ [Xa ; Xb ] :
(2.17)
The roots are functionals on the Cartan subalgebra satisfying ad Hi (E ) = [Hi ; E ] = (Hi )E ;
(2.18)
where Hi is in the Cartan subalgebra. The eigenvectors E are called the root vectors. These are exactly the raising and lowering operators E± for the weight vectors -. There are canonical commutation relations de4ning the system of roots belonging to each simple rank r-algebra. These are summarized below: 2 [Hi ; Hj ] = 0;
[Hi ; E ] = i E ;
[E ; E− ] = i Hi :
(2.19)
1 The rank of an algebra is de4ned through the secular equation (see Section 6.1). For a non-semisimple algebra, the maximal number of mutually commuting generators can be greater than the rank of the algebra. 2 For the reader who wants to understand more about the origin of the structure of Lie algebras, we recommend Chapter 7 of Gilmore [31].
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One can prove the fundamental relation [35,36] 2 · = −(p − q) ; 2
(2.20)
where is a root, - is a weight, and p, q are positive integers such that E |- + p = 0, E− |- − q = 0. 3 This relation gives rise to the strict properties of root lattices, and permits the complete classi4cation of all the complex (semi)simple algebras. Eq. (2.20) is true for any representation, but has particularly strong implications for the adjoint representation. In this case - is a root. As a consequence of Eq. (2.20), the possible angle between two root vectors of a simple Lie algebra is limited to a few values: these turn out to be multiples of /=6 and /=4 (see e.g. [36, Chapter VI]). The root lattice is invariant under reQections in the hyperplanes orthogonal to the roots (the Weyl group). As we will shortly see, this is true not only for the root lattice, but for the weight lattice of any representation. Note that the roots are real-valued linear functionals on the Cartan subalgebra. Therefore they are in the space dual to H0 . A subset of the positive roots span the root lattice. These are called simple roots. Obviously, since the roots are in the space dual to H0 , the number of simple roots is equal to the rank of the algebra. The same relation (2.20) determines the highest weights of all irreducible representations. Setting p = 0, choosing a positive integer q, and letting run through the simple roots, = i (i = 1; : : : ; r), we 4nd the highest weights -i of all the irreducible representations corresponding to the given value of q [36]. For example, for q = 1 we get the highest weights of the r fundamental representations of the group, each corresponding to a simple root i . For higher values of q we get the highest weights of higher-dimensional representations of the same group. The set of all possible simple root systems are classi4ed by means of Dynkin diagrams, each of which correspond to an equivalence class of isomorphic Lie algebras. The classical Lie algebras SU(n + 1; C), SO(2n + 1; C), Sp(2n; C) and SO(2n; C) correspond to root systems An , Bn , Cn , and Dn , respectively. In addition there are 4ve exceptional algebras corresponding to root systems E6 , E7 , E8 , F4 and G2 . Each of these complex algebras in general has several real forms associated with it (see Section 4). These real forms correspond to the same Dynkin diagram and root system as the complex algebra. Since we will not make reference to Dynkin diagrams in the following, we will not discuss them here. The interested reader can 4nd suVcient material for example in the book by Georgi [36]. 3
Here the scalar product · can be de4ned in terms of the metric on the Lie algebra. For the adjoint representation, - is a root and 2K(H ; H ) 2 · 2(H ) = ≡ ; 2 K(H ; H ) (H )
(2.21)
where K denotes the Killing form (see Section 3.4). There is always a unique element H in the algebra such that K(H; H ) = (H ) for each H ∈ H0 (see for example [35, Chapter 10]). In general for a linear form - on the Lie algebra, 2 · 2-(H ) = : 2 (H )
(2.22)
Then - is a highest weight for some representation if and only if this expression is an integer for each positive root .
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The (semi)simple complex algebra G decomposes into a direct sum of root spaces [35]: G = H0 ⊕ G ;
53
(2.23)
where G is generated by {E± }. This will be evident in the example given below. Example. The root system An−1 corresponds to the complex Lie algebra SL(n; C) and all its real forms. In a later section we will see how to construct all the real forms associated with a given complex Lie algebra. Let us see here explicitly how to construct the root lattice of SU(3; C), which is one of the real forms of SL(3; C). The generators are determined by the commutation relations. In physics it is common to write the commutation relations in the form [Ti ; Tj ] = ifijk Tk
(2.24)
(an alternative form is to de4ne the generators as Xi = iTi and write the commutation relations as [Xi ; Xj ] = −fijk Xk ) where fijk are structure constants for the algebra SU(3; C). a Using the notation g = eit Ta for the group elements (with t a real and a sum over a implied), the generators Ta in the fundamental representation of this group are hermitean: 4 0 1 0 0 −i 0 1 0 0 1 1 1 1 0 0 i 0 0 0 −1 0 T1 = ; T2 = ; T3 = ; 2 2 2 0 0 0 0 0 0 0 0 0 0 0 1 0 0 −i 0 0 0 1 ; T5 = 1 0 0 0 ; T6 = 1 0 0 1 ; 0 0 0 T4 = 2 2 2 1 0 0 i 0 0 0 1 0 0 0 0 1 0 0 1 1 0 1 0 : ; T8 = √ 0 0 −i (2.25) T7 = 2 2 3 0 i 0 0 0 −2 In high energy physics the matrices 2Ta are known as Gell–Mann matrices. The generators are normalized in such a way that tr (Ta Tb ) = 12 6ab . Note that T1 , T2 , T3 form an SU(2; C) subalgebra. We take the Cartan subalgebra to be H0 = {T3 ; T8 }. The rank of this group is r = 2. 4
Note that we have written an explicit factor of i in front of the generators in the expression for the group elements. This is often done for compact groups; since the Killing form (Section 3.4) has to be negative de4nite, the coordinates of the algebra spanned by the generators must be purely imaginary. Here we use this notation because it is conventional. If we absorb the factor of i into the generators, we get antihermitean matrices Xa = iTa ; we will do this in the example in Section 3.1 to comply with Eq. (3.1). Of course, the matrices in the algebra are always antihermitean.
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Let us 4rst 4nd the weight vectors of the fundamental representation. To this end we look for the eigenvalues -i of the operators in the abelian subalgebra H0 : 1 1 1 1 1 1 T3 (2.26) 0 = 2 0 ; T8 0 = 2√3 0 ; 0 0 0 0 therefore the eigenvector (1 0 0)T corresponds to the state |- where 1 1 ; √ - ≡ (-1 ; -2 ) = 2 2 3
(2.27)
is distinguished by its eigenvalues under the operators Hi of the Cartan subalgebra. In the same way we 4nd that (0 1 0)T and (0 0 1)T correspond to the states labelled by weight vectors 1 1 1 - = − ; √ ; - = 0; − √ ; (2.28) 2 2 3 3 respectively. -, - , and - are the weights of the fundamental representation 7 = D and they form an equilateral triangle in the plane. The highest weight of the representation D is - = 12 ; 2√1 3 . There is also another fundamental representation DW of the algebra SU(3; C), since it generates a group of rank 2. Indeed, from Eq. (2.20), for p = 0, q = 1, there is one highest weight -i , and one fundamental representation, for each simple root i . The highest weight -W of the representation DW is 1 1 ;− √ -W = : (2.29) 2 2 3 The highest weights of the representations corresponding to any positive integer q can be obtained as soon as we know the simple roots. Then, by operating with lowering operators on this weight, we obtain other weights, on which we can further operate with lowering operators until we have obtained all the weights in the representation. For an example of this procedure see [36, Chapter IX]. Let us see now how to obtain the roots of SU(3; C). Each root vector E corresponds to either a raising or a lowering operator: E is the eigenvector belonging to the root i ≡ (Hi ) under the adjoint representation of Hi , like in Eq. (2.32). Each raising or lowering operator is a linear combination of generators Ti that takes one state of the fundamental representation to another state of the same representation: E± |- = N± ; - |- ± . Therefore the root vectors will be diOerences of weight vectors in the fundamental representation. We 4nd the raising and lowering operators E± to be E±(1; 0) = E 1
√1
√ 3 ± 2; 2
E
2
(T1 ± iT2 ) ;
=
√ 3 1 ± −2; 2
√1
=
2
(T4 ± iT5 ) ;
√1
2
(T6 ± iT7 ) :
(2.30)
These generate the subspaces G in Eq. (2.23). In the fundamental representation, we 4nd using the Gell–Mann matrices that these are matrices with only one non-zero element. For example,
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the raising operator E that 0 1 1 0 0 E+(1; 0) = √ 2 0 0
corresponds to the root = (1; 0) is 0 0 (2.31) : 0 This operator takes us from the state |- = − 12 ; 2√1 3 to the state |- = 12 ; 2√1 3 . The components of the root vectors of SU(3; C) are the eigenvalues i of these under the adjoint representation of the Cartan subalgebra. That is, Hi |E ≡ ad Hi (E ) ≡ [Hi ; E ] = i |E :
(2.32)
This way we easily 4nd the roots: we can either explicitly use the structure constants of SU (3) c Tc (note the explicit factor of i due to our conventions regarding the in [Ta ; Tb ] = ifabc Tc = −iCab generators) or we can use an explicit representation for Hi , E like in Eqs. (2.25), (2.30), (2.31), to calculate the commutators: ad H1 (E±(1; 0) ) = [H1 ; E±(1; 0) ] = T3 ; √12 (T1 ± iT2 ) = √12 (iT2 ± T1 ) = ±E±(1; 0) ≡ 1± E±(1; 0) ; ad H2 (E±(1; 0) ) = [H2 ; E±(1; 0) ] = T8 ; √12 (T1 ± iT2 ) = 0 ≡ 2± E±(1; 0) :
(2.33)
The root vector corresponding to the raising operator E+(1; 0) is thus = ( 1+ ; 2+ ) = (1; 0) and the − − root vector corresponding to the lowering operator E−(1; 0) is − = These root ( 1 ; 2 ) = (−1; 0). 1 √ 1 vectors are indeed the diOerences between the weight vectors - = 2 ; 2 3 and - = − 12 ; 2√1 3 of the fundamental representation. √ √ In the same way we 4nd the other root vectors ± 12 ; ± 23 , ∓ 12 ; ± 23 , and (0; 0) (with multiplicity 2), by operating with H1 and H2 on the remaining E± ’s and on the Hi ’s. The last root with multiplicity 2 has as its components the eigenvalues under H1 , H2 of the states |H1 and |H2 : Hi |Hj = [Hi ; Hj ] = 0; i, j ∈ {1;
2}. The root vectors
form a regular hexagon in the plane. The positive √
√
roots are (1; 0), 1 = 12 ; 23 and 2 = 12 ; − 23 . The latter two are simple roots. (1; 0) is not simple because it is the sum of the other positive roots. There are two simple roots, since the rank of SU (3) is 2 and the root lattice is two-dimensional. The root lattice of SU (3) is invariant under reQections in the hyperplanes orthogonal to the root vectors. This is true of any weight or root lattice; the symmetry group of reQections in hyperplanes orthogonal to the roots is called the Weyl group. It is obtained from Eq. (2.20): since for any root and any weight -, 2( · -)= 2 is the integer q − p, 2( · -) - = - − (2.34) 2 is also a weight. Eq. (2.34) is exactly the above mentioned reQection, as can easily be seen. 2.6. The Weyl chambers The roots are linear functionals on the Cartan subalgebra. We may denote the Cartan subalgebra by H0 and its dual space by H0∗ . A Weyl reQection like the one in (2.34) can be de4ned not only
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for the weights or roots - in the space H0∗ , but for an arbitrary vector q ∈ H0∗ or, in all generality, for a vector q in an arbitrary 4nite-dimensional vector space: s (q) = q − ∗ (q) :
(2.35)
Note that q ∈ H0∗ is in the space dual to H0 and may denote a root. In (2.35) the function ∗ (q) is a linear functional on H0∗ such that ∗ ( ) = 2. We will be concerned only with the crystallographic case when ∗ (q) is integer. We denote the hyperplanes in H0∗ where the function ∗ (q) vanishes by H ( ) : H ( ) = {q ∈ H0∗ : ∗ (q) = 0} ;
(2.36)
( )
where H is orthogonal to the root , and s (q) is a reQection in this hyperplane. By identifying the dual spaces H0 and H0∗ (this is possible since they have the same dimension), we can consider hyperplanes like the ones in (2.36) in the space H0 . The role of the linear functional ∗ (q) is then played by 2q · 2q(H ) ; (2.37) ∗ (q) = = 2 (H ) where (H ) = K(H ; H ). Here K is the Killing form (a metric on the algebra to be de4ned in Section 3.4) and H is the unique element in H0 such that K(H; H ) = (H ). The open subsets of H0 where roots are nonzero are called Weyl chambers. Consequently, the walls of the Weyl chambers are the hyperplanes in H0 where the roots q(H ) are zero. 2.7. The simple root systems We have just shown by an example, in Section 2.5, how to obtain a root system of type An . In general, for any simple algebra the commutation relations determine the Cartan subalgebra and raising and lowering operators, that in turn determine a unique root system, and correspond to a given Dynkin diagram. In this way we can classify all the simple algebras according to the type of root system it possesses. The root systems for the four in4nite series of classical nonexceptional Lie groups can be characterized as follows [36] (denote the r-dimensional space spanned by the roots by V and let {e1 ; : : : ; en } be a canonical basis in Rn ): An−1 : Let V be the hyperplane in Rn that passes through the points (1; 0; 0; : : : ; 0); (0; 1; 0; : : : ; 0); : : : ; (0; 0; : : : ; 0; 1) (the endpoints of the ei , i = 1; : : : ; n). Then the root lattice contains the vectors {ei − ej ; i = j}. Bn : Let V be Rn ; then the roots are {±ei ; ±ei ± ej ; i = j}. Cn : Let V be Rn ; then the roots are {±2ei ; ±ei ± ej ; i = j}. Dn : Let V be Rn ; then the roots are {±ei ± ej ; i = j}. The root lattice BCn , that we will discuss in conjunction with restricted root systems, is the union of Bn and Cn . It is characterized as follows: BCn : Let V be Rn ; then the roots are {±ei ; ±2ei ; ±ei ± ej ; i = j}. Because this system contains both ei and 2ei , it is called non-reduced (normally the only root collinear with is − ). However, it is irreducible in the usual sense, which means it is not the direct sum of two disjoint root systems Bn and Cn . This can be seen from the root multiplicities (cf. Table 1).
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The semisimple algebras are direct sums of simple ones. That means the simple constituent algebras commute with each other, and the root systems are direct sums of the corresponding simple root systems. Therefore, knowing the properties of the simple Lie algebras, we also know the semisimple ones.
3. Symmetric spaces In the previous section, we have reminded ourselves of some elementary facts concerning root spaces and the classi4cation of the complex semisimple algebras. In this section we will de4ne and discuss symmetric spaces. A symmetric space is associated to an involutive automorphism of a given Lie algebra. As we will see, several diOerent involutive automorphisms can act on the same algebra. Therefore we normally have several diOerent symmetric spaces deriving from the same Lie algebra. The involutive automorphism de4nes a symmetric subalgebra and a remaining complementary subspace of the algebra. Under general conditions, the complementary subspace is mapped onto a symmetric space through the exponential map. In the following subsections we make these statements more precise. We discuss how the elements of the Lie group can act as transformations on the elements of the symmetric space. This naturally leads to the de4nition of two coordinate systems on symmetric spaces: the spherical and the horospheric coordinate systems. The radial coordinates associated to each element of a symmetric space through its spherical or horospheric decomposition will be of relevance when we discuss the radial parts of diOerential operators on symmetric spaces in Section 6. In the same section we explain why these operators are important in applications to physical problems, and in Part II we will discuss some of their uses. In all of this paper we will distinguish between compact and non-compact symmetric spaces. In order to give a precise notion of compactness, we will de4ne the metric tensor on a Lie algebra in terms of the Killing form in Section 3.4. The latter is de4ned as a symmetric bilinear trace form on the adjoint representation, and is therefore expressible in terms of the structure constants. We will give several examples of Killing forms later, as we discuss the various real forms of a Lie algebra. The metric tensor serves to de4ne the curvature tensor on a symmetric space (Section 5.1). It is also needed in computing the Jacobian of the transformation to radial coordinates. This Jacobian is relevant in calculating the radial part of the Laplace–Beltrami operator (see Section 6.2). We will close this section with a discussion of the general algebraic form of coset representatives in Section 3.5. 3.1. Involutive automorphisms An automorphism of a Lie algebra G is a mapping from G onto itself such that it preserves the algebraic operations on the Lie algebra. For example, if : is an automorphism, it preserves multiplication: [:(X ); :(Y )] = :([X; Y ]), for X , Y ∈ G. Suppose that the linear automorphism : : G → G is such that :2 = 1, but : is not the identity. That means that : has eigenvalues ±1, and it splits the algebra G into orthogonal eigensubspaces corresponding to these eigenvalues. Such a mapping is called an involutive automorphism.
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Suppose now that G is a compact simple Lie algebra, : is an involutive automorphism of G, and G = K ⊕ P where :(X ) = X
for X ∈ K;
:(X ) = −X
for X ∈ P :
(3.1)
From the properties of automorphisms mentioned above, it is easy to see that K is a subalgebra, but P is not. In fact, the commutation relations [K; K] ⊂ K;
[K; P] ⊂ P;
[P; P] ⊂ K
(3.2)
hold. A subalgebra K satisfying (3.2) is called a symmetric subalgebra. If we now multiply the elements in P by i (the “Weyl unitary trick”), we construct a new noncompact algebra G∗ = K ⊕ iP. This is called a Cartan decomposition, and K is a maximal compact subalgebra of G∗ . The coset spaces G=K and G ∗ =K are symmetric spaces. Example. Suppose G = SU (n; C), the group of unitary complex matrices with determinant +1. The algebra of this group then consists of complex antihermitean 5 matrices of zero trace (this follows by diOerentiating the identities UU † = 1 and det U = 1 with respect to t where U (t) is a curve passing a through the identity at t = 0); a group element is written as g = et Xa with t a real. Therefore any matrix X in the Lie algebra of this group can be written X =A+iB, where A is real, skew-symmetric, and traceless and B is real, symmetric and traceless. This means the algebra can be decomposed as G=K⊕P, where K is the compact connected subalgebra SO(n; R) consisting of real, skew-symmetric and traceless matrices, and P is the subspace of matrices of the form iB, where B is real, symmetric, and traceless. P is not a subalgebra. Referring to the example for SU(3; C) in Section 2.5 we see, setting Xa = iTa , that the {Xa } split into two sets under the involutive automorphism : de4ned by complex conjugation : = K. This splits the compact algebra G into K ⊕ P, since P consists of imaginary matrices: 0 1 0 0 0 1 0 0 0 1 1 1 −1 0 0 0 0 0 0 0 1 K = {X2 ; X5 ; X7 } = ; 2 ; 2 ; 2 0 0 0 −1 0 0 0 −1 0 P = {X1 ; X3 ; X4 ; X6 ; X8 } 0 1 0 1 i i 1 0 0; 0 = 2 2 0 0 0 0
1
i √ 0 2 3 0 5
0 1 0
0 ; −2
0 −1 0
0
0
i 0 ; 2 0 0 1
0 0 0
1
0
i 0 ; 2 0 0 0
0
0
0
1 ;
1
0
0
See the footnote in Section 2.5.
(3.3)
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K spans the real subalgebra SO(3; R). Setting X2 ≡ L3 , X5 ≡ L2 , X7 ≡ L1 , the commutation relations for the subalgebra are [Li ; Lj ] = 12 jijk Lk . The Cartan subalgebra iH0 = {X3 ; X8 } is here entirely in the subspace P. Going back to the general case of G = SU(n; C), we obtain from G by the Weyl unitary trick the non-compact algebra G∗ = K ⊕ iP. iP is now the subspace of real, symmetric, and traceless matrices B. The Lie algebra G∗ = SL(n; R) is then the set of n × n real matrices of zero trace, and generates the linear group of transformations represented by real n × n matrices of unit determinant. The involutive automorphism that split the algebra G above was de4ned to be complex conjugation T −1 : =K. The involutive automorphism that splits G∗ is de4ned by :(g)=(g ˜ ) for g ∈ G ∗ , as we will now see. On the level of the algebra, :(g) ˜ = (gT )−1 means :(X ˜ ) = −X T . Suppose now g = etX ∈ G ∗ with X real and traceless and t a real parameter. If now X is an element of the subalgebra K, we then have :(X ˜ ) = +X , i.e. −X T = X and X is skew-symmetric. If instead X ∈ iP, we have T :(X ˜ ) = −X = −X , i.e. X is symmetric. The decomposition G∗ = K ⊕ iP is the usual decomposition of a SL(n; R) matrix in symmetric and skew-symmetric parts. G=K = SU (n; C)=SO(n; R) is a symmetric space of compact type, and the related symmetric space of non-compact type is G ∗ =K = SL(n; R)=SO(n; R). 3.2. The action of the group on the symmetric space Let G be a semisimple Lie group and K a compact symmetric subgroup. As we saw in the preceding paragraph, the coset spaces G=K and G ∗ =K represent symmetric spaces. Just as we have de4ned a Cartan subalgebra and the rank of a Lie algebra, we can de4ne, in an exactly analogous way, a Cartan subalgebra and the rank of a symmetric space. A Cartan subalgebra of a symmetric space is a maximal abelian subalgebra of the subspace P (see Section 5.2), and the rank of a symmetric space is the number of generators in this subalgebra. If G is connected and G = K ⊕ P where K is a compact symmetric subalgebra, then each group element can be decomposed as g = kp (right coset decomposition) or g = pk (left coset decomposition), with k ∈ K = eK , p ∈ P = eP . P is not a subgroup, unless it is abelian and coincides with its Cartan subalgebra. However, if the involutive automorphism that splits the algebra is denoted :, one can show [37, Chapter 6] that gp:(g−1 ) ∈ P. This de4nes G as a transformation group on P. Since :(k −1 ) = k −1 for k ∈ K, this means p = kpk −1 ∈ P
(3.4)
if k ∈ K, p ∈ P. Now suppose there are no other elements in G that satisfy :(g) = g than those in K. This will happen if the set of elements satisfying :(g) = g is connected. Then P is isomorphic to G=K. Also, G acts transitively on P in the manner de4ned above (cf. Section 2.3). The tangent space of G=K at the origin (identity element) is spanned by the subspace P of the algebra. 3.3. Radial coordinates In this paragraph we de4ne two coordinate systems frequently used on symmetric spaces. Let G = K ⊕ P be a Cartan decomposition of a semisimple algebra and let H0 ⊂ P be a maximal abelian
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subalgebra in the subspace P. De4ne M to be the subgroup of elements in K such that M = {k ∈ K: kHk −1 = H; H ∈ H0 } :
(3.5)
This set is called the centralizer of H0 in K. Under conjugation by k ∈ K, each element H of the Cartan subalgebra is preserved. Further, denote M = {k ∈ K: kHk −1 = H ; H; H ∈ H0 } :
(3.6)
This is a larger subgroup than M that preserves the Cartan subalgebra as a whole, but not necessarily each element separately, and is called the normalizer of H0 in K. If K is a compact symmetric subgroup of G, one can show [37, Chapter 6] that every element p of P G=K is conjugated with some element h = eH for some H ∈ H0 by means of the adjoint representation 6 of the stationary subgroup K: p = khk −1 = kh:(k −1 ) ;
(3.8)
where k ∈ K=M and H is de4ned up to the elements in the factor group M =M . This factor group coincides with the Weyl group that was de4ned in Eq. (2.34): since the space H0 can be identi4ed with its dual space H0∗ , we can identify M =M with the Weyl group of the restricted root system (see Section 5.2). The eOect of the Weyl group is to transform the algebra H0 ⊂ P into another Cartan subalgebra H0 ⊂ P conjugate with the original one. This amounts to a permutation of the roots of the restricted root lattice corresponding to a Weyl reQection. Eq. (3.8) means that every element g ∈ G can be decomposed as g = pk = k hk −1 k = k hk , and this is very much like the Euler angle decomposition of SO(n). Thus, if x0 is the 4xed point of the subgroup K, an arbitrary point x ∈ P can be written x = khk −1 x0 = khx0 :
(3.9)
The coordinates (k(x); h(x)) are called spherical coordinates. k(x) is the angular coordinate and h(x) is the spherical radial coordinate of the point x. Eq. (3.8) de4nes the so-called spherical decomposition of the elements in the coset space. Of course, a similar reasoning is true for the space P ∗ G ∗ =K. This means every matrix p in the coset space G=K can be diagonalized by a similarity transformation by the subgroup K, and the radial coordinates are exactly the set of eigenvalues of the matrix p. These “eigenvalues” are not necessarily real numbers. This is easily seen in the example in Eq. (3.3). It can also be seen in the adjoint representation. Suppose the algebra G = K ⊕ P is
6
Note that eK H e−K = ead K H ≡
∞ (ad K)n H : n! n=0
(3.7)
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compact. From Eq. (2.13), in the adjoint representation Hi ∈ H0 has the form 0 ::: .. : . : 0 i ; Hi = − i .. . ;i −;i
61
(3.10)
where the matrix is determined by the structure constants ([Hi ; Hj ] = 0, [Hi ; E± ] = ± i E± : : : and ± i ; : : : ; ±;i are the roots corresponding to Hi ). Since the Killing form must be negative (see Section 3.4) for a compact algebra, the coordinates of the Cartan subalgebra must be purely imaginary and the group elements corresponding to H0 must have the form 1 ::: .. . : : 1 it · H (3.11) = e eit · .. . e−it · with t = (t 1 ; t 2 ; : : : t r ) and t i real parameters. In particular, if the eigenvalues are real for p ∈ P ∗ , they are complex numbers for p ∈ P. Example. In the example we gave in the preceding subsection, the coset space G ∗ =K = SL(n; R)= SO(n) P ∗ = eiP consists of real positive-de4nite symmetric matrices. Note that G = K ⊕ P implies that G can be decomposed as G =PK and G ∗ as G ∗ =P ∗ K. The decomposition G ∗ =P ∗ K in this case is the decomposition of a SL(n; R) matrix in a positive-de4nite symmetric matrix and an orthogonal one. Each positive-de4nite symmetric matrix can be further decomposed: it can be diagonalized by an SO(n) similarity transformation. This is the content of Eq. (3.8) for this case, and we know it to be true from linear algebra. Similarly, according to Eq. (3.8) the complex symmetric matrices in G=K = SU (n; C)=SO(n) P = eP can be diagonalized by the group K = SO(n) to a form where the eigenvalues are similar to those in Eq. (3.11). In terms of the subspace P of the algebra, Eq. (3.8) amounts to saying that any two Cartan subalgebras H0 , H0 of the symmetric space are conjugate under a similarity transformation by K,
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and we can choose the Cartan subalgebra in any way we please. However, the number of elements that we can diagonalize simultaneously will always be equal to the rank of the symmetric space. There is also another coordinate system valid only for spaces of the type P ∗ ∼ G ∗ =K. This coordinate system is called horospheric and is based on the so called Iwasawa decomposition [37] of the algebra: G = N + ⊕ H0 ⊕ K :
(3.12)
Here K; H0 ; N+ are three subalgebras of G. K is a maximal compact subalgebra, H0 is a Cartan subalgebra, and N+ = G (3.13) ∈ R+
is an algebra of raising operators corresponding to the positive roots (H ) ¿ 0 with respect to H0 (G is the space generated by E ). As a consequence, the group elements can be decomposed g=nhk, in an obvious notation. This means that if x0 is the 4xed point of K, any point x ∈ G ∗ =K can be written as x = nhkx0 = nhx0 :
(3.14)
The coordinates (n(x); h(x)) are called horospheric coordinates and the element h = h(x) is called the horospheric projection of the point x or the horospheric radial coordinate. 3.4. The metric on a Lie algebra A metric tensor can be de4ned on a Lie algebra [30,31,35,37]. For our purposes, it will eventually serve to de4ne the curvature of a symmetric space and be useful in computing the Jacobian of the transformation to radial coordinates. In Sections 6 and 8 we will see the importance of this Jacobian in physical applications in connection with the radial part of the Laplace–Beltrami operator. If {Xi } form a basis for the Lie algebra G, the metric tensor is de4ned by gij = K(Xi ; Xj ) ≡ tr(ad Xi ad Xj ) = Cisr Cjrs :
(3.15)
The symmetric bilinear form K(Xi ; Xj ) is called the Killing form. It is intrinsically associated with the Lie algebra, and since the Lie bracket is invariant under automorphisms of the algebra, so is the Killing form. Example. The generators X7 ≡ L1 , X5 ≡ L2 , X2 ≡ L3 of SO(3) given in Eq. (3.3) obey the commutation relations [Li ; Lj ] = Cijk Lk = 12 jijk Lk . From Eq. (3.15), the metric for this algebra is gij = − 12 6ij . The generators and the structure constants can be normalized so that the metric takes the canonical form gij = −6ij . Just like we de4ned the Killing form K(Xi ; Xj ) for the algebra G in Eq. (3.15) using the adjoint representation, we can de4ne a similar trace form K7 and a metric tensor g7 for any representation 7 by g7; ij = K7 (Xi ; Xj ) = tr(7(Xi )7(Xj )) ;
(3.16)
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63
where 7(X ) is the matrix representative of the Lie algebra element X . If 7 is an automorphism of G, K7 (Xi ; Xj ) = K(Xi ; Xj ). Suppose the Lie algebra is semisimple (this is true for all the classical Lie algebras except the Lie algebras GL(n; C), U (n; C)). According to Cartan’s criterion, the Killing form is non-degenerate for a semisimple algebra. This means that det gij = 0, so that the inverse of gij , denoted by gij , exists. Since it is also real and symmetric, it can be reduced to canonical form gij =diag(−1; : : : ; −1; 1; : : : ; 1) with p −1’s and (n − p) +1’s, where n is the dimension of the algebra. p is an invariant of the quadratic form. In fact, for any real form of a complex algebra, the trace of the metric, called the character of the particular real form (see below and in [31]) distinguishes the real forms from each other (though it can be degenerate for the classical Lie algebras [31]). The character ranges from −n, where n is the dimension of the algebra, to +r, where r is its rank. All the real forms of the algebra have a character that lies in between these values. In Section 4.1 we will see several explicit examples of Killing forms. A famous theorem by Weyl states that a simple Lie group G is compact, if and only if the Killing form on G is negative de
9xi (I ) 9xj (I ) ; 9xr (M ) 9xs (M )
(3.17)
where gij (I ) is the metric at the origin (identity element) of the coset space. (3.17) follows from the invariance of the line element ds2 = gij d xi d xj under translations. If {Xi } is a basis in the tangent space, and dM = exp(d xi Xi ) is a coset representative in4nitesimally close to the identity, we need to know how d xi transforms under translations by the coset representative M . We will not discuss that here, but some generalities can be found for example in Chapter 9, paragraph V.4. of Ref. [31]. In general, it is not an easy problem unless the coset has rank 1. Example. The line element ds2 on the radius-1 2-sphere SO(3)=SO(2) in polar coordinates is ds2 = d 2 + sin2 d!2 . The metric at the point ( ; !) is 1 0 1 0 gij = ; gij = ; (3.18) 0 sin2 0 sin−2 where the rows and columns are labelled in the order , !. The distance between points on the symmetric space is de4ned as follows. The length of a vector X = i t i Xi in the tangent space P (this object is well-de4ned because P is endowed with a de4nite metric) is identi4ed with the length of the geodesic connecting the identity element in the coset space with the element M = exp(X ) [31].
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3.5. The algebraic structure of symmetric spaces Except for the two algebras SL(n; R) and SU∗ (2n) (and their dual spaces related by the Weyl trick), for which the subspace representatives of K, P and iP consist of square, irreducible matrices (for SL(n; R), we saw this in the example in Section 3.1 and for SU(n; C) explicitly in Eq. (3.3)), the matrix representatives of the subalgebra K and of the subspaces P and iP in the fundamental representation consist of block-diagonal matrices X ∈ K, Y ∈ P, Y ∈ iP of the form [31] ˜ 0 C 0 C A 0 (3.19) ; Y = X= ; Y= 0 B −C † 0 C˜ † 0 in the Cartan decomposition. Here A† = −A, B† = −B and C˜ = iC. In fact, for any 4nite-dimensional representation, the matrix representatives of K and P are antihermitean (thus they become antisymmetric if the representation of P is real) and as a consequence, those of iP are hermitean (symmetric in case the representation of iP is real) [31]. This is true irrespective of whether the matrix representatives are block-diagonal or square. The exponential maps of the subspaces P and iP are isomorphic to coset spaces G=K and G ∗ =K, respectively (see for example [30,37]). The exponential map of the algebra maps the subspaces P and iP into unitary and hermitean matrices, respectively. In the fundamental representation, these spaces are mapped onto [31] 0 C I − XX † X ; exp(P) = exp = −C † 0 I − XX † −X † 0 C˜ I + X˜ X˜ † X˜ ; (3.20) exp(iP) = exp = † † C˜ † 0 X˜ I + X˜ X˜ ˜ where X is a spherical and X˜ a hyperbolic function of the submatrix C (C): √ sin C † C sinh C˜ † C˜ X =C √ : ; X˜ = C˜ C †C C˜ † C˜
(3.21)
This shows explicitly that the range of parameters parametrizing the two cosets is bounded for the compact coset and unbounded for the noncompact coset, respectively. We already saw an explicit example of these formulas in Section 2.3. 4. Real forms of semisimple algebras In this section we will introduce the tools needed to 4nd all the real forms of any (semi)simple algebra. The same tools will then be used in the next section to 4nd the real forms of a symmetric space. When thinking of a real form, it is convenient to visualize it in terms of its metric. As we saw in Section 3.4 the trace of the metric is called the character of the real form and it distinguishes the real forms from each other. In the following subsection we discuss various real forms of an algebra and we see how to go from one form to another. In each case, we compute the metric and
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65
the character explicitly. We also give the simplest possible example of this procedure, the rank-1 algebra. In Section 4.2 we enumerate the involutive automorphisms needed to classify all real forms of semisimple algebras and again, we illustrate it with two examples. 4.1. The real forms of a complex algebra In general a semisimple complex algebra has several distinct real forms. Recall from Section 2.5 that a real form of an algebra is obtained by taking linear combinations of its elements with real coeVcients. The real forms of the complex Lie algebra G c i Hi + c E (ci ; c complex) ; (4.1) i
where H0 = {Hi } is the Cartan subalgebra and {E± } are the sets of raising and lowering operators, can be classi4ed according to all the involutive automorphisms of G obeying :2 = 1. Two distinctive real forms are the normal real form and the compact real form. The normal real form of the algebra (4.1), which is also the least compact real form, consists of the subspace in which the coeVcients ci , c are real. The metric in this case with respect to the bases {Hi ; E± } is (with appropriate normalization of the elements of the Lie algebra to make the entries of the metric equal to ±1) 1 .. . 1 0 1 (4.2) gij = 1 0 .. . 0 1 1 0 where the r 1’s on the diagonal correspond to the elements of the Cartan subalgebra (r is obviously the rank of the algebra), and the 2 × 2 matrices on the diagonal correspond to the pairs E± of raising and lowering operators. This structure reQects the decomposition of the algebra G into a direct sum of the root spaces: G = H0 ⊕ G . This metric tensor can be transformed to diagonal form, if we choose the generators to be (E − E− ) (E + E− ) √ √ K= ; iP = Hi ; : (4.3) 2 2 Example. In our example with SU(3; C), K and iP are exactly the subspaces spanned by {X2 ; X5 ; X7 } and {iX1 ; iX3 ; iX4 ; iX6 ; iX8 } (cf. Eq. (3.3)), and (E − E− ) and −i(E + E− ) are exactly the Gell–Mann matrices (cf. Eq. (2.30)).
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Then gij takes the form 1 .. . 1 1 gij = 0
0 −1 ..
. 1 0
; 0 −1
(4.4)
where the entries with a minus sign correspond to the generators of the compact subalgebra K, the 4rst r entries equal to +1 correspond to the Cartan subalgebra, and the remaining ones to the operators in iP not in the Cartan subalgebra. This is the diagonal metric tensor corresponding to the normal real form. The character of the normal real form is plus the rank of the algebra. The compact real form of G is obtained from the normal real form by the Weyl unitary trick: (E − E− ) i(E + E− ) √ √ K= ; P = iHi ; : (4.5) 2 2 The character of the compact real form is minus the dimension of the algebra, and the metric tensor is gij = diag(−1; : : : ; −1). Example. We will use as an example the well-known SU(2; C) algebra with Cartan subalgebra H0 = {J3 } and raising and lowering operators {J± }. We have chosen the normalization such that the nonzero entries of gij are all equal to 1: J3 =
1 √ =; 2 2 3
J± = 14 (=1 ± i=2 ) ;
(4.6)
where in the de4ning representation of SU(2; C) 1 0 0 1 0 =3 = ; =1 = ; =2 = 0 −1 1 0 i
−i 0
:
(4.7)
The normalization is such that [J3 ; J± ] = ± √12 J± ;
[J+ ; J− ] =
√1
2
J3 :
(4.8)
In Eq. (2.16) we constructed the adjoint representation of this algebra, albeit with a diOerent normalization. Using the present normalization to set the entries of the metric equal to 1, we see − + + 3 3 √1 that the nonzero structure constants are C3+ = −C+3 = −C3−− = C− 3 = C+− = −C−+ = 2 . The entries of the metric are given by Eq. (3.15), gij = K(Ji ; Jj ) = Cisr Cjrs with summation over repeated indices,
M. Caselle, U. Magnea / Physics Reports 394 (2004) 41 – 156
so we see that the metric of the normal real form SU(2; R) in this basis is 1 0 0 gij = 0 0 1 ; 0 1 0
67
(4.9)
where the rows and columns are labelled by 3; +; − respectively. This corresponds to Eq. (4.2). To pass now to a diagonal metric, we just have to set >3 = J3 ; 1 J+ + J − √ = √ =1 ; 2 2 2 i J+ − J − = √ =2 ; >2 = √ 2 2 2 >1 =
(4.10)
like in Eq. (4.3). The commutation relations then become [>1 ; >2 ] = − √12 >3 ;
[>2 ; >3 ] = − √12 >1 ;
[>3 ; >1 ] =
√1
2
>2 :
(4.11)
These commutation relations characterize the algebra SO(2; 1; R). From here we 4nd the structure 3 3 1 1 2 2 constants C12 = −C21 = C23 = −C32 = −C31 = C13 = − √12 and the diagonal metric of the normal real form with rows and columns labelled 3, 1, 2 (in order to comply with the notation in Eq. (4.4)) is 1 0 0 gij = (4.12) 0 1 0 ; 0 0 −1 which is to be compared with Eq. (4.4). According to Eq. (4.3), the Cartan decomposition of G∗ is G∗ = K ⊕ iP where K = {>2 } and iP = {>3 ; >1 }. The Cartan subalgebra consists of >3 . Finally, we arrive at the compact real form by multiplying >3 and >1 with i. Setting i>1 = >˜ 1 , >2 = >˜ 2 , i>3 = >˜ 3 the commutation relations become those of the special orthogonal group: [>˜ 1 ; >˜ 2 ] = − √12 >˜ 3 ;
[>˜ 2 ; >˜ 3 ] = − √12 >˜ 1 ;
[>˜ 3 ; >˜ 1 ] = − √12 >˜ 2 :
(4.13)
The last commutation relation in Eq. (4.11) has changed sign whereas the others are unchanged. 2 2 C31 , C13 , and consequently g33 and g11 change sign and we get the metric for SO(3; R): −1 0 0 : 0 −1 0 gij = (4.14) 0 0 −1 This is the compact real form. The subspaces of the compact algebra G = K ⊕ P are K = {>˜ 2 } and P = {>˜ 3 ; >˜ 1 }. Weyl’s theorem states that a simple Lie group G is compact, if and only if the Killing form on G is negative de4nite; otherwise it is non-compact. In the present example, we see this explicitly.
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4.2. The classi
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:2 ) A general matrix in the Lie algebra SU(n; C) can be written in the form A B X= ; −B† C
69
(4.16)
where A, C are complex p × p and q × q matrices satisfying A† = −A, C † = −C, tr A + tr C = 0 (since the determinant of the group elements must be +1), and B is an arbitrary complex p × q matrix (p + q = n). In Eq. (4.16), the matrices are all linear combinations of submatrices in A, B and C √ 1 √ both subspaces K = (E i − E− i ) and P = {iHj ; i= 2(E i + E− i )}. The action of the involution 2 :2 = Ip; q on X is A −B −1 Ip; q XIp; : (4.17) q = B† C Therefore, we see that the subspaces K and P are given by the matrices 0 B A 0 ∈ P : ∈K ; † 0 C 0 −B
(4.18)
Indeed, we see that Ip; q transforms the Lie algebra elements in K into themselves, and those in P into minus themselves. The transformation by Ip; q mixes the subspaces K and P, and splits the algebra in a diOerent way into K ⊕ P . The matrices A iB (4.19) ∈ K ⊕ iP † C −iB de4ne the noncompact real form G ∗ . This algebra is called SU(p; q; C) and its maximal compact subalgebra K is SU(p) ⊗ SU(q) ⊗ U(1). :3 ) By the involutive automorphism :3 :1 = Jp; p K one constructs in a similar way (for details see [31]) a third noncompact real form (for even n = 2p) G ∗ = K ⊕ iP associated to the algebra G = SU(2p; C). G ∗ is the algebra SU∗ (2p) and its maximal compact subalgebra is USp(2p). 7 7
The algebra SU∗ (2p) is represented by complex 2p × 2p matrices of the form A B X= −B∗ −A∗
(4.20)
where tr A + tr A∗ = 0. USp(2p) denotes the complex 2p × 2p matrix algebra of the group with both unitary and symplectic symmetry (USp(2p; C) can also be denoted U(p; Q) where Q is the 4eld of quaternions). A matrix in the algebra USp(2p; C) can be written as A B X= ; (4.21) −B† −AR where A† = −A, BR = B, and the superscript R denotes reQection in the minor diagonal.
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This procedure, summarized in the formula below, exhausts all the real forms of the simple algebras. :1
G∗ = K ⊕ iP ; ∗
: GC → G = K ⊕ P →2 G = K ⊕ iP ; :3
(4.22)
∗
G = K ⊕ iP :
Example. Note that it may not always be possible to apply all the above involutions :1 , :2 , :3 to the algebra. For example, complex conjugation :1 does not do anything to SO(2n + 1; R), because it is represented by real matrices, neither is :3 a symmetry of this algebra, since the adjoint representation is odd-dimensional and :3 has to act on a 2p × 2p matrix. The only possibility that remains is :2 = Ip; q . For a second, even more concrete example, let us look at the algebra SO(3; R), belonging to the root lattice B1 . This algebra is spanned by the generators L1 , L2 , L3 given in Section 2.3. A general element of the algebra is t3 t2 t2 t3 1 3 1 1 3 1 1 −t ⊕ X =t · L= = (4.23) −t t t 2 2 : 2 2 1 2 1 −t −t −t −t This splitting of the algebra is caused by the involution I2; 1 acting on the representation: t3 t 3 −t 2 t2 1 1 1 3 1 3 1 1 = −t −t 1 1 I2; 1 XI2;−11 = t −t 2 2 2 1 2 1 −1 −1 −t −t t t (4.24) and it splits it into SO(3) = K ⊕ P = SO(2) ⊕ SO(3)=SO(2). Exponentiating, as we saw in Section 2.3, the coset representative is a point on the 2-sphere (t 1 )2 + (t 2 )2 2 sin : : t 1 2 (t ) + (t 2 )2 : : x 2 2 2 1 )2 + (t 2 )2 : : y M = (4.25) sin (t = 1 ; x + y + z = 1 : : : t (t 1 )2 + (t 2 )2 : : z : : cos (t 1 )2 + (t 2 )2 By the Weyl unitary trick we now get the noncompact real form G∗ = K ⊕ iP: SO(2; 1) = SO(2) ⊕ SO(2; 1)=SO(2). This algebra is represented by t3 it 2 it 2 t3 3 3 −t ⊕ (4.26) it 1 it 1 = −t ; −it 2 −it 1 −it 2 −it 1
M. Caselle, U. Magnea / Physics Reports 394 (2004) 41 – 156
and after exponentiation of the coset generators 1 2 2 2 2 sinh (t ) + (t ) : : it (t 1 )2 + (t 2 )2 : : M = sinh (t 1 )2 + (t 2 )2 = : : it 1 1 2 2 2 (t ) + (t ) : 1 2 2 2 : : cosh (t ) + (t )
:
ix
71
:
iy ;
:
z
(ix)2 + (iy)2 + z 2 = 1 :
(4.27)
The surface in R3 consisting of points (x; y; z) satisfying this equation is the hyperboloid H 2 . ˜ P=SO(2)⊕ ˜ Similarly, we get the isomorphic space SO(1; 2)=SO(2) by applying I1; 2 : SO(1; 2)= K⊕i SO(1; 2)=SO(2) and in terms of the algebra −it 3 −it 2 1 3 1 1 it : t X˜ = ⊕ (4.28) 2 2 −t 1 it 2
5. The classi-cation of symmetric spaces In this section we introduce the curvature tensor and the sectional curvature of a symmetric space, and we extend the family of symmetric spaces to include also Qat or Euclidean-type spaces. These are identi4ed with the subspace P of the Lie algebra itself, and the group that acts on it is a semidirect product of the subgroup K and the subspace P. As we will learn, to each compact subgroup K corresponds a triplet of symmetric spaces with positive, zero and negative curvature. The classi4cation of these symmetric spaces is in exact correspondence with the new classi4cation of random matrix models to be discussed in Part II. These spaces exhaust the Cartan classi4cation and have a de4nite metric. They are listed in Table 1 together with some of their properties. In Section 5.2 we introduce restricted root systems. In the same way as a Lie algebra corresponds to a given root system, the “algebra” (subspace P or iP) of each symmetric space corresponds to a restricted root system. These root systems are of primary importance in the physical applications to be discussed in Part II. The restricted root system can be of an entirely diOerent type from the root system inherited from the complex extension algebra, and its rank may be diOerent. We work out a speci4c example of a restricted root system as an illustration. In spite of their importance, we have not been able to 4nd any explicit reference in the literature that explains how to obtain the restricted root systems. Instead, we found that they are often referred to in tables and in mathematical texts without explicitly mentioning that they are restricted, which could easily lead to confusion with the inherited root systems. In Ref. [31] the root system that is associated to each symmetric space is the one inherited from the complex extension algebra, whereas for example in Table B1 of Ref. [9] and in [38] the restricted root systems are listed. There are also symmetric spaces with an inde4nite metric, so called pseudo-Riemannian spaces, corresponding to a maximal noncompact subgroup H . For completeness, we will brieQy discuss how these are obtained as real forms of symmetric spaces corresponding to compact symmetric subgroups.
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Table 1 Irreducible symmetric spaces of positive and negative curvature originating in simple Lie groups Root space
Restricted root space Cartan class G=K (G)
G ∗ =K (G C =G)
m o m l ms
AN −1
AN −1
A
SU (N )
SL(N; C) SU (N )
2
0
0
AN −1
AI
SU (N ) SO(N )
SL(N; R) SO(N )
1
0
0
AN −1
AII
SU (2N ) USp(2N )
SU ∗ (2N ) USp(2N )
4
0
0
AIII
SU (p; q) SU (p + q) 2 SU (p) × SU (q) × U (1) SU (p) × SU (q) × U (1)
1
2(p − q)
BCq (p ¿ q) Cq (p = q) BN
BN
B
SO(2N + 1)
SO(2N + 1; C) SO(2N + 1)
2
0
2
CN
CN
C
USp(2N )
Sp(2N; C) USp(2N )
2
2
0
CN
CI
USp(2N ) SU (N ) × U (1)
Sp(2N; R) SU (N ) × U (1)
1
1
0
CII
USp(2p + 2q) USp(2p) × USp(2q)
USp(2p; 2q) USp(2p) × USp(2q)
4
3
4(p − q)
DN
D
SO(2N )
SO(2N; C) SO(2N )
2
0
0
CN
DIII-even
SO(4N ) SU (2N ) × U (1)
SO∗ (4N ) SU (2N ) × U (1)
4
1
0
BCN
DIII-odd
SO(4N + 2) SU (2N + 1) × U (1)
SO∗ (4N + 2) SU (2N + 1) × U (1)
4
1
4
BDI
SO(p + q) SO(p) × SO(q)
SO(p; q) SO(p) × SO(q)
1
0
p−q
BCq (p ¿ q) Cq (p = q) DN
BN
(p+q=2N +1)
Bq
(p¿q)
DN
(p+q=2N )
Dq
(p=q)
In the fourth and 4fth columns the symmetric spaces G=K and G ∗ =K are listed for all the entries except the simple Lie groups themselves, for which the symmetric spaces G and G C =G are listed. Note that there are also zero curvature spaces corresponding to nonsemisimple groups and isomorphic to the subspace P of the algebra, when P is an abelian invariant subalgebra. These are not listed in the table, but can be constructed as explained in Section 5.1. The root multiplicities listed pertain to the restricted root systems of the pairs of dual symmetric spaces with positive and negative curvature.
This does not require any new tools than the ones we have already introduced, namely the involutive automorphisms. 5.1. The curvature tensor and triplicity Suppose that K is a maximal compact subalgebra of the non-compact algebra G∗ in the Cartan decomposition G∗ =K⊕iP, where iP is a complementary subspace. K and P (alternatively K and iP)
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73
satisfy Eq. (3.2): [K; K] ⊂ K;
[K; P] ⊂ P;
[P; P] ⊂ K ;
(5.1) G ∗ =K
K is called a symmetric subalgebra and the coset spaces exp(P) G=K and exp(iP) are globally symmetric Riemannian spaces. Globally symmetric means that every point on the manifold can be moved to any other point by a particular group operation (we discussed this in Section 2.3; for a rigorous de4nition of globally symmetric spaces see Helgason [30, paragraph IV.3]). In the same way, the metric can be de4ned in any point of the manifold by moving the metric at the origin to this point, using a group operation (cf. Eq. (3.17) in Section 3.4). The Killing form restricted to the tangent spaces P and iP at any point in the coset manifold has a de4nite sign. The manifold is then called “Riemannian”. The metric can be taken to be either plus or minus the Killing form so that it is always positive de4nite (cf. Section 3.4). A curvature tensor with components Rijkl can be de4ned on the manifold G=K or G ∗ =K in the usual way [30,32]. It is a function of the metric tensor and its derivatives. It was proved for instance in [30], Chapter IV, that the components of the curvature tensor at the origin of a globally symmetric coset manifold is given by the expression n Rnijk Xn = [Xi ; [Xj ; Xk ]] = Cim Cjkm Xn ;
(5.2)
where {Xi } is a basis for the Lie algebra. The sectional curvature at a point p is equal to K = g([[X; Y ]; X ]; Y ) ;
(5.3)
where g is an arbitrary symmetric and nondegenerate metric (such a metric is also called a pseudoRiemannian structure, or simply a Riemannian structure if it has a de4nite sign) on the tangent space at p, invariant under the action of the group elements. In (5.3), g(Xi ; Xj ) ≡ gij and {X; Y } is an orthonormal basis for a two dimensional subspace S of the tangent space at the point p (assuming it has dimension ¿ 2). The sectional curvature is equal to the gaussian curvature on a two-dimensional manifold. If the manifold has dimension ¿ 2, (5.3) gives the sectional curvature along the section S. Eqs. (5.2) and (5.3), together with Eq. (5.1) show that the curvature of the spaces G=K and ∗ G =K has a de4nite and opposite sign ([30, paragraph V.3]). Thus, we see that if G is a compact semisimple group, to the same subgroup K there corresponds a positive curvature space P G=K and a dual negative curvature space P ∗ G ∗ =K. The reason for this is exactly the same as the reason why the sign changes for the components of the metric corresponding to the generators in iP as we go to the dual space P. We remind the reader that the sign of the metric can be chosen positive or negative for a compact space. The issue here is that the sign changes in going from G ∗ =K to G=K. Example. We can use the example of SU (2) in Section 4.1 to see that the sectional curvature is the opposite for the two spaces G=K and G ∗ =K. If we take {X; Y } = {>3 ; >1 } as the basis in the space iP and {>˜ 3 ; >˜ 1 } (>˜ i ≡ i>i ) as the basis in the space P, we see by comparing the signs of the entries of the metrics we computed in Eqs. (4.12) and (4.13) that the sectional curvature K at the origin has the opposite sign for the two spaces SO(2; 1)=SO(2) and SO(3)=SO(2). Actually, there is also a zero-curvature symmetric space X 0 = G 0 =K related to X + = G=K and = G ∗ =K, so that we can speak of a triplet of symmetric spaces related to the same symmetric
X−
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subgroup K. The zero-curvature spaces were discussed in [9] and in Ch. V of Helgason’s book [30], where they are referred to as “symmetric spaces of the euclidean type”. That their curvature is zero was proved in Theorem 3.1 of [30, Chapter V]. The Qat symmetric space X 0 can be identi4ed with the subspace P of the algebra. The group G 0 is a semidirect product of the subgroup K and the invariant subspace P of the algebra, and its elements g = (k; a) act on the elements of X 0 in the following way: g(x) = kx + a;
k ∈ K; x; a ∈ X 0
(5.4)
if the x’s are vectors, and g(x) = kxk −1 + a;
k ∈ K; x; a ∈ X 0
(5.5)
if the x’s are matrices. We will see one example of each below. The elements of the algebra P now de4ne an abelian additive group, and X 0 is a vector space with euclidean geometry. In the above scenario, the subspace P contains only the operators of the Cartan subalgebra and no others: P = H0 , so that P is a subalgebra of G0 . The algebra G0 = K ⊕ P belongs to a nonsemisimple group G 0 , since it has an abelian ideal P : [K; K] ⊂ K, [K; P] ⊂ P, [P; P] = 0. Note that K and P still satisfy the commutation relations (5.1). In this case the coset space X 0 is Qat, since by (5.1), Rnijk = 0 for all the elements X ∈ P. Eq. (5.2) is valid for any space with a Riemannian structure. Indeed, it is easy to see from Eqs. (5.2), (5.3) that Rnijk = K = 0 if the generators are abelian. Even though the Killing form on nonsemisimple algebras is degenerate, it is trivial to 4nd a nondegenerate metric on the symmetric space X 0 that can be used in (5.3) to 4nd that the sectional curvature at any point is zero. For example, as we pass from the sphere to the plane, the metric becomes degenerate in the limit as [L1 ; L2 ] ∼ L3 → [P1 ; P2 ] = 0 (see the example below). Obviously, we do not inherit this degenerate metric from the tangent space on R2 like in the case of the sphere, but the usual metric for R2 , gij = 6ij provides the Riemannian structure on the plane. Examples. An example of a Qat symmetric space is E2 =K, where G 0 = E2 is the euclidean group of motions of the plane R2 : g(x) = kx + a, g = (k; a) ∈ G 0 where k ∈ K = SO(2) and a ∈ R2 . The generators of this group are translations P1 , P2 ∈ H0 = P and a rotation J ∈ K satisfying [P1 ; P2 ] = 0, [J; Pi ] = −jij Pj , [J; J ] = 0, in agreement with Eq. (5.1) de4ning a symmetric subgroup. The abelian algebra of translations 2i=1 t i Pi , t i ∈ R, is isomorphic to the plane R2 , and can be identi4ed with it. The commutation relations for E2 are a kind of limiting case of the commutation relations for SO(3) ∼ SU(2) and SO(2; 1). If in the limit of in4nite radius of the sphere S 2 we identify >˜ 1 with P1 , >˜ 2 with P2 , and >˜ 3 with J , we see that the commutation relations resemble the ones described in 3 3 Eqs. (4.11) and (4.13)—we only have to set [>˜ 1 ; >˜ 2 ] = 0, which amounts to setting C12 = −C21 → 0. From here we get the degenerate metric of the nonsemisimple algebra E2 : −1 ; 0 gij = (5.6) 0 where the only nonzero element is g33 . This is to be confronted with Eqs. (4.12) and (4.14) which are the metrics for SO(2; 1) and SO(3). This is an example of contraction of an algebra.
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75
An example of a triplet {X + ; X 0 ; X − } corresponding to the same subgroup K = SO(n) is: (1) X + = SU (n; C)=SO(n), the set of symmetric unitary matrices with unit determinant; it is the space exp(P) where P are real, symmetric and traceless n × n matrices. (Cf. the example in Section 3.1.) (2) X 0 is the set P of real, symmetric and traceless n × n matrices and the non-semisimple group G 0 is the group whose action is de4ned by g(x)=kxk −1 +a, g=(k; a) ∈ G 0 where k ∈ K =SO(n) and x; a ∈ X 0 . The involutive automorphism maps g = (k; a) ∈ G 0 into g = (k; −a). (3) X − = SL(n; R)=SO(n) is the set of real, positive, symmetric matrices with unit determinant; it is the space exp(iP) where P are real, symmetric and traceless n × n matrices. We remark that the zero-curvature symmetric spaces correspond to the integration manifolds of many known matrix models with physical applications. The pairs of dual symmetric spaces of positive and negative curvature listed in each row of Table 1 originate in the same complex extension algebra [31] with a given root lattice. This “inherited” root lattice is listed in the 4rst column of the table. In our example in Section 4.2 this was the root lattice of the complex algebra GC = SL(n; C). The same root lattice An−1 characterizes the real forms of SL(n; C): as we saw in the example these are the algebras SU(n; C), SL(n; R), SU(p; q; C) and SU∗ (2n), and we have seen how to construct them using involutive automorphisms. However, also listed in Table 1 is the restricted root system corresponding to each symmetric space. This root system may be diOerent from the one inherited from the complex extension algebra. Below, we will de4ne the restricted root system and see an explicit example of one such system. While the original root lattice characterizes the complex extension algebra and its real forms, the restricted root lattice characterizes a particular symmetric space originating from one of its real forms. The root lattices of the classical simple algebras are the in4nite sequences An , Bn , Cn , Dn , where the index n denotes the rank of the corresponding group. The root multiplicities mo , ml , ms listed in Table 1 (where the subscripts refer to ordinary, long and short roots, respectively) are characteristic of the restricted root lattices. In general, in the root lattice of a simple algebra (or in the graphical representation of any irreducible representation), the roots (weights) may be degenerate and thus have a multiplicity greater than 1. This happens if the same weight - = (-1 ; : : : ; -r ) corresponds to diOerent states in the representation. In that case one can arrive at that particular weight using diOerent sets of lowering operators E− on the highest weight of the representation. Indeed, we saw in the example of SU (3; C) in Section 2.5, that the roots can have a multiplicity diOerent from 1. The same is true for the restricted roots. The sets of simple roots of the classical root systems (brieQy listed in Section 2.7) have been obtained for example in [31,36]. In the canonical basis in Rn , the roots of type {±ei ± ej ; i = j} are ordinary while the roots {±2ei } are long and the roots {±ei } are short. Only a few sets of root multiplicities are compatible with the strict properties characterizing root lattices in general. 5.2. Restricted root systems The restricted root systems play an important role in connection with matrix models and integrable Calogero–Sutherland models (these models will be introduced in Section 7). We will discuss this
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M. Caselle, U. Magnea / Physics Reports 394 (2004) 41 – 156
in detail in Part II. In this subsection we will explain how restricted root systems are obtained and how they are related to a given symmetric space. 8 As we have repeatedly seen in the examples using the compact algebra SU(n; C) (in particular in Section 4.2), the algebra SU(p; q; C) (p + q = n) is a non-compact real form of the former. This means they share the same rank-(n − 1) root system An−1 . However, to the symmetric space SU (p; q; C)=(SU (p) ⊗ SU (q) ⊗ U (1)) one can associate another rank-r root system, where r = min(p; q) is the rank of the symmetric space. For some symmetric spaces, it is the same as the root system inherited from the complex extension algebra (see Table 1 for a list of the restricted root systems), but this need not be the case. For example, the restricted root system is, in the case of SU (p; q; C)=(SU (p) ⊗ SU (q) ⊗ U (1)), BCr . When it is the same and when it is diOerent, as well as why the rank can change, will be obvious from the example we will give below. In general the restricted root system will be diOerent from the original, inherited root system if the Cartan subalgebra is a subset of K. The procedure to 4nd the restricted root system is then to de4ne an alternative Cartan subalgebra that lies partly (or entirely) in P (or iP). To achieve this, we 4rst look for a diOerent representation of the original Cartan subalgebra, that gives the same root lattice as the original one (i.e., An−1 for the SU(p; q; C) algebra). In general, this root lattice is an automorphism of the original root lattice of the same kind, obtained by a permutation of the roots. Unless we 4nd this new representation, we will not be able to 4nd a new, alternative Cartan subalgebra that lies partly in the subspace P. Once this has been done, we take a maximal abelian subalgebra of P (the number of generators in it will be equal to the rank r of the symmetric space G=K or G ∗ =K) and 4nd the generators in K that commute with it. These generators will be among the ones that are in the new representation of the original Cartan subalgebra. These commuting generators now form our new, alternative Cartan subalgebra that lies partly in P, partly in K. Let us call it A0 . The new root system is de4ned with respect to the part of the maximal abelian subalgebra that lies in P. Therefore its rank is normally smaller than the rank of the root system inherited from the complex extension. We can de4ne raising and lowering operators E in the whole algebra G that satisfy [Xi ; E ] = i E
(Xi ∈ A0 ∩ P) :
(5.7)
The roots i de4ne the restricted root system. Example. Let us now look at a speci4c example. We will start with the by now familiar algebra SU(3; C). As before, we use the convention of regarding the Ti ’s as the generators, awithout the a factor of i (recall that the algebra consists of elements of the form t X = i a a a t Ta ; cf. the footnote in conjuction with Eq. (2.25)). In Section 2.5 we explicitly constructed its root lattice A2 . Let us write down the generators again: 0 1 0 0 −i 0 1 0 0 1 1 1 T1 = 1 0 0 ; T2 = i 0 0 ; T3 = 0 −1 0 ; 2 2 2 0 0 0 0 0 0 0 0 0
8
The authors are indebted to Prof. Simon Salamon for explaining how the restricted root systems are obtained.
M. Caselle, U. Magnea / Physics Reports 394 (2004) 41 – 156
0
0
1 0 2 1 0 1 T7 = 0 2 0 T4 =
1
0
0;
0
0
0
0
−i ; 0
0 i
T5 =
0
1 0 2 i
0
−i
0 ; 0
0
0 1 1 T8 = √ 0 2 3 0
0 1 0
T6 =
0
0
1 0 2 0
0
77
0
0
1 ;
1
0
0 : −2
(5.8)
The splitting of the SU(3; C) algebra in terms of the subspaces K and P was given in Eq. (3.3): K = {iT2 ; iT5 ; iT7 };
P = {iT1 ; iT3 ; iT4 ; iT6 ; iT8 } :
(5.9)
The Cartan subalgebra is {iT3 ; iT8 }. The raising and lowering operators were given in (2.30) in terms of Ti : E±(1; 0) = E 1
√1
√ 3 ± 2; 2
E
2
(T1 ± iT2 ) ;
=
√ 3 1 ± −2; 2
√1
=
2
(T4 ± iT5 ) ;
√1
2
(T6 ± iT7 ) :
(5.10)
Now let us construct the Cartan decomposition of G ∗ = K ⊕ iP = SU(2; 1; C). We know from Section 4.2 that K and P are given by matrices of the form 0 B A 0 (5.11) ∈ P ; ∈ K ; 0 C −B† 0 where A and C are antihermitean and tr A + tr C = 0. Combining the generators to form this kind of block-structures (or alternatively, using the involution :2 = I2; 1 ) we need to take linear combinations of the Xi ’s, with real coeVcients, and we then see that the subspaces K and iP are spanned by 0 1 0 1 1 0 1 0 i 1 i i √ 1 0 −1 0 0 −1 0 1 K = ; ; ; 2 2 2 3 2 0 0 0 −2 = {iT1 ; iT2 ; iT3 ; iT8 } 1 1 i 0 iP = ; 2 2 1 0 1 = {T4 ; T5 ; T6 ; T7 } ;
−1 0
1 0 ; 2
0 0
1
i 1 ; 2
−1 0
0
1
(5.12)
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M. Caselle, U. Magnea / Physics Reports 394 (2004) 41 – 156
where the block-structure is evidenced by leaving blank the remaining zero entries. K spans the algebra of the symmetric subgroup SU (2) ⊗ U (1) and iP spans the complementary subspace corresponding to the symmetric space SU (2; 1)=(SU (2) ⊗ U (1)). iP is spanned by matrices of the form 0 B˜ : (5.13) B˜ † 0 We see that the Cartan subalgebra iH0 = {iT3 ; iT8 } lies entirely in K . It is easy to see that by using the alternative representation 1 1 1 1 ; T8 = √ 0 −2 T3 = (5.14) 2 2 3 −1 1 of the Cartan subalgebra (note that this is a valid representation of SU(3; C) generators) while the other Ti ’s are unchanged, we still get the same root lattice A2 . The eigenvectors under the adjoint representation, the E ’s, are still given by Eq. (5.10). However, their eigenvalues (roots) are permuted under the new adjoint representation of the Cartan subalgebra, so that they no longer correspond to the root subscripts in (5.10). This permutation is a Weyl reQection; more speci4cally, it is the √ 3 1 reQection in the hyperplane orthogonal to the root − 2 ; 2 . Now we choose the alternative Cartan subalgebra to consist of the generators T4 , T8 : A0 = {T4 ; T8 };
[T4 ; T8 ] = 0;
iT4 ∈ P ; iT8 ∈ K :
(5.15)
(Note that unless we 4rst take a new representation of the original Cartan subalgebra, we are not able to 4nd the alternative Cartan subalgebra that lies partly in P .) The restricted root system is now about to be revealed. We de4ne raising and lowering operators E in the whole algebra according to E± 1 ∼ (T5 ± iT3 );
E
1
±2
∼ (T6 ± iT2 ); E˜ ± 1 ∼ (T7 ± iT1 ) :
(5.16)
2
The ± subscripts are the eigenvalues of T4 ∈ iP in the adjoint representation: [T4 ; E± 1 ] = ±E±1 ;
[T4 ; E 1 ] = ± 12 E 1 ; [T4 ; E˜ ± 1 ] = ± 12 E˜ ± 1 : ±2
±2
2
2
(5.17)
These roots form a one-dimensional root system of type BC1 . We see that the multiplicity of the long roots is 1 and the multiplicity of the short roots is 2 = 2(p − q). This result is general (cf. Table 1). If we had ordinary roots, their multiplicity would be 2, but for this low-dimensional group we can have only three pairs of roots. Note that we can rescale the lengths of all the roots together by rescaling the operator T4 in (5.17), but their characters as long and short roots cannot change. The root system BC1 is with respect to the part of the Cartan subalgebra lying in iP only, thus it is called restricted. According to Eq. (3.8), every element p of P G=K is conjugated with some element h = eH (H ∈ H0 ) through p=khk −1 , where k ∈ K=M and H is de4ned up to the elements in the factor group M =M . Thus, the decomposition p=khk −1 is not unique. The factor group M =M transforms a Cartan subalgebra H0 ⊂ P into another Cartan subalgebra H0 ⊂ P conjugate with the original one. This
M. Caselle, U. Magnea / Physics Reports 394 (2004) 41 – 156
79
amounts to a permutation of the roots of the restricted root lattice corresponding to Weyl reQections. The factor group M =M then coincides with the Weyl group of the restricted root system. If we 4x the Weyl chamber of H , H is unique and k is de4ned up to transformations by the subgroup M . 5.3. Real forms of symmetric spaces Involutive automorphisms were used to split the algebra G into orthogonal subspaces to obtain the real forms G, G∗ , G ∗ : : : of a complex extension algebra GC . By re-applying the same involutive automorphisms to the spaces K, P, and iP, these spaces with a de4nite metric tensor can in turn be split into subspaces with eigenvalue +1 and −1 under this new involutive automorphism =. Thus, :: G → K ⊕ P;
G∗ = K ⊕ iP ;
=: K → K1 ⊕ K2 ;
H = K1 ⊕ iK2 ;
=: P → P1 ⊕ P2 ;
M = P1 ⊕ iP2 ;
=: iP → iP1 ⊕ iP2 ;
iM = iP1 ⊕ P2 :
(5.18)
As we already know, K is a compact subgroup, and exp(P) and exp(iP) de4ne symmetric spaces with a de
9
Note that not all the theorems governing symmetric spaces corresponding to maximal compact subgroups apply to the case at hand. A prime example is the decomposition involving radial coordinates in Section 3.3. We will not discuss the symmetric spaces involving maximal non-compact subgroups in any detail in this paper.
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apply I1; 2 to G∗ : I1; 2 XI1;−21 =
1 −1
=
1 t3 2 it 2
t3
it 2
1
1 3 1 −t −1 it 2 2 1 −1 −1 −it −it −t 3 −it 2 −t 3 1 3 1 ⊕ it 1 = 2 t 2 1 −it it 2 −it 1
−it 2
it 1
= (K1 ⊕ K2 ) ⊕ i(P1 ⊕ P2 ) ;
(5.19)
where in this example, K1 is empty. The spaces K1 , K2 , P1 , P2 consist of the generators in G with the following combinations of eigenvalues under the two successive involutions, :=: K1 : + +;
K2 : + −;
P1 : − +;
P2 : − − :
Thus we see that K1 is empty and the others are spanned by 1 1 1 −1 ; P1 = 1 K2 = ; 2 2 −1
(5.20) 1 P2 = 2
1 −1
: (5.21)
The new symmetric space is obtained by doing the Weyl unitary trick on the split spaces (K1 ⊕ K2 ) and (P1 ⊕ P2 ): it 3 1 ; H = K1 ⊕ iK2 = −it 3 2 M = P1 ⊕ iP2 =
1 2
it 2 −it 2
−t 1
t1 :
(5.22)
The second involution = (plus the Weyl trick) gives rise to a non-compact subgroup H =SO(1; 1) and to the symmetric space M ∼ exp M and its dual M ∗ ∼ exp(iM). The coset M ∼ SO(2; 1)=SO(1; 1) is represented by : : ix 2 2 2 : y exp M = (5.23) : ; (ix) + y + z = 1 : −ix −y z The real forms of the simple Lie groups do not include all the possible Riemannian symmetric coset spaces. For example, the compact Lie group G is itself such a space, and so is its dual G C =G
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(here the algebra GC = G∗ ⊕ iG∗ is the complex extension of all the real forms G∗ ). By starting with a compact algebra G and applying to it all the combinations of the two involutive automorphisms :, =, we construct, in the way just described, all the remaining pseudo-Riemannian symmetric spaces associated to the corresponding root system. A complete list of these spaces can be found in Table 9.7 of Ref. [31]. Note that all the properties of the Lie algebra G (Killing form, rank, and so on) can be transferred to the vector subspaces P, iP [31]. The only diOerence is that the subspaces are not closed under commutation. In this section of the paper we have discussed symmetric spaces of positive, zero and negative curvature that will be relevant for matrix models of the circular, gaussian, and transfer matrix type, respectively. In Part II we will de4ne and discuss various types of random matrix ensembles and their applications to various physical problems, and we will associate them to the corresponding symmetric spaces in Table 1. 6. Operators on symmetric spaces The diOerential operator uniquely determined by the simplest Casimir operator on a symmetric space (and especially its radial part) plays an important role both in mathematics and in the physical applications of symmetric spaces. Its eigenfunctions provide a complete basis for the expansion of an arbitrary square-integrable function on the symmetric space, and are therefore important in their own right. Their importance in the applications to be discussed in Part II is evident when considering that the radial part of the Laplace–Beltrami operator on an underlying symmetric space determines the dynamics of the transfer matrix eigenvalues of the Dorokhov–Mello–Pereyra–Kumar equation (DMPK equation for short) in the theoretical description of quantum wires, and maps onto the Hamiltonians of integrable Calogero–Sutherland models. Here we will de4ne some concepts related to the Laplace–Beltrami operator and discuss its eigenfunctions. 6.1. Casimir operators Let G be a semisimple rank-r Lie algebra. A Casimir operator (invariant operator) Ck (k=1; : : : ; r) associated with the algebra G is a homogeneous polynomial operator that satis4es [Ck ; Xi ] = 0
(6.1)
for all Xi ∈ G. The simplest (quadratic) Casimir operator associated to the adjoint representation of the algebra G is given by C = gij Xi Xj ;
(6.2)
where gij is the inverse of the metric tensor de4ned in (3.15) and the generators Xi are in the adjoint representation. More generally, it can be de4ned for any representation 7 of G by C7 = g7ij 7(Xi )7(Xj ) ;
(6.3)
where g7ij is the inverse of the metric (3.16) for the representation 7 (cf. Section 3.4). The Casimir operators lie in the enveloping algebra obtained by embedding G in the associative algebra de4ned
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by the relations X (YZ) = (XY )Z;
[X; Y ] = XY − YX
(6.4)
(note that in general, XY makes no sense in the algebra G). The number of functionally independent Casimir operators is equal to the rank r of the group. Other Casimir operators can be formed by taking polynomials of the independent Casimir operators Ck (k = 1; : : : ; r). Since the Casimir operators commute with all the elements in G, they make up the center of the associative algebra (6.4). Note that Casimir operators are de4ned for semisimple algebras, where the metric tensor has an inverse. This does not prevent one from 4nding operators that commute with all the generators of non-semisimple algebras. For group E3 of rotations {J1 ; J2 ; J3 } example, for the euclidean and translations {P1 ; P2 ; P3 }, P2 = Pi Pi and P · J = Pi Ji commute with all the generators (cf. the comments in the paragraph following Eq. (5.5)). Also the operators that commute with all the generators of a nonsemisimple algebra are often referred to as Casimir operators. All the independent Casimir operators of the algebra G can be obtained as follows. Suppose 7 is an n-dimensional representation of the rank-r Lie algebra G. The secular equation for the algebra G is de4ned as the eigenvalue equation dim G n det t i 7(Xi ) − In = (− )n−k ’k (t i ) = 0 ; (6.5) i=1
k=0 i
where the ’k (t ) are functions of the real coordinates t i . In general, they will not all be functionally independent (for example, ’0 (t i ) is a constant). There will be r functionally independent coeVcients ’k (t i ) multiplying the powers of − [31]. When writing down the secular equation, it is easiest to take a low-dimensional representation. By making the substitution t i → Xi in the functionally independent coeVcients, they become the functionally independent Casimir operators of the algebra G: t i →X
’k (t i ) → i Cl (Xi ) :
(6.6)
Example. The generators L1 , L2 , L3 of the SO(3) algebra were given explicitly in the adjoint representation in Eqs. (2.5), (2.6) in Section 2.3. The secular equation for this algebra is then − t 3 =2 t 2 =2 1 det(t · L − I3 ) = −t 3 =2 (6.7) − t 1 =2 = (− )3 + (− ) t2 = 0 : 4 2 −t =2 −t 1 =2 − The equation has one functionally independent coeVcient, which is proportional to the trace of the matrix (t · L)2 . It equals ’1 (t) = 14 t2 . The rank of SO(3) is 1 and the only Casimir operator is C1 ∼ L2 = L21 + L22 + L23
(6.8)
obtained by the substitution t i → Li in ’1 (t). The Casimir operator can also be obtained from Eq. (6.2) by using the metric gij = − 12 6ij for SO(3) given in the example in Section 3.4. We know from elementary quantum mechanics that [L2 ; L1 ] = [L2 ; L2 ] = [L2 ; L3 ] = 0
(6.9)
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83
is an immediate consequence of the commutation relations, so we see that this operator indeed commutes with all the generators. Even though the commutation relations are not the same in polar coordinates or after a general coordinate transformation, (6.9) will nevertheless be true. Example. SU (3) is a rank-2 group and therefore its characteristic equation will have two independent coeVcients. If we denote a general SU(3) matrix (aij ) we get the characteristic equation a12 a13 a11 − a22 − a23 det a21 a31 a32 a33 − = (− )3 + (− )2 (a11 + a22 + a33 ) + (− )(a11 a22 + a22 a33 + a33 a11 − a12 a21 − a23 a32 − a31 a13 ) + (a11 (a22 a33 − a23 a32 ) + a12 (a23 a31 − a21 a33 ) + a13 (a32 a21 − a31 a22 )) = 0 :
(6.10)
The term proportional to (− )2 vanishes, because the trace of any matrix in the SU(3) algebra is zero. The two independent coeVcients are then ’2 (aij ) and ’3 (aij ). Substituting the values in terms of the coordinates t i of the algebra i t i Ti for the aij (for example, a11 = t 3 + √13 t 8 , a12 = t 1 + it 2 , etc.), we see that the expression for ’2 (t i ) becomes i
’2 (t ) =
8
(t i )2
(6.11)
i=1
and therefore the substitution (6.6) gives the 4rst Casimir operator C1 =
H12
+
H22
+
(E E− + E− E ) =
8
Ti2
(6.12)
i=1
as expected. Making the same substitution in ’3 (t i ) gives the second Casimir operator for SU (3), which has a more complicated form. 6.2. Laplace operators The Casimir operators can be expressed as diOerential operators in the local coordinates on the symmetric space. This is due to the fact that each in4nitesimal generator X ∈ G is a contravariant vector 4eld on the group manifold. An element in the Lie algebra can be written as 9 X (x)9 ≡ X (x) ; (6.13) X= 9x where x are local coordinates [30,35] (for example, L1 = (r × P)1 = x2 93 − x3 92 ). That the generators transform as lower index objects follows from the commutation relations.
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Example. As an example we take the group SO(3). Under a rotation R = R(t 1 ; t 2 ; t 3 ) = exp the vector x = xi eˆ i ∈ R3 transforms as R
i
x→x = x eˆi ;
t k Lk , (6.14)
where the transformation laws for the components and the natural basis vectors are i
x = Rij xj ;
eˆi = eˆ j Rji
(6.15)
and R−1 = RT . The one-parameter subgroups of SO(3) are rotations R(t n ) = exp(t n Ln );
(n = 1; 2; 3)
(6.16)
(no summation) where Ln are SO(3) generators. It is easy to show using the commutation relations for Ln (given after Eq. (2.7)) that under in4nitesimal rotations the Ln transform like the lower index objects eˆ i : RLi R−1 = Lj Rji :
(6.17)
Expressed in local coordinates as diOerential operators, the Casimirs are called Laplace operators. In analogy with the Laplacian in Rn , n 92 2 (6.18) P =D= 9xi 2 i=1 which is invariant under the group En of rigid motions (isometries) of Rn , the Laplace operators on (pseudo-)Riemannian manifolds are invariant under the group of isometries of the manifold. The isometry group of the symmetric space P G=K is G, since G acts transitively on this space and preserves the metric, so the Laplace operators are invariant under the group operations g ∈ G. The number of independent Laplace operators on a Riemannian symmetric coset space is equal to the rank of the space. As we de4ned in Section 3.2, the rank of a symmetric space is the maximal number of mutually commuting generators Hi in the subspace P (cf. also Section 5.2). If X , X ; : : : ∈ K and Xi , Xj ; : : : ∈ P, it is also equal to the number of functionally independent solutions to the equation dim P n k t 7(Xk ) − In = (− )n−l ’l (t k ) = 0 ; (6.19) det k=1
l=0
where n is the dimension of the representation 7 and where now in the determinant we sum over all Xk ∈ P. This is equivalent to setting the coordinates t E for all the XE ∈ K equal to zero in the secular equation. In the example in the preceding paragraph, the rank of the symmetric space SO(3)=SO(2) (the 2-sphere) is 1, which in this case is also the rank of the group SO(3). The Laplace–Beltrami operator on a symmetric space is the special second order Laplace operator de4ned (when acting on a function (0-form) f) as 9 ij 1 9 DB f = gij Di Dj f = gij (9i 9j − Fijk 9k )f = g |g| j f; g ≡ det gij : (6.20) i 9x |g| 9x Here Di denotes the covariant derivative on the symmetric space and gij are the components of the inverse of the metric tensor. (The metric has an inverse because it is non-degenerate on a semisimple algebra and can be mapped over the entire symmetric space. For euclidean type spaces, we have the
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85
usual metric 6ij .) Di is de4ned in the usual way [30,32,34], for example it acts on the components xj of a contravariant vector 4eld in the following way: Di xj = 9i xj + Fkij xk ;
(6.21)
where Fkij are ChristoOel symbols (connection coeVcients). The last term represents the change in xj due to the curvature of the space. We remind the reader that on a Riemannian manifold, the Fkij are expressible in terms of the metric tensor, hence the formula in Eq. (6.20). Example. Let us calculate the Laplace–Beltrami operator on the symmetric space SO(3)=SO(2) in polar coordinates using (6.20) and the metric at the point ( ; !) given in the second example of Section 3.4: 1 0 1 0 ij ; g = : (6.22) gij = 0 sin2 0 sin−2 Substituting in the formula and computing derivatives we obtain the Laplace–Beltrami operator on the sphere of radius 1: DB = 92 + cot 9 + sin−2 92! :
(6.23)
2
Of course this operator is exactly L . We can check this by computing Lx = y9z − z9y , Ly = z9x − x9z , and Lz = x9y − y9x in spherical coordinates (setting r = 1) and then forming the operator L2x + L2y + L2z , remembering that all the operators have to act also on anything coming after the expression for each L2i . We 4nd that L2 in spherical coordinates, expressed as a diOerential operator, is exactly the Laplace–Beltrami operator. In general, a Laplace–Beltrami operator can be split into a radial part DB and a transversal part. The radial part acts on geodesics orthogonal to some submanifold S, typically a sphere centered at the origin [39]. Example. For the usual Laplace–Beltrami operator in R3 expressed in spherical coordinates, DB = 92r + 2r −1 9r + r −2 (92 + cot 9 + sin−2 92! )
(6.24)
the 4rst two terms DB = 92r + 2r −1 9r
(6.25)
constitute the radial part with respect to a sphere centered at the origin and the expression in parenthesis multiplied by r −2 is the transversal part. The transversal part is equal to the projection of DB on the sphere of radius r and equals the Laplace–Beltrami operator on the sphere, given for r = 1 in Eq. (6.23). This is a general result. For any Riemannian manifold V and an arbitrary submanifold S, the projection on S of the Laplace–Beltrami operator on V is the Laplace–Beltrami operator on S (see Helgason [39, Chapter II, paragraph 3]). The radial part of the Laplace–Beltrami operator on a symmetric space has the general form r 1 9 (j) 9 DB = (j) J (j = 0; −; +) ; (6.26) J =1 9q 9q where r is the dimension of the maximal abelian subalgebra H0 in the tangent space P (the rank of the symmetric space) and J (j) is the Jacobian, to be given in Eq. (6.30), of the transformation
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to radial coordinates. The sum goes over the labels of the independent radial coordinates de4ned in Section 3.3: q = log h(x) = (q1 ; : : : ; qr ) where h(x) is the exponential map of an element in the Cartan subalgebra and q are canonical coordinates on H0 (in [9] they were denoted (q; ) ≡ q · ; we will see in a moment that they are indeed given by q · where is a restricted root). The adjoint representation of a general element H in the maximal abelian subalgebra H0 follows from a form similar to Eq. (3.11) (with or without a factor of i depending on whether we have a compact or noncompact space), but now the roots are in the restricted root lattice. For a noncompact space of type P ∗ log h = H = q · H =
0 ..
. 0 q· ..
. −q ·
≡
0 ..
:
. 0 q ..
.
(6.27)
q −; Hence q = q · and H h=e =
1 ..
:
. 1 eq
..
. eq
−;
(6.28)
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Example. For the simple rank-1 algebra corresponding to the compact group SU (2), the above formulas take the form (cf. Eq. (2.16)) 1 0 : ; h = ei H1 = ei 1 (6.29) H = H1 = −1 e− i The radial coordinate is q = (q1 ) = . There is a general theory for the radial parts of Laplace–Beltrami operators [39]. It is of interest to consider the radial part of the Laplace–Beltrami operator on a manifold V with respect to a submanifold W of V that is transversal to the orbit of an element w ∈ W under the action of a subgroup of the isometry group of V . Of special interest to us is the case in which the manifold is a symmetric space G=K and the Lie subgroup is K. (j) The Jacobian J = |g| (where g is the metric tensor at an arbitrary point of the symmetric space) of the transformation to radial coordinates takes the form ! J (0) (q) = (q )m ; ∈ R+
J (−) (q) =
!
(sinh(q ))m ;
∈ R+
J (+) (q) =
!
(sin(q ))m ;
(6.30)
∈ R+
for the various types of symmetric spaces with zero, negative and positive " curvature, respectively (see [39, Chapter I, paragraph 5]). In these equations the products denoted ∈R+ are over all the positive roots of the restricted root lattice and m is the multiplicity of the root . The multiplicities m were listed in Table 1. A remark on Eq. (6.30) is in order here. Strictly speaking, in the euclidean case we have not de4ned any restricted root lattice. The formula for the Jacobian J (0) (q) for the zero-curvature space is understood as the in4nitesimal version of the formula pertaining to the negative-curvature space. In the proof [39] of the formula for the zero-curvature case we consider the Jacobian for a mapping from K=M × H0 onto P (where M is the centralizer in K of the Cartan subalgebra H0 ), whereas in the proof of the formula for the negative-curvature symmetric space we consider the Jacobian for a mapping from K=M × eiH0 onto a subset of the symmetric space eiP . m can in both cases be interpreted as the dimension of the subspace of raising operators corresponding to the root in the same algebra (this is, in the case of X − , the root multiplicity of the restricted root ; see [39, Chapter I, paragraph 5]). In the example illustrating the construction of restricted root systems in = T + iT and E ˜ 1=2 = T7 + iT1 for the Section 5.2 this space was spanned by the raising operators E1=2 6 2 1 root = 2 , whereas for the root = 1 it was spanned by only one raising operator E1 = T5 + iT3 . The above can perhaps also be understood in terms of the limiting procedure discussed in Section 8.3.2 (cf. also equations (6.42) and (6.43) in Section 6.3). The m ’s will in the following be referred to as “root multiplicities” also for the zero curvature spaces, keeping the above in mind.
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Example. On the hyperboloid H 2 with metric 1 0 1 0 ij ; g = ; gij = 0 sinh2 0 sinh−2 Eqs. (6.2) and (6.30) give the radial part of DB for H 2 . The radial coordinate is agreement with (6.30) 1 9 sinh 9 = (92 + coth 9 ) : J (−) = |g| = sinh ; DB = sinh
(6.31) so we get, in (6.32)
In the same way we can also easily derive Eqs. (6.25) and (6.33). In particular, comparing with (6.32) we immediately get the radial part of the Laplace–Beltrami operator acting on the two-sphere S 2 transversally to a one-sphere S 1 centered on the north pole: DB = (92 + cot 9 )
(6.33)
which is exactly the radial part appearing in Eq. (6.23). The eigenfunctions of the radial part of the Laplace–Beltrami operator on a symmetric space are called zonal spherical functions. In the applications of symmetric spaces to random matrix theory (for example in quantum transport), the properties of these zonal spherical functions are of central importance. For this reason we shall devote the following subsection (and the appendix at the end of this review) to a detailed discussion of their properties. The so-called DMPK operator will be discussed in Section 8. The diOerential equation involving this operator describes the evolution of the distribution of the set of eigenvalues of the transfer matrix of a quantum wire with an increasing length of the wire. One of the most interesting applications of symmetric spaces in the random matrix theory of quantum transport lies in the identi4cation of the DMPK operator with a simple transformation of the Laplace–Beltrami operator on the symmetric space de4ning the random matrix universality class. We will discuss this in more detail in Part II of this review (see Section 8.3.5). 6.3. Zonal spherical functions The properties of the so-called zonal spherical functions are important for the research results to be discussed in Part II. Since there is a natural mapping from the Hamiltonians of integrable Calogero– Sutherland systems onto the Laplace–Beltrami operators of the underlying symmetric spaces, these eigenfunctions play an important role in the physics of integrable systems. But they are also relevant in transport problems in connection with the DMPK equation for a quantum wire. The known asymptotic expressions for these eigenfunctions allows one to solve this equation in general or in the asymptotic regime, because of the simple mapping from the DMPK evolution operator to the radial part of the Laplace–Beltrami operator. (For an example of their use see [40–42].) When 7 is an irreducible representation of an algebra, the associated Casimir operators Ck; 7 are multiples of the identity operator [35,37] (Schur’s lemma). This means that it has eigenvalues and eigenfunctions. Since the Casimir operators (and consequently the Laplace operators) form a commutative algebra, they have common eigenfunctions. There exists an extensive theory regarding invariant diOerential operators and their eigenfunctions [39]. Of particular interest are the diOerential
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89
operators on a group G or on a symmetric space G=K that are left-invariant under the group G and right-invariant under a maximal compact subgroup K. Suppose the smooth complex-valued function ! (x) is an eigenfunction of such an invariant diOerential operator D on the symmetric space G=K: D! (x) = ED ( )! (x) :
(6.34)
Here the eigenfunction is labelled by the parameter and ED ( ) is the eigenvalue. If in addition ! (kxk ) = ! (x) (x ∈ G=K; k ∈ K) and ! (e) = 1 (e = identity element), the function ! is called spherical. A spherical function satis4es [39] # ! (xky) d k = ! (x)! (y) ; (6.35) K
where d k is the normalized Haar measure on the subgroup K. We will see examples of this formula below. The common eigenfunctions of the Laplace operators on the symmetric space G=K are invariant under the subgroup K. They are termed zonal spherical functions. Because of the bi-invariance under K, these functions depend only on the radial coordinates h: ! (x) = ! (h) :
(6.36)
Example. Let us study for a moment the eigenfunctions of the Laplace operator on G=K = SO(3)= SO(2). We know from quantum mechanics that the eigenfunctions of L2 are the associated Legendre polynomials Pl (cos ), and −l(l + 1) is the eigenvalue under L2 (our de4nition of L diOers by a factor of i from the de4nition common in quantum mechanics): L2 Pl (cos ) = −l(l + 1)Pl (cos )
(6.37)
where cos is the z-coordinate of the point P = (x; y; z) on the sphere of radius 1. In spherical coordinates, P = (sin cos !; sin sin !; cos ). As we can see, the eigenfunctions are functions of the radial coordinate only. The subgroup that keeps the north pole 4xed is K = SO(2) and its algebra contains the operator Lz = 9! . Indeed, Pl (cos ) is unchanged if the point P is rotated around the z-axis. Example. In terms of Euler angles, a general SO(3)-rotation takes the form R( ; ; E) = g( )k()h(E) cos 0 −sin cos 1 0 = 0 sin sin 0 cos 0
−sin cos 0
0
cos E
0 0
1
sin E
0
−sin E
1
0
0
cos E
;
(6.38)
where g and h are rotations around the y axis by the angles and E respectively, and k is a rotation around the z-axis by the angle . Under such a rotation, the north pole (0; 0; 1) goes into (−cos cos sin E−sin cos E; −sin sin E; −sin cos sin E+cos cos E). This means that Eq. (6.35)
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takes the form # 2/ 1 Pl (−sin cos sin E + cos cos E) d = Pl (cos )Pl (cos E) : 2/ 0
(6.39)
(To avoid confusion, note that the eigenvalue l in Eq. (6.37) is not equal to . In fact, = l + 1=2.) Example. For the symmetric space G=K = E2 =SO(2) the spherical functions are the plane waves: (r) = eikr ;
(6.40)
where k is a complex number. If g, h denote translations in the x-direction by a distance b, a respectively, and k is a rotation around the origin of magnitude !, then the transformation g(b)k(!)h(a) moves the point x ∈ R2 by a distance a2 + b2 + 2ab cos !. Therefore we obtain from (6.35) # 2/
1 a2 + b2 + 2ab cos ! d! = (a) (b) : (6.41) 2/ 0 We introduce a parameter a into the Jacobians (6.30) as in Ref. [9], ! J (0) (q) = (q )m ; ∈ R+
J (−) (q) =
!
(a−1 sinh(aq ))m ;
∈ R+
J (+) (q) =
!
(a−1 sin(aq ))m :
(6.42)
∈ R+
The parameter a corresponds to a radius. For example, for the sphere SO(3)=SO(2) it is the radius of the 2-sphere. The various spherical functions corresponding to the spaces of positive, negative and zero curvature are then related to each other by the simple transformations [9] (− ) !(0) (q) = lim ! (q) ; a→0
(− ) !(+) (q) = ! (q)|a→ia :
(6.43)
There exist integral representations of spherical functions for the various types of spaces G=K [9,39]. We will list an integral representation only of ! (−) (q) below, recalling that formulas for the other types of spherical functions can be obtained by (6.43). If ! (−) (x) is spherical and h is the spherical radial part of x, # ! (−) (x) = ! (−) (h) = e(i −7)H (kx) d k : (6.44) K
Here is a complex-valued linear function on the maximal abelian subalgebra H0 of iP and 7 is the function de4ned below in Eq. (6.46). In Eq. (6.44) they act on the unique element H (kx) ∈ H0 such that kx = neH (kx) k in the Iwasawa decomposition introduced in Section 3.3. It was shown by Harish-Chandra [43] that two functions ! (−) (x) and !I(−) (x) are identical if and only if = sI, where s denotes a Weyl reQection. The Weyl group is the group of reQections in hyperplanes orthogonal to the roots and was de4ned in Section 2.5, Eq. (2.34), and discussed further in Section 2.6.
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Eq. (6.44) may seem a bit cryptic, but it becomes much more clear if one uses the explicit expression for the Iwasawa decomposition. It essentially becomes the integral over the product of all the possible lower principal minors (raised to suitable powers) of the matrix keah k −1 (where a is the free parameter introduced above). The explicit expression can be found in [9]. This integral representation is of great importance, because in some cases it can be exactly integrated leading to explicit expressions for the zonal spherical functions. This point will be discussed in more detail in the appendix at the end of this review. The eigenvalues of the radial part of the Laplace–Beltrami operator corresponding to the eigenfunctions on zero, negative and positive curvature symmetric spaces are given by the following equations (see [9,39, Chapter IV, paragraph 5]): 2 (0) DB !(0) = − ! ; 2 (− ) 2 DB ! = − 2 − 7 ! (−) ; a 2 (+) 2 DB ! = − 2 + 7 !(+) ; a
where 7 is the function de4ned by 1 7= m : 2 +
(6.45)
(6.46)
∈R
Example. Take the symmetric space SO(3)=SO(2). From Table 1 we see that this space has p − q = 2 − 1 = 1 short root of length 1. Then 2 1 1 (6.47) 72 = · 12 · | |2 = 2 4 and setting a = 1, the eigenvalue is − 2 + 1=4 = −l(l + 1). Remarkably enough, for a few classes of symmetric spaces explicit expressions for the zonal spherical functions can be obtained. We will make use of this result in Section 9.4.1 when discussing the exact solution of the DMPK equation in the = 2 case. These solutions play an important role in several branches of physics. For this reason we decided to discuss them in some detail in the appendix of this review. 6.4. The analog of Fourier transforms on symmetric spaces Much of the material presented in this subsection is taken from the book by Wu-Ki Tung [44]. A continuous smooth (C ∞ ) spherical function f is said to be elementary if it is an eigenfunction of any diOerential operator that is invariant under left translations by G and right translations by K. Thus the eigenfunctions of the Laplace operators are elementary. The elementary spherical functions are related to irreducible representation functions for the group G. The irreducible representation functions are the matrix elements of the group elements g in the matrix representation 7. Let us clarify this statement by an example well-known from quantum mechanics.
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Example. The angular momentum basis for SO(3) is de4ned by L2 |lm = l(l + 1)|lm L3 |lm = m|lm L± |lm = l(l + 1) − m(m ± 1)|lm ;
(6.48)
where l labels the representation. The irreducible representation functions are, in the angular momentum basis |lm, the matrix elements Dl (R)mm such that R|lm = |lm Dl (R)mm ;
(6.49)
where R = exp(t · L) is a general SO(3) rotation. It is known that for R ∈ SO(3), if , , E are the Euler angles of the rotation R = R( ; ; E), these matrix elements take the form Dl (R)mm = Dl ( ; ; E)mm = e−i m dl ()mm e−iEm ;
dl ()mm ≡ lm |e−iL2 |lm :
(6.50)
The associated Legendre functions Plm (cos ) and the special functions Ylm ( ; !) called spherical harmonics are essentially this kind of matrix elements: $ (l + m)! l m Plm (cos ) = (−1)m d ( )0 ; (l − m)! % 2l + 1 l [D (!; ; 0)m0 ]∗ : Ylm ( ; !) = (6.51) 4/ The irreducible representation functions Dl (R)mm satisfy orthogonality and completeness relations. In fact, they form a complete basis in the space of square integrable functions de4ned on the group manifold. This is the Peter–Weyl theorem. From here the corresponding theorems follow for the special functions of mathematical physics. Example. For SO(3) the orthonormality condition reads # (2l + 1) d= Dl† (R)mn Dl (R)nm = 6ll 6nn 6mm ; Dl† (R)mn ≡ [Dl (R)nm ]∗ ;
(6.52)
where R = R( ; ; E) is an SO(3) rotation expressed in Euler angles and d= is the invariant group integration measure normalized to unity, d= = d d(cos ) dE=8/2 . That the irreducible representation functions form a complete basis for the square-integrable functions on the SO(3) group manifold can be expressed as n flm Dl (R)mn ; (6.53) f(R) = lmn
where f(R) is square-integrable. Using (6.52) we obtain # n flm = (2l + 1) d= Dl† (R)nm f(R) :
(6.54)
If R( ; ; E) = R(!; ; 0) we get the special case of the spherical harmonics on the unit sphere (setting 0 4/=(2l + 1)flm ≡ f˜ ): lm (6.55) f˜lm Ylm ( ; !) ; f( ; !) = lm
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f˜lm =
#
∗ f( ; !)Ylm ( ; !) d(cos ) d!
93
(6.56)
and further, for R( ; ; E) = R(0; ; 0) we get the completeness relation for the associated Legendre polynomials Pl (cos ) = Yl0 4/=(2l + 1) fl Pl (cos ) ; (6.57) f( ) = l
(2l + 1) fl = 2
#
f( )Pl∗ (cos ) d(cos ) ;
(6.58)
0 . These are analogous to Fourier transforms. In the above example, we considered a where fl = fl0 symmetric space with positive curvature. For a space with zero or negative curvature, we have an integral instead of a sum in (6.57): # # ! ( j) (q) ( j) 2 ˜ ˜ f( ) d ; (6.59) f(q) = f( )! (q) d-( ) ˙ w |c( )|2 # ˜ (6.60) f( ) = f(q)[! ( j) (q)]∗ J (j) (q) dq ;
where q are canonical radial coordinates. The integration measure d-( ) was determined by HarishChandra [43] to be well-de4ned and proportional to w2 |c( )|−2 d , where c( ), to be discussed in detail below, is a known function whose inverse is analytic (in this context, see also [9,39]) and w is the order of the Weyl group (the number of distinct Weyl reQections). J (j) (q) dq is the invariant measure on the space of radial coordinates. In Eq. (6.59) the arbitrary square-integrable function f(q) is expressed in terms of the complete set of basis functions ! ( j) (q). One can show [39,43] that the dimension of the space of eigenfunctions of DB is less than or equal to w. It is a remarkable fact that the eigenfunctions of the radial part of the Laplace–Beltrami operator DB have the property of being eigenfunctions of the radial part of any left-invariant diOerential operator on the symmetric space as well [39, Chapter IV]. In the asymptotic expansions that we will discuss in Section 9.4 of Part II of this review in the context of quantum transport, a crucial role is played by the function c( ) in Eq. (6.59). It encodes all the information relating the transport problem to an underlying symmetric space. The explicit form of c( ) is ! c( ) = c ( ) (6.61) ∈ R+
with c ( ) =
F(i =2) ; F(m =2 + i =2)
(6.62)
where F denotes the Euler gamma function, is a generic root belonging to the restricted root lattice of the symmetric space, m denotes its multiplicity and the product is restricted to the sublattice R+ of positive restricted roots only. We shall see an explicit example of these functions in Eqs. (9.23) and (9.24) for the three symmetric spaces which are relevant for the random matrix description of quantum wires.
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An important feature of the zonal spherical functions is that they satisfy, for large values of |h|, the following asymptotic expression for spaces of zero and negative curvature [9,43]: c(s )e(is −7)(H ) ; (6.63) ! (h) ∼ s ∈W H
where h = e is a spherical coordinate, H is an element of the maximal abelian subalgebra, is a complex-valued linear function on the maximal abelian subalgebra, and the function 7 was de4ned in Eq. (6.46). This expansion will play a major role later in this review. We will refer to it in the following as the “Harish-Chandra asymptotic expansion”. 7. Integrable models related to root systems As mentioned in the introduction, an important role in our analysis is played by the class of integrable models known as Calogero–Sutherland (CS) models, which turn out to be deeply related to the theory of symmetric spaces. These models describe n particles in one dimension, identi4ed by their coordinates q1 ; : : : ; q n and interacting (at least in the simplest version of the models) through a pair potential v(qi − qj ). The Hamiltonian of such a system is given by n 1 2 2 p + g v(q ) ; H= 2 i=1 i ∈ R+ n 9 q i i ; (7.1) pi = −i i ; q = q · = 9q i=1 where the coordinate q is q = (q1 ; : : : ; q n ), p1 ; : : : ; pn are the particle momenta, and the particle mass is set to unity. In Eq. (7.1) R+ is the subsystem of positive roots of the root system R = { 1 ; : : : ; I } related to a speci4c simple Lie algebra or symmetric space, and n is the dimension of the maximal abelian subalgebra H0 and of its dual space H0∗ . The components of the positive root = k ∈ R+ are 1k ; : : : ; nk . he number of positive roots is I=2, where I is the total number of roots. In general, the coupling constants g are the same for equivalent roots, namely those that are connected with each other by transformations of the Weyl group W of the root system (see Sections 2.5, 3.3 and 5.2). Several realizations of the potential v(q ) have been studied in the literature (for a review see Ref. [9]): vI (K) = K−2 ; vII (K) = sinh−2 (K) ; vIII (K) = sin−2 (K) ; vIV (K) = P(K) ; vV (K) = K−2 + !2 K2 ; vVI (K) = eK :
(7.2)
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Here P(q) denotes the Weierstrass P-function, to be de4ned in Eq. (10.20). We will mainly be interested in the 4rst three realizations. The potential expressed in terms of the Weierstrass P-function will be discussed in Section 10.3, while we will not deal with the last two cases which we reported here only for completeness. The reader is referred to [9] for a discussion of these two potentials. The most relevant features of the Calogero–Sutherland models are: • Under rather general conditions (see [9] for a detailed discussion) they are completely integrable, in the sense that they possess n commuting integrals of motion. • For particular values of the coupling constants (i.e. for those related to the root multiplicities by Eq. (7.6) which we will discuss below) the Calogero–Sutherland Hamiltonians can be mapped onto the radial parts of the Laplace–Beltrami operators on suitably chosen symmetric spaces. As we will see, these spaces have negative curvature for the sinh-type models and positive curvature for the sin-type models.
7.1. The root lattice structure of the CS models In the original formulation of the CS model, the interaction among the particles was simply pairwise [8]. Only later it was realized that this particular choice was the signature of an underlying structure, namely the root lattices of Lie algebras of type An (see the example below). Also, the model could be extended to any root lattice canonically associated to a simple Lie algebra or symmetric space, keeping its relevant properties: complete integrability and mapping to the radial part of a Laplace–Beltrami operator for special values of the couplings [9]. This corresponds exactly to the choice of Hamiltonian introduced in Eq. (7.1). Let us look at two examples which may clarify this construction. In these examples we use the root lattices An and Cn and the potential of type II. Remember that the coupling constants g are the same for equivalent roots, thus we expect a single coupling constant in the 4rst example and only two coupling constants in the second example. In order to 4x the notation, let us denote with {e1 ; : : : ; en } a canonical basis in the space Rn . An : As mentioned in Section 2.7, the An root system is contained in the hyperplane in Rn+1 with equation x1 +x2 +· · ·+x n+1 =1. The root system R is given by R={ 1 ; : : : ; I }={ei −ej ; i = j}. In this case W , the Weyl group, is the permutation group of the set {ei }. The corresponding CS Hamiltonian is in this case n
H=−
1 92 g2 + : 2 i 2 2 i=1 9(q ) sinh (qi − qj ) i¡j
(7.3)
The arguments of the sinh-function are qk = q · k = (q1 ; : : : ; q n ) · (ei − ej ) = qi − qj (i ¡ j) where k is a positive root of the root lattice R. As anticipated we have only pairwise interaction and a single coupling constant appears in the model. This is the model originally considered in [8]. Cn : The Cn root system is R = {±2ei ; ±ei ± ej ; i = j}. The Weyl group is the product of the permutation group and the group of transformations that changes the sign of the vectors {ei }.
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The corresponding Hamiltonian is n
H=−
1 92 gl2 + + 2 2 i=1 9(qi )2 sinh (2qi ) i¡j i
go2 go2 + 2 i 2 sinh (q − qj ) sinh (qi + qj )
:
(7.4)
As anticipated, we have diOerent coupling constants for the ordinary roots (corresponding to pairwise interaction) and for the long roots. In the following sections we will come back to this particular choice of Hamiltonian which turns out to be of great importance in the random matrix description of quantum wires. 7.2. Mapping to symmetric spaces The most interesting property of these Hamiltonians, which is a direct consequence of the underlying structure of the symmetric space, is that for the potentials of type I, II and III there exists an exact transformation of the Hamiltonian H into the radial part of the Laplace–Beltrami operator on the symmetric space corresponding to the root lattice R of H. In particular the target spaces have negative curvature for the sinh-type models and positive curvature for the sin-type models. Denote the radial part of the Laplace–Beltrami operator by DB ; the transformation is then given by (see appendix D of Ref. [9] for a proof) 1 (D ± 72 )K−1 (q) (+ for II; − for III; 7 = 0 for I) 2 B if and only if the coupling constants g in H take the following root values: H = K(q)
g 2 =
m (m + 2m2 − 2)| |2 : 8
(7.5)
(7.6)
In Eq. (7.6) m is the multiplicity of the root and | | its length, and 7 in Eq. (7.5) is the vector de4ned in (6.46) 1 7= m : (7.7) 2 + ∈R
For the ordinary roots Eq. (7.6) simpli4es to g 2 =
m (m − 2)| |2 : 8
(7.8)
The function K(q) in (7.5) is related to the Jacobian of the transformation to radial coordinates (cf. Eq. (6.30)). It is given by ! [q ]m =2 I ; ∈ R+ ! [sinh(q )]m =2 II ; (7.9) K(q) = ∈ R+ ! [sin(q )]m =2 III ∈ R+
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for quantum systems with potentials of type I, II, and III, respectively. The radial part of the Laplace–Beltrami operator, DB , on the symmetric space can be expressed in terms of K(q) as follows (see Eq. (6.26) in Section 6.2): 1 9 2 9 K (q) i (7.10) DB = 2 i K (q) i 9q 9q and the symmetric space has zero, negative, and positive curvature for the cases I, II, and III, respectively. At this point a number of results can be obtained for the corresponding quantum systems merely by using the theory of symmetric spaces. A detailed collection of results pertaining to spectra, wave functions, and integral representations of wave functions can be found in the original article [9], and we will not further elaborate on them here. The crucial point in this relation for our analysis is another one. From Eq. (7.8) we see that whenever the ordinary root multiplicity is m = 2, the interaction between particles in the Hamiltonian vanishes. The degrees of freedom decouple and the problem can be solved explicitly. Such a decoupling could hardly have been inferred by looking at the original form of the radial part of the Laplace–Beltrami operator, Eq. (6.26). This decoupling is also the reason why in these cases explicit expressions for the zonal spherical functions exist (see Section 6.3 and, for a more detailed discussion, the appendix at the end of this review). Part II In the second part of this review we will use the background material presented in Part I to make evident the close relationships between symmetric spaces and random matrix ensembles. To this end, we discuss in detail the identi4cations that have to be made between matrix ensembles and symmetric space characteristics in Section 8. We show that the integration manifolds of random matrix theories with physical applications in the description of complex nuclei, gauge 4eld theories and mesoscopic systems, can be exactly identi4ed with the irreducible symmetric spaces based on non-exceptional groups that were classi4ed by Cartan, and their euclidean counterparts. We identify the random matrix eigenvalues with the spherical radial coordinates frequently used in the theory of Lie groups, and show that the Jacobians determining the probability distribution functions of random matrix eigenvalues are exactly the Jacobians appearing in Eq. (6.42) determined by the metric g on the symmetric space: J = |g|. The structure of the restricted root lattice associated to the symmetric space determines this Jacobian completely through the multiplicities of its roots. In connection with the discussion of transfer matrix ensembles, we de4ne the Dorokhov–Mello–Pereyra–Kumar (DMPK) equation, and in Section 8.5 we include a general discussion of Fokker–Planck equations associated to random matrix ensembles. We also discuss the so-called Coulomb gas analogy used in random matrix theory. We show that the DMPK operator connected to any random matrix ensemble is simply related to the radial part of the Laplace–Beltrami operator (de4ned in Section 6) on the appropriate symmetric space. We conclude by summarizing the results in Table 2. We devote Section 9 to some consequences of the identi4cations between random matrix theories and symmetric spaces. In Section 9.1 we discuss the new classi4cation of disordered systems that the equivalence gives rise to, as a natural consequence of the Cartan classi4cation of symmetric spaces. In Sections 9.2–9.3 we will study how the properties of the symmetric spaces are reQected in the orthogonal polynomials associated to a random matrix partition function, and see how the reQection
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or translation symmetries of the Jacobians are direct consequences of the corresponding properties of the restricted root lattices. An introduction to the relationship between the restricted root lattices of symmetric spaces and integrable Calogero–Sutherland models describing interacting many-particle systems in one dimension was given in Section 7. Olshanetsky and Perelomov [9] showed that the dynamics of these systems are related to free diOusion on a symmetric space. This relationship is due to the fact that the Hamiltonians of integrable Calogero–Sutherland models map onto the radial parts of the Laplace– Beltrami operators of the underlying symmetric spaces, as explained in Section 7 of Part I. Since the Dorokhov–Mello–Pereyra–Kumar equation for a disordered conductor can be mapped onto a SchrNodinger-like equation in imaginary time ([45], see also [41]) featuring a Calogero– Sutherland type Hamiltonian with root values of the coupling constants, information on the solutions of the DMPK equation [40,41] can be extracted from known properties of the zonal spherical functions of the underlying symmetric space. As we have seen in Section 6, these are the eigenfunctions of the radial part of the Laplace–Beltrami operator. Some results relating to this will be described in detail in Section 9.4, as well as an application of the theory of symmetric spaces in a quantum transport problem involving the magnetoconductance. Finally in Section 10 we discuss some possible extensions of the present results and some non-standard applications of the symmetric space formalism. We go beyond the Cartan classi4cation and discuss non-Cartan parametrization of symmetric spaces in Section 10.1. In Section 10.2 we discuss generalizations of the DMPK equation and the alternative clustered solutions to the DMPK equation that are a consequence of the exact integrability of Calogero–Sutherland models. Finally, in the last Section 10.3 we discuss the Weierstrass P-function and show that it describes three types of Calogero–Sutherland potential in its various limits, corresponding to symmetric spaces of diOerent curvature. These limiting potentials correspond in the Calogero–Sutherland model to particles interacting on a circle, hyperbola, or line, respectively, reQecting the corresponding triplet of symmetric spaces underlying the respective Calogero–Sutherland models. 8. Random matrix theories and symmetric spaces 8.1. Introduction to the theory of random matrices 8.1.1. What is random matrix theory? Random matrix theory has evolved into a rich and versatile 4eld with applications in several branches of physics and mathematics. In the theory of random matrices one studies the statistical properties of the eigenvalues of large matrices with randomly distributed elements. Historically, large random matrices were 4rst employed by Wigner [47] and Dyson [46] to describe the energy levels in complex nuclei, where they modelled the Hamiltonian of the system. Disregarding the continuous part of the spectrum, one may represent the Hamiltonian of such a system quantum mechanically by a large hermitean matrix acting on a 4nite-dimensional Hilbert space. In principle, the eigenvalues of the Hamiltonian then give us the energy levels of the nucleus, and the eigenvectors give us the eigenstates. However, for a system involving hundreds of nucleons (or, in a small metal sample, hundreds of electrons) we do not know the form of the Hamiltonian, and even if we did, the huge number of degrees of freedom involved would prevent us from solving the SchrNodinger equation exactly.
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Choosing instead to represent the Hamiltonians of an ensemble of such systems by an ensemble of large hermitean matrices, whose elements are random variables with some given distribution, we obtain a statistical theory for its eigenvalues 1 ; : : : ; N , where N is a large number equal to the size of the random matrices. 10 If the random matrices in the chosen ensemble have the same global symmetry properties as the actual Hamiltonian, this statistical theory describes also the eigenvalues of the Hamiltonian, to the extent that they depend only on symmetry, i.e. they are universal. Even systems with just a few degrees of freedom lend themselves to such a description. In general, a description in terms of random matrices is possible whenever we are dealing with a chaotic or disordered system (see [28] for a discussion). Such random matrices were studied by Dyson in a series of papers [46]. He showed that they fall into one of three classes named the orthogonal, unitary and symplectic ensemble, depending on whether or not the Hamiltonian possesses time-reversal invariance and rotational invariance. This classi4cation was reviewed in Chapter 2 of the book by Mehta [11]. The corresponding random matrices are real symmetric, complex hermitean, or self-dual quaternion, respectively (see below). 8.1.2. Some of the applications of random matrix theory In typical applications, large matrices with given symmetry properties and randomly distributed elements substitute a physical quantum operator. This operator could be a Hamiltonian like in the example of nuclear energy levels or in quantum chaotic scattering (in the latter case an eOective Hamiltonian appears in the scattering matrix), or it could be for instance a scattering matrix, transfer matrix, or Dirac operator. Then one studies the statistics of its eigenvalue spectrum and extracts the universal behavior. An early application of the Wigner–Dyson ensembles was in the theory of scattering in chaotic quantum systems. With the help of random matrices one is able to study the statistics of resonance poles and scattering phase shifts in nuclei, atoms, and molecules, as well as microwave cavities and ballistic systems (for a self-contained review see [48]). A major area where random matrices are employed is in the description of the infrared limit of gauge theories, notably QCD. In these applications an integration over an appropriate random matrix ensemble replaces the integration over gauge 4eld con4gurations in the partition function. This is achieved by substituting a suitable random matrix for the Dirac operator appearing in the fermion determinant. Because the magnitude of the quark condensate is proportional to the spectral density of the Dirac operator at the origin of the spectrum, random matrices are useful in this context in obtaining insights concerning the spontaneous breaking of chiral symmetry. This is one of the most fundamental phenomena in QCD, since it determines the hadronic mass spectrum. As it is a phenomenon that takes place at very low energy, it is inaccessible to perturbation theory and therefore one of the most diVcult to study from a theoretical point of view. The random matrix approach brings considerable simpli4cation and has indeed contributed to the understanding of this phenomenon. The corresponding random matrix ensembles are called chiral ensembles, because of the chiral symmetry of the Dirac operator. We will discuss chiral random matrix theories in more detail in Section 8.3.3. Random matrix theories are also largely and successfully applied in the theoretical description of mesoscopic systems, where they may be used to model the properties of scattering and transfer 10
For symplectic ensembles the eigenvalues are two-fold degenerate.
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matrices. A mesoscopic conductor is a micrometer-sized metal grain or wire at low temperature with randomly distributed impurities that act as scattering centers for electrons originating from current leads (alternatively, random scattering takes place at the edges and depends on the shape of the conductor). Examples include disordered wires, quantum dots, and normal metal–superconductor heterostructures. The theory of quantum transport is concerned with the statistics of the transmission eigenvalues in such systems. The transmission eigenvalues are related to the transfer matrix in a way which will be discussed in detail in Section 8.3.4. These eigenvalues directly determine the physical conductance. We will discuss quantum wires in some detail below. We will also brieQy discuss applications in normal metal–superconductor heterostructures as well as in ballistic quantum dots. The ensembles used in these applications are related to yet other symmetric spaces than those pertinent to the ensembles used in the study of the usual mesoscopic systems. The reason is, of course, that the symmetries of the physical operator whose eigenvalues we study are diOerent. The same chiral ensembles used in the description of chiral gauge theories are also realized in the Hamiltonians of random Qux and random hopping problems [49,50] in the theory of quantum transport. Some important applications of hermitean matrix models which will not be treated here, are in the description of random surfaces in the 4eld of quantum gravity (for an excellent introduction see [19]), where a N −2 expansion amounts to a genus expansion of the random surface, and in string theory. The wide range of applications is one of the most fascinating aspects of random matrix theory. Notably, the classical random matrix theories have no adjustable parameters. The reason for their success in accurately characterizing such a wide range of systems (nuclei, disordered metals, chaotic systems) lies in the universal behavior of the eigenvalue correlators in an appropriate double scaling limit. This is the subject of the following section. 8.1.3. Why are random matrix models successful? The statistical properties of a sequence of apparently random numbers like nuclear energy levels may be either of the local or of the global type. The local and global spectral characteristics are completely disconnected. Systems with identical global characteristics may have diOerent local characteristics, and vice versa. A typical example of a global, or macroscopic, property is the mean level (eigenvalue) density, or one-point correlation function, in the limit of large N . It is often denoted 7( ). For the gaussian Wigner–Dyson ensembles the mean level density approaches a semicircle in the large N limit. In general its shape depends on the random matrix potential determining the probability distribution of individual matrix elements. A global characteristic spectral property does not change appreciably on the scale of a few level spacings. An example of a local spectral property is the spacing between two successive levels. It Quctuates from level to level. One amazing feature of random matrix models is that some local spectral characteristics are independent of the distribution of individual matrix elements in the limit of large N , if we simultaneously rescale the eigenvalues in an appropriate way. This is called a double scaling limit. In this limit local spectral characteristics are determined solely by the global symmetries of the random matrices. This so-called microscopic universality is manifest only in the appropriate double scaling limit, where the average level spacing becomes much smaller than the scale on which it varies.
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The n-point correlation function, or joint probability density for n eigenvalues 7n ( 1 ; : : : ; n ) (1 6 n 6 N ) is central in the theory of random matrices and is the most important example of such a microscopically universal quantity. It is de4ned as follows: 7n ( 1 ; : : : ; n )d 1 : : : d n is the probability of 4nding one eigenvalue in each of the intervals [ i ; i + d i ] (1 6 i 6 n). This means that if one rescales the eigenvalues by the average level spacing 11 and simultaneously takes the limit N → ∞, the n-point function 7n for each ensemble becomes a universal function characteristic of the ensemble. There are also other spectral characteristics that depend on symmetry only. The number variance (i.e., the variance related to the number of eigenvalues in a certain interval), the so-called spectral rigidity D3 (this quantity was de4ned for example in [11,51]), the distribution of the 4rst eigenvalue, and the distribution of spacings between adjacent levels are examples of such quantities. In a hermitean matrix model the eigenvalues are real. The eigenvalue density (one-point function) of such a model is typically zero outside some interval or intervals around the origin. As we learnt above, its macroscopic shape depends on the probability distribution, while locally its microscopic shape is universal. Spectrum edge universality, i.e. universality of scaled correlation functions near the edge of the eigenvalue spectrum, is known to diOer from universality in the bulk of the spectrum. One normally distinguishes between three microscopic scaling regimes: the bulk, the hard edge (vicinity of the origin) and the soft edge (tail) of the spectrum. The rescaling of the eigenvalues must be chosen appropriately in each regime, and in practice amounts to a rescaling by an appropriate integer or fractal power of N [52–55], which is proportional to the inverse local density of eigenvalues and thus proportional to the average local level spacing. This procedure in turn amounts to magnifying a local region in such a way that the behavior of the eigenvalue correlators is universal. In case of a (multi)critical hermitean matrix model, the macroscopic spectral density develops extra (multiple) zeros on the limit between a one-cut (single-interval) support and a multi-band support. This is achieved by 4ne-tuning couplings in the random matrix potential determining the probability distribution. In this case the scaling is more subtle and the calculations more involved due to subleading terms in the large-N expansion [53]. Several attempts at theoretically demonstrate universality can be found in Refs. [52–56]. It is also supported by a wealth of numerical [57] evidence. In [52], the universality of correlation functions near the origin of the eigenvalue spectrum of chiral unitary and unitary ensembles of random matrices in the microscopic limit was investigated. Universality was shown to follow from the fact that the recursion relation for the orthogonal polynomials determining the kernel from which all correlators are derived, reduces to a universal diOerential equation (in this case for Bessel functions) in the appropriate microscopic limit. 8.2. The basics of matrix models In this subsection we will give a few basic de4nitions and results regarding the simplest kinds of hermitean matrix models. Similar results apply for all the matrix models we will study in this paper, and the purpose here is to introduce the reader not at all familiar with matrix models into the subject. 11
In general one can consider also other scales.
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A general hermitean matrix model is de4ned by a partition function # Z ∼ dS P(S)
(8.1)
where S is a square N ×N hermitean matrix with randomly distributed elements Sij and dS is a Haar measure. For example, for the gaussian unitary ensemble consisting of complex hermitean matrices (this ensemble describes a system with no time-reversal invariance) the matrix elements are complex numbers, Sij = Sijr + iSiji (except the ones on the diagonal that must be real). dS is then given by ! ! dS = dSijr dSkli (8.2) i 6j
k¡l
and the probability P(S) dS that a system described by the gaussian unitary ensemble will belong to the volume element dS is invariant under the automorphism S → U −1 SU
(8.3)
of the ensemble to itself, where U is any unitary N × N matrix. 12 This amounts to a change of basis. The invariance of P(S) dS under such automorphisms restricts P(S) to depend on the traces of the 4rst N powers of S. For the gaussian matrix models reviewed for example in the book by Mehta, there was another requirement, namely that the matrix elements should all be statistically independent. This excludes everything except the traces of the 4rst two powers, and these may occur only in an exponential [11]. However, this requirement is not physically motivated and was subsequently relaxed as other types of matrix models were introduced. A probability distribution of the form P(S) ˙ e−c tr V (S)
(8.4)
is often used. Here c is a constant and V (S) is a matrix potential, typically a polynomial with a 4nite number of terms. By doing an appropriate similarity transformation (8.3) on the ensemble of random matrices, the Haar measure dS and potential V (S) can be expressed in terms of the eigenvectors and eigenvalues of the matrix S. This amounts to (block)diagonalizing S. At the same time the Haar measure is factorized into a part that depends only on the eigenvectors (the “angular” part) and a part that is a function of the eigenvalues { i } of S only. Assume for simplicity that the random matrix potential is given by cV (S) = S 2 :
(8.5)
This kind of potential is called gaussian, for obvious reasons. The Jacobian of the similarity transformation depends on { i } only and is given by [11] ! J ({ i }) ∼ ( i − j )2 (8.6) i¡j
for the gaussian unitary ensemble. For the other gaussian ensembles one has a similar Jacobian (see Eq. (8.29) below). The part of P(S) dS that depends on the eigenvalues then takes the form N ! 2 P({ i }) d 1 : : : d N ˙ ( i − j )2 e− j=1 j d 1 : : : d N ; (8.7) i¡j 12
If S is time-reversal invariant U will be either a real orthogonal matrix or a symplectic matrix, depending on whether rotational symmetry is also present. In this case we are dealing with the orthogonal or symplectic ensemble.
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whereas the “angular” degrees of freedom do not appear in the integrand at all and can be integrated out to give a constant in front of the integral. This model is easily solvable. This is the strength of the random matrix description of disordered systems. The eigenvalue correlators (k-point functions) are de4ned as # ! N N! 7k ( 1 ; : : : ; k ) = d j P({ 1 ; : : : ; N }) (8.8) (N − k)! j=k+1
and can be calculated exactly by rewriting the Jacobian as a product of Vandermonde determinants of a set of polynomials orthogonal with respect to the measure e−cV ( ) . This procedure was reviewed in [11]. For the gaussian unitary ensemble these polynomials are Hermite polynomials Hn ( ) and satisfy # +∞ 2 Hm ( )Hn ( )e− d = hn 6mn ; (8.9) −∞
where hn is a normalization. In practice one de4nes a kernel KN ( i ; j ) that turns out to be universal and whose determinant gives all the correlation functions [11]. De4ning so-called oscillator wave functions that for the gaussian unitary ensemble take the form ’j ( ) = hj−1=2 Hj ( )e−
2
=2
;
(8.10)
this kernel is given by KN ( i ; j ) =
N −1
’k ( i )’k ( j ) :
(8.11)
k=0
It can be evaluated explicitly using the ChristoOel–Darboux formula for the sum of products of orthogonal polynomials. The k-point function is then 7k ( 1 ; : : : ; k ) = det[KN ( i ; j )]16i; j6k :
(8.12)
DiOerent authors use diOerent conventions for the normalization constants involved in these formulas. So far we have not said anything about the integration manifold in Eq. (8.1). As we have seen, we start out with some given class G of matrices S, and the integral is a priori over this class. However, by doing the similarity transformation in Eq. (8.3), we can perform the integration over the subgroup K to which the matrix U belongs without further ado. We then end up with an integral over a smaller manifold G=K, the manifold of the random matrix eigenvalues, and as we will see in the following subsection, this manifold turns out to be a symmetric space for all the commonly used hermitean random matrix ensembles. This gives rise to the deep and useful connection between random matrix theories and symmetric spaces that is the subject of this review. Let us now look more in detail at some explicit examples of random matrix theories. In the following subsections we will identify in turn their integration manifolds as symmetric spaces, their eigenvalues as radial coordinates on the respective symmetric spaces, and their Dyson and boundary indices (to be de4ned) as determined by the multiplicities of the corresponding restricted root lattices. As we will see, the Jacobian of the similarity transformation to the space of random matrix eigenvalues and eigenvectors is determined explicitly by these root multiplicities and the curvature of the symmetric space manifold.
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8.3.1.2. De
S → VSW ( = 2)
(8.15)
where U is unitary ( = 1) or unitary with quaternion elements ( = 4) and V , W are unitary. Diagonalization. In each case the matrix S can be diagonalized by a similarity transformation like the ones in Eq. (8.15). The eigenvalues lie on a unit circle and take the form ei!i (thereof the name “circular ensembles”). They are doubly degenerate in the symplectic case = 4. Probability distribution of the eigenvalues. A simple calculation [11] shows that the Jacobian of the transformation from the space of unitary matrices to eigenvalue space is given by ! J (!i ) ˙ |ei!i − ei!j | : (8.16) i¡j
It gives rise to correlations between eigenvalues. Since the original distribution function P(S) of the random matrices is constant, the Jacobian coincides with the probability distribution function of the
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eigenvalues: P (!i ) ∼ J (!i ) :
(8.17)
8.3.1.4. The symmetric spaces associated to the circular ensembles. In this section our main interest is identifying the integration manifold of the three circular ensembles. Orthogonal ensemble. Every symmetric unitary matrix S can be written as S = U TU ;
(8.18)
where U is a generic unitary matrix. However, this mapping is not one-to-one. If we assume that S=U T U =V T V , then it is easy to see that the matrix relating the two expressions, R=VU −1 , is unitary and satis4es RT R = 1. Hence R must be real and orthogonal. Thus we see that the manifold of the unitary symmetric matrices is actually the coset U (N )=O(N ), due to the above mentioned degeneracy. From the point of view of the physical properties of the ensemble nothing changes if we perform the restriction to an irreducible symmetric space. 13 Then the manifold becomes SU (N )=SO(N ). We will systematically perform the reduction to irreducible symmetric spaces in the following. Unitary ensemble. Finding the integration manifold is trivial in the = 2 case, where we simply have unitary matrices without any further constraint and the manifold is simply the group U (N ). Like for = 1, if one is interested in irreducible symmetric spaces a U (1) factor must be taken out, and the manifold becomes SU (N ). Symplectic ensemble. This case strictly resembles the = 1 case discussed above. Any self-dual unitary quaternion matrix can be written as S = U RU
(8.20)
where U this time is a 2N × 2N unitary matrix. By following the same reasoning as in the case of the orthogonal ensemble, we see that the same matrix S can be obtained using a new unitary matrix V obtained from the previous one by the transformation V = BU where B is constrained by BR B = BBR = 1. By de4nition of the duality operator R this means that B is a symplectic matrix. Thus, extracting a U (1) factor, the manifold coincides with SU (2N )=Sp(2N ). By comparing with Table 1 in Section 5.2 in Part I of the present review, we see that the integration manifolds of the three circular ensembles are exactly the 4rst three coset spaces (described in the Cartan notation as A, AI and AII) of positive curvature in the list of possible irreducible symmetric spaces. This is the main result of this paragraph. 8.3.2. Gaussian ensembles 8.3.2.1. Physical applications of the gaussian ensembles. Some of the physical applications of the gaussian ensembles were discussed already in the introduction to this section, in Section 8.1.
13
In the partition function # dS P (S) Z∼ G=K
extracting such a U (1) factor from the integration manifold just amounts to rede4ning Z by a constant.
(8.19)
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In general gaussian ensembles describe hamiltonian ensembles. This is because they correspond to Qat symmetric spaces, as we will see below. The corresponding positive curvature ensembles are the ensembles of the scattering matrices of the same systems, since in general the scattering matrix is expressed as S = eiH . As we have seen, applications include spectra resulting from complicated many-body interactions like those taking place in neutron resonances in atomic nuclei, the electronic energy levels inside tiny metal grains at low temperature (that depend on the shape of the grain), or generally the spectra of electrons moving in a random potential with no further symmetries present. The energy spectrum of classically chaotic systems is another example of apparently gaussian random behavior. A realization of such a system is a chaotic billiard. The motion of the billiard ball is determined by the shape of the billiard table (which is chosen to be irregular in some way) and it is drastically diOerent for small variations in the initial trajectory. Corresponding quantum billards may also be considered, represented by a free quantum particle con4ned to a 4nite part of space. Its discrete energy spectrum will be determined by the Laplacian on the space, and may possess varying degrees of randomness. In all these systems the statistical properties of the energy spectrum can be described by a gaussian random matrix ensemble. 8.3.2.2. De
(8.21)
where H is a hermitean N × N matrix and V (H ) is a quadratic potential. We can de4ne a partition function # Z = dH P (H ) ; (8.22) where dH is an invariant Haar measure. It can be shown [11] that the form of P (H ) is automatically restricted to the form P (H ) = exp(−a tr H 2 + b tr H + c) ;
(8.23)
(a ¿ 0) if one postulates statistical independence of the matrix elements Hij . Note that P (H ) can be cast in the form P (H ) ∼ e−a tr H
2
(8.24)
by simply completing the square in the exponent. Depending on the nature of H we can distinguish three cases labelled by = 1; 2, and 4: • In the orthogonal ensemble ( = 1) H is an N × N hermitean symmetric random matrix. It immediately follows that the probability distribution P1 (H ) and the integration measure dH are invariant under all real orthogonal transformations of H . This ensemble is used when the random matrix H models the time-reversal and rotation invariant Hamiltonian of a system with integral spin [46,11, Chapter 2]. • In the unitary ensemble ( = 2) H is an N × N hermitean random matrix and P2 (H ) dH is invariant under all unitary transformations of H . A system without time-reversal invariance has this type of Hamiltonian [46,11].
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• In case of the symplectic ensemble ( = 4) H is an N × N hermitean self-dual random matrix with quaternionic elements. It can be shown that the entries of such a matrix are all real quaternions which can be expressed in terms of 2×2 matrices as Q=a0 :0 +i˜a ·˜:, where the ai are real numbers and (:0 ; ˜:) = (1; :1 ; :2 ; :3 ) are the 2 × 2 unit and Pauli matrices. P4 (H ) dH is invariant under all symplectic transformations of H . The Hamiltonian of a system with time-reversal invariance, no rotational invariance and half-odd integral spin is of this type [46,11]. As we can see, the symmetry classes are distinguished by their behavior under time reversal (TR) and spin rotation (SR). 8.3.2.3. Properties of the gaussian ensembles. As we did for the circular ensembles, let us review some general properties of the gaussian ensembles. Statistical independence of the entries. With the choice of potential in Eq. (8.24) the three ensembles have independently distributed elements, since the potential tr H 2 = ij |Hij |2 . Invariance. As implied above, P (H ) and the integration measure dH are separately invariant under the transformation H → UHU −1 ;
(8.25)
where U is an orthogonal, unitary or symplectic N × N matrix depending on the value of . However, it is important to notice that the symmetry group of P (H ) dH is larger and consists of rotations by the matrix U like in Eq. (8.25), and addition by square hermitean matrices: H → UHU −1 + H :
(8.26)
This will play an important role in the identi4cation of these ensembles with a symmetric space. Diagonalization. For each matrix H there is a matrix U that maps it onto its eigenvalues: H = UOU −1 ;
O = diag( 1 ; : : : ; N ) ;
(8.27)
where i (i = 1; : : : ; N ) are real eigenvalues (if = 4, they are twofold degenerate.) A simple calculation shows that the Jacobian of the transformation from the space of hermitean matrices H to eigenvalue space is given by J ({ i }) ˙
!
| i − j | ;
(8.28)
i¡j
where = 1; 2, and 4 in the orthogonal, unitary and symplectic case, respectively (see [11,61, Chapter 3]). Probability distribution of the eigenvalues. Under the transformation (8.27), the integration measure dH factorizes into an integral over the " symmetry subgroup K and an integral over the eigenvalues with integration measure J ({ i }) i d i . The integral over the subgroup can be performed immediately and gives a constant equal to the volume of the subgroup. With this in mind and the
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random matrix potential given by V (H ) = 12 H 2 , 14 the joint probability density for the eigenvalues becomes N 2 ! P ({ i }) ˙ | i − j | e−=2 j=1 j : (8.29) i¡j
The proportionality constant is not relevant if one normalizes the integral to unity (however, if needed, exact normalization constants for many integrals like the one in (8.29) were given by Hua in [61]). The spectral correlations are due to the Jacobian. Eq. (8.29) can be rewritten in the general form P ({ i }) ∼ e−(
i
V ( i )−
i¡j
ln| i − j |)
:
(8.30)
The logarithmic pair potential leads to repulsion between the eigenvalues. This is the so-called spectral rigidity which is one of the most important features of random matrix theory. It is not present in uncorrelated Poisson distributions. The interpretation of (8.30) is that the probability " of 4nding the i’th eigenvalue in the interval between i and i + d i is proportional to P ({ i }) i d i . 8.3.2.4. The symmetric spaces associated to the gaussian ensembles. Let us now rephrase the above results in a group theoretical language. This will allow us to make contact with the analysis in Section 5. Our goal here is to show that the integration manifolds of the above ensembles are symmetric spaces of zero curvature. This is essentially due to the translational invariance of Eq. (8.26). Let us start with the = 1 case. As a 4rst step let us limit ourselves to traceless matrices only. This is the equivalent in the present context of the choice that we made in the previous subsection when we eliminated the U (1) factor from the coset. Also in the present case the reason is that we want to obtain a description in terms of irreducible symmetric spaces. The remaining degree of freedom acts trivially and is usually neglected. We now try to describe the set of all real, symmetric, and traceless matrices as a symmetric space X = G=K. By comparing with the example in Section 3.1 in Part I of the review, this can easily be done by identifying it with the algebra subspace SL(N; R)=SO(N). This simply amounts to taking a matrix from the set of generic, traceless, real N × N matrices, write it as a sum of a symmetric and an antisymmetric matrix, and then eliminate the antisymmetric part. The resulting space is exactly the set of all real, symmetric and traceless matrices that we were looking for. By taking only traceless matrices, we have eliminated an R+ factor from the algebra GL(N; R), corresponding to the absolute value of the determinant, and we end up with the algebra SL(N; R), of which we then choose only the symmetric generators. Following the discussion of Qat symmetric spaces in Section 5.1, the set of all real, symmetric and traceless matrices (let us call it P) can be realized as a zero-curvature symmetric space by the following two steps: (1) The group G 0 of aVne transformations of P = SL(N; R)=SO(N) into itself is given by the semidirect product of the subgroup K = SO(N ) and the invariant algebra subspace P. The action of G 0 on P is given by g(p) = kpk −1 + a, where g = (k; a) ∈ G 0 , k ∈ K, p; a ∈ X 0 = G 0 =K = P. 14
This form of the potential gave rise to the name “gaussian ensembles”. Of course, a gaussian random matrix potential can be used in any context, as it often is, because it is the simplest one to deal with.
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But this aVne transformation is exactly of the same type as the symmetry transformation of Eq. (8.26) that leaves the gaussian ensembles invariant (in fact, for the algebra subspace under discussion it is this transformation for = 1). (2) The algebra subspace P now forms an abelian subalgebra (identical to the Cartan subalgebra) of the algebra G0 , because it is additive: [P; P] = 0. Therefore G0 = K ⊕ P has an abelian ideal, and is nonsemisimple. The curvature tensor of these spaces is zero by Eqs. (5.1) and (5.2) of Section 5.1. Also, the commutation relations (3.2) de4ning a symmetric space are satis4ed for the algebra elements in K and P. Hence X 0 = G 0 =K = P is a Qat symmetric space with a euclidean geometry. By performing a similar analysis for = 2; 4 we obtain the following general result: The gaussian ensembles labelled by = 1; 2, and 4 consist of hermitean square matrices belonging to algebra subspaces SL(N; R)=SO(N), SL(N; C)=SU(N), and SU∗ (2N)=USp(2N), respectively. The algebra SL(N; C) is the algebra of hermitean, complex and traceless N × N matrices, and the subalgebra SU(N) de4nes the subgroup of invariance of the unitary ensemble. The algebras SU∗ (2N) and USp(2N) were formally de4ned in a footnote in Section 4.2. Again, they are exactly the algebra of hermitean, self-dual N × N quaternion matrices and the subalgebra of unitary symplectic matrices. From Table 1 we see that these three symmetric spaces correspond to algebra subspaces in Cartan classes A, AI and AII. The integration manifolds of the circular ensembles are the positive curvature coset spaces corresponding to the same Cartan classes. 8.3.2.5. Remarks concerning Dat symmetric spaces. The Qat symmetric spaces may be seen as limiting cases of their curved counterparts. Below we make a few observations that illustrate this. As we discussed in Section 5.1, the symmetric spaces are naturally organized in triplets of negative, zero and positive curvature. The simplest way to look at the zero curvature symmetric spaces is to see them as limiting cases of one of the other two symmetric spaces (with nonzero curvature) in the same triplet. This limiting procedure is related to group contraction. As the simplest example of this, we can take the group SO(3). As we saw in Sections 4.1 and 5.1, the algebras SO(3; R), SO(2; 1; R) and E2 are all related to each other. The coset SO(3)=SO(2) can be identi4ed with the unit 2-sphere S 2 , a positive curvature space (cf. Section 2.3). By performing the Weyl unitary trick on the algebra SO(3) we obtain the non-compact algebra SO(2; 1; R). As we showed in Section 4.2, the coset space SO(2; 1; R)=SO(2) corresponds to the hyperboloid H 2 , a negative curvature symmetric space. The commutation relations for the nonsemisimple euclidean group E2 are a limiting case of the commutation relations for SO(3) and SO(2; 1) in the limit of in4nite radius (cf. Section 5.1). The symmetric space E2 =SO(2) is thus a zero curvature symmetric space, isomorphic to the euclidean plane and a limiting case of the corresponding positive and negative curvature spaces in the same triplet. A simple and important example of a zero curvature symmetric space, which has several analogies with the case at hand, is the Qat euclidean space in four dimensions. It is well known that this space ˜ can be realized as the coset of the euclidean Poincare’ group P˜ with respect to SO(4): P ∼ P=SO(4). The translations of the Poincare’ group play the role of P, they are isomorphic to euclidean space and have all the characteristics of a symmetric space of vanishing curvature. The above observation that the zero curvature spaces can be obtained as limits of positive curvature spaces can be exempli4ed as follows. We can realize the euclidean Poincare’ group as a suitable limit of the SO(5) group.
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In this limit the coset SO(5)=SO(4) (which is nothing else than the four dimensional unit sphere) exactly becomes the euclidean four-dimensional space. We can rephrase the above result in still another (more geometric) way. The zero curvature symmetric spaces can be seen as tangent spaces of their curved partners. This is clear both from the above example and from the algebraic structure of the space. 8.3.3. Chiral ensembles A natural generalization of the previous results is the extension to rectangular random matrices. The study of this class of ensembles was initiated in the mid-1980s by Cicuta et al. [62] and further discussed in Ref. [63] where they were named Laguerre ensembles (due to the fact that the orthogonal polynomials associated to them are Laguerre polynomials). An updated review on this subject can be found for instance in [27]. 8.3.3.1. Physical applications of the chiral ensembles. One of the most relevant applications of the chiral ensembles is in the study of the infrared limit of gauge theories. In particular, the universal properties of Dirac spectra that depend only on the global symmetry of the euclidean Dirac operator iD , , can be reproduced by substituting the integral over gauge 4elds in the euclidean partition function by an integral over an appropriate random matrix ensemble. The work by Verbaarschot et al. [64] related to this subject dates back to the 1990s. It is in this framework (for reasons which will soon be clear) that these ensembles came to be named “chiral”. In this paragraph we discuss some of their characteristics with emphasis on their connection with the theory of symmetric spaces. Chiral ensembles are also relevant in connection with random Qux and random hopping problems. In this subsection we will discuss two papers by Mudry et al. [49,50] where chiral ensembles are realized. Chiral random matrix theory and QCD. The interest in the low-lying spectrum of the Dirac operator is due to the Banks–Casher formula [65], /7(0) qq W = (8.31) V relating the magnitude of the quark condensate in QCD in a 4nite volume V to the spectral density of the Dirac operator at the origin. The part of the Dirac spectrum near the origin is therefore of interest in studying the mechanism of spontaneous breaking of chiral symmetry, since the quark condensate is an order parameter for the chiral transition. Since we are interested in the connection with symmetric spaces in this review, we will not further report the results of these important applications. We only give a sketchy description below and refer to the literature cited for more detailed information. Due to the presence of chiral symmetry ({E5 ; iD , } = 0) in four space–time dimensions, the Dirac operator has the block structure 0 W iD ; (8.32) , = W† 0 where W is rectangular, say p × q (p ¿ q), thus I ≡ p − q corresponds to the number of fermionic zero modes. In QCD, these may describe the zero modes in the 4eld of an instanton. Then I
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equals the winding number (topological charge). Zero modes may also originate in the bulk of the spectrum due to repulsion between eigenvalues. The total size N = p + q of the matrix in (8.32) corresponds to the 4nite space–time volume V in the gauge theory. In the calculations to be sketched below, the rectangular submatrices W in the euclidean Dirac operator iD , will be replaced by random matrices T chosen from an appropriate ensemble. The symmetries of the chiral ensembles of random matrices (i.e. the ones having a block-structure like the one in Eq. (8.32)) are chosen on the basis of the fermion representation (fundamental, adjoint) and the number of quark colors. The various possibilities lead to the presence (or absence) of an anti-unitary symmetry [iD , ; Q] = 0 ;
(8.33)
where Q is an anti-unitary operator. The presence or absence of the symmetry (8.33) and the explicit form of Q determines whether the Dirac operator can be represented as a matrix with real ( = 1), complex ( = 2) or quaternion real ( = 4) elements [66]. The corresponding random matrices are in Cartan class BDI, AIII, and CII. In the gauge theory partition function in euclidean space # ! Z = DAe−S[A] det(D (8.34) , + mf ) f
(where the integral is over the gauge 4eld con4gurations, S[A] is a gauge 4eld action, the product of fermion determinants is over the Qavor degree of freedom and mf is the fermion mass for Qavor f) the integral over gauge 4elds is then substituted with a gaussian average over a random matrix T . Since the Dirac operator has the block-structure in Eq. (8.32), we can substitute W with T to get a much simpler model # ! mf iT −(q>2 =2)tr T † T Z = DT e (8.35) det iT † mf f with the same symmetries as the original partition function (DT is a Haar measure and the size of T is p × q). From this random matrix theory we obtain the eigenvalue density and correlators, in particular their universal microscopic limit. Both the random matrix theory and the 4eld theory map onto the same low-energy eOective partition function in the mesoscopic regime describing static Goldstone modes in a 4nite volume. This partition function is expressed as an integral over a coset manifold G =K related to spontaneous symmetry breaking and the emergence of Goldstone modes. Here G is the original symmetry group of the Lagrangian and K is the unbroken subgroup. It is worth noting that these Goldstone manifolds obtained in [66–68] for = 1, 2, and 4 for QCD in both three (non-chiral ensembles) and four (chiral ensembles) space–time dimensions are saddle-point manifolds that appear in the large-N limit and should not be confused with the symmetric spaces G=K identi4ed with the original ensembles. These Goldstone manifolds are the fermionic parts MF of the symmetric supermanifolds MB × MF discussed by Zirnbauer in [4]. In addition to the 4nite volume partition function of static Goldstone modes expressing the quark mass dependence, one can obtain the Qavor symmetry (or parity in odd space–time dimensions) breaking pattern. One can also derive sum rules constraining the eigenvalues of the Dirac operator in a 4nite volume [66–68]. Chiral ensembles in the context of random Dux and random hopping problems. It is of interest in the theory of quantum transport to study how the conductance of a system behaves analytically
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as one or more of the dimensions of the system go to in4nity. This approach is called the scaling theory of localization, and deals with the conditions under which a system is metallic, insulating, or undergoes a transition between the two states. Such a transition can be disorder-induced, i.e. it can depend on impurities in the metal. Another important factor is dimensionality. We postpone a further discussion of transport problems to the subsection on transfer matrix ensembles. In this paragraph we will only mention some results relating to chiral ensembles. In the random magnetic Qux problem one studies the localization properties of a spinless electron moving in a plane perpendicular to a static magnetic 4eld of random amplitude and vanishing mean. In two dimensions, it is diVcult to reach a conclusion about the localization properties of the states based on numerical data. Therefore it is easier to study a quasi-one dimensional con4guration. This was done by Mudry et al. in Ref. [49], where an M × N lattice (M N 1) corresponding to a thick quantum wire with weak disorder was studied. Although the study of such wires is a vast 4eld, we will limit ourselves here to reporting only one of the results of this interesting paper, so as not to go beyond our scope, which is to give examples of physical manifestations of chiral random matrix symmetry classes. In the case at hand, the system is governed by a Hamiltonian H (whose detailed form the interested reader can look up in the original paper) through a SchrNodinger equation H = j . Away from the band center j = 0, the localization properties of the particle are those of the standard gaussian unitary ensemble. In other words, the Hamiltonian H of the system is in Cartan class A. Exactly at the band center j = 0, an additional symmetry of the transfer matrix, called chiral or particle–hole symmetry, changes the symmetry class of the Hamiltonian into the chiral unitary symmetry class AIII. (Note that the presence of the magnetic 4eld breaks time-reversal symmetry, so the chiral orthogonal and the chiral symplectic class cannot be realized in this system.) Thus the random magnetic Qux problem provides a physical realization of the chiral unitary ensemble. We will refer to the above results again in connection with transfer matrices, when we enumerate some of the physical manifestations of symmetric spaces of negative curvature. Related to the random Qux problem is the random hopping problem, in which a particle hops on a lattice with random hopping amplitudes. Also in this problem one 4nds that the point at the center of the band is special. A delocalization transition in such a system in one dimension goes back to work by Dyson (see the discussion and references in [50]). A quasi-one dimensional hopping model with weak staggering was investigated in [50]. The lattice consisted of N coupled chains of length L, where LN . In the vicinity of the Fermi energy the lattice model was approximated by a continuum model. For = 1; 2; 4 respectively, the Hamiltonian has the symmetries of a chiral ensemble and belongs to Cartan classes BDI, AIII, and CII, respectively. For our analysis the important point here is that the Hamiltonian in each case is a realization of a chiral symmetry class, just like the Dirac operator in four-dimensional QCD and the Hamiltonian at the band center in the random Qux problem. For details and other results of the quasi-1d random hopping problem we refer to the original paper. 8.3.3.2. Some properties of the chiral ensembles. Invariance and diagonalization. Just like in the circular and gaussian ensembles, the rectangular random matrix T above can be diagonalized through a transformation with unitary matrices U and V : T = UOV −1 :
(8.36)
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The matrix O is a diagonal real matrix with positive elements corresponding to the non-zero eigenvalues of T . The partition function in (8.35) is invariant under this transformation. Probability distribution of the eigenvalues. It can be shown that the Jacobian for the transformation from matrix to eigenvalue space for the chiral ensembles is given by ! ! (I+1)−1 JI ({ i }) ˙ | i2 − j2 | k ; (8.37) i¡j
k
where { i } are the eigenvalues of the rectangular matrix T . In terms of the new variables xi ≡ i2 (8.37) takes the well-known form ! ((I+1)−2)=2 ! JI ({xi }) ˙ |xi − xj | xi : (8.38) i¡j
k
Here I is the number of zero modes of the chiral matrices. 8.3.3.3. The symmetric spaces associated to the chiral ensembles. The chiral block-structure of the random matrices already gives a hint that they must belong to the subspace P of some algebra (see Section 3.5). Depending on the number of degrees of freedom of the matrix elements (i.e. the value of : 1, 2, or 4), it is not hard to guess that this subspace is identi4ed respectively with SO(p; q)=(SO(p) ⊗ SO(q)), SU(p; q)=(SU(p) ⊗ SU(q)), or USp(p; q)=(USp(p) ⊗ USp(q)) (in this case p, q have to be even 15 ). Obviously, these algebra subspaces are symmetric spaces of the euclidean (zero-curvature) type. They correspond to Cartan classes BDI, AIII and CII in Table 1. Above we have enumerated several physical realizations of these symmetry classes. 8.3.4. Transfer matrix ensembles The transfer matrix ensembles appear in the theory of quantum transport, in the random matrix theory description of the so-called quantum wires. In these pages we will only discuss the part of the theory which is relevant for our purpose, the study of the mapping between random matrix theory and symmetric spaces. For further information on the experimental and theoretical issues we refer the reader to the excellent introductory review on random matrix theory and quantum transport by Beenakker [26]. 8.3.4.1. Physical context of the use of transfer matrix ensembles. The natural theoretical framework for describing mesoscopic systems is the Landauer theory [69]. Within this approach Fisher and Lee proposed the following expression for the conductance in a two-probe geometry (a 4nite disordered section of wire to which current is supplied by two semi-in4nite ordered leads): 2e2 G = G0 Tr(tt † ) ≡ G0 ; (8.39) Tn ; G0 = h n where t is the N × N transmission matrix of the conductor (see Eq. (8.14)), N is the number of scattering channels at the Fermi level and T1 ; T2 ; : : : ; TN are the eigenvalues of the matrix tt † . The Ti ’s are usually referred to as transmission eigenvalues. The constants e and h denote the electronic charge and Planck’s constant, respectively. 15
Our notation is such that USp(2p) ≡ USp(2p; C) ≡ U (p; Q).
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The transmission eigenvalues Ti are related to the scattering matrix and to the transfer matrix, which de4ne equivalent descriptions of the impurity scattering process. In the next paragraph we will de4ne the relationship between the physical degrees of freedom Ti and the transfer matrix. 8.3.4.2. De
(8.42)
(8.43)
(8.44)
where u, u , v, v are unitary N × N matrices (related by complex conjugation: u = u∗ , v = v∗ if M ∈ Sp(2N; R) or M ∈ SO∗ (4N ), see below) and O=diag( 1 ; : : : ; N ). In case spin–rotation symmetry is broken, the number of degrees of freedom in (8.44) is doubled and the matrix elements are real quaternions. 8.3.4.3. The symmetric spaces associated to the transfer matrix ensembles. Transfer matrices are strongly constrained by various physical requirements. As a result they belong to one of three diOerent symmetric spaces. We discuss in some detail below how these symmetric spaces are obtained. The physical requirements of Qux conservation, presence or absence of time-reversal symmetry, and presence or absence of spin–rotation symmetry lead to conditions on the transfer matrix. These
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conditions determine the group G to which M belongs. Flux conservation leads to the following condition on M [59] 1 0 M † >z M = >z ; >z = ; (8.45) 0 −1 i.e. M preserves the (2N × 2N ) metric >z . This means that M belongs to the pseudo-unitary group SU (N; N ) (M has to be continuously connected to the unit matrix so we take the connected component of U (N; N )). If the Hamiltonian is invariant under time-reversal, the condition on the transfer matrix is [59] 0 1 ∗ M = >x M>x ; >x = : (8.46) 1 0 It is easy to check that together with the condition of Qux conservation, this implies 0 −1 T M JM = J; J = >x >z = : 1 0
(8.47)
But J is the skew-symmetric form invariant under the non-compact symplectic group Sp(2N; R) (see e.g. [2,35,70]), thus M belongs to this group in case time-reversal symmetry is present. If the Hamiltonian contains a spin–orbit interaction (i.e. in case of magnetic impurities in the conductor), spin–rotation symmetry is broken. In this case the presence of an extra spin degree of freedom doubles the number of components in the vectors I , I , O, O of incoming/outgoing wave amplitudes (the components become spinors). Formally, the same conditions (8.45), (8.46) on M are valid for Qux conservation and time-reversal symmetry, but M is now a matrix of N × N real quaternion elements [71]. If time-reversal symmetry is broken, we get M ∈ SU (2N; 2N ) like before. If it is conserved, a condition analogous to (8.46) is valid, with the only diOerence that the matrices now act on an N -dimensional vector space of real quaternions. In this case M belongs to the group SO∗ (4N ). This is the connected component of the group of linear transformations that preserve a skew-symmetric bilinear form on a quaternionic vector space (see, e.g. [72, paragraph 7.2]; cf. also [2]). Invariance of the transfer matrix ensembles. As we can check using the parametrization Eq. (8.44), rotating M by a matrix W ∈ U (N ), SU (N )×SU (N )×U (1), or U (2N ) (if M ∈ Sp(2N; R), SU (N; N ), or SO∗ (4N ), respectively), gives a new transfer matrix M = WMW −1 = U FV with the same physical degrees of freedom { 1 ; : : : ; N }, since the matrix F is unchanged. This means that in each case F belongs to a coset space G=K, where M ∈ G and W ∈ K. These three ensembles of transfer matrices, corresponding to diOerent physical symmetries, are usually named (in analogy with the nomenclature of the standard gaussian ensembles) the “orthogonal” (Sp(2N; R)=U (N )), “unitary” (SU (N; N )=SU (N ) × SU (N ) × U (1)) and “symplectic” (SO∗ (4N )=U (2N )) ensemble. This nomenclature is a bit unfortunate, since the stability subgroups do not correspond to it. These coset spaces are irreducible and noncompact. Thus they are symmetric spaces of negative curvature, as is also evident from Table 1. They correspond to Cartan classes CI, AIII, and CII, respectively. New transfer matrix ensembles in the context of random tight-binding models. In connection with the chiral ensembles we discussed the Hamiltonian at the band center in a random magnetic Qux problem. It is of the chiral symmetry class AIII [49]. This means that the transfer matrix for
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the same physical system belongs to Cartan class A, i.e. it is a physical realization of the coset space SL(N; C)=SU (N ). We also discussed the random hopping problem [50] in the same context and saw that the Hamiltonian for = 1; 2; 4 has chiral symmetry. This means the transfer matrices of the systems belong to coset spaces SL(N; R)=SO(N ) (Cartan class AI), SL(N; C)=SU (N ) (Cartan class A), and SU ∗ (2N )=USp(2N ) (Cartan class AII). These are new realizations of transfer matrix ensembles not found in the applications discussed above. In the same type of problems, also other transfer matrix ensembles are manifest. We refer to [6,49,50,73,76,87] for more details. Of course, to each hamiltonian ensemble corresponds a transfer matrix ensemble and a scattering ensemble, so in a certain sense, all the boxes in Table 2 should be 4lled with “physical” ensembles. As we have already noted, the correspondence between hamiltonian ensembles and the transfer matrix ensembles for the same systems was given in Table I of Ref. [6]. By inspection of the parametrization of Eq. (8.44), we realize that the (generalized) eigenvalues i are not exactly the radial coordinates in the three symmetric spaces. By a change of coordinates (see Eq. (8.51) below), we will be able to “disentangle” the coordinates in such a way that this parametrization, and the DMPK equation to be discussed in the next paragraph, become much more tractable. 8.3.5. The DMPK equation What is needed at this point to complete the transfer matrix description of the quantum wire, is the explicit expression for the probability distribution of the { i } as a function of an external parameter, the length L of the quantum wire. The standard approach to this rather nontrivial problem has been to 4nd some dynamical principle so as to obtain an “equation of motion” for the probability distribution of the { i } as a function of L, and then (hopefully) to solve the equation and obtain the probability distribution. The construction of the equation for the probability distribution was completed during the 1980s, at least in the case of quasi-one-dimensional wires (LW where W is the thickness of the wire), by Dorokhov [74], and independently by Mello et al. [59] (for = 1) by studying the in4nitesimal transfer matrix describing the addition of a thin slice to the wire. The resulting evolution equation for the eigenvalue distribution P({ i }; s), where s is the dimensionless length of the wire, is usually known as the Dorokhov–Mello–Pereyra–Kumar (DMPK) equation. The only assumptions which are needed to obtain this equation are: (1) that the conductor is weakly disordered, so that the scattering in the thin slice can be treated by using perturbation theory, and (2) that the Qux incident in one scattering channel is, on average, equally distributed among all outgoing channels. It is exactly this second assumption which restricts the DMPK equation to the quasi-1D regime, where the 4nite time scale for transverse diOusion can be neglected. The results of [59] were subsequently generalized to = 2; 4 in Refs. [75,71]. The DMPK equation can be written as 9P = DP ; 9s
(8.48)
where s is the wire length measured in units of the mean free path l: s ≡ L=l, and D can be written in terms of the { i } as follows: N
D=
2 9 9 i (1 + i )J ( ) J ( )−1 ; E i=1 9 i 9 i
(8.49)
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with E ≡ N + 2 − . is the symmetry index of the ensemble of scattering matrices, in analogy with the well-known Wigner–Dyson classi4cation, and J ( ) ≡ J ({ n }) is given by ! J ( ) = | j − i | : (8.50) i¡j
The solution of this equation will be discussed in the forthcoming subsections. For the moment, let us only point out that J ( ), which is the Jacobian of the transformation from random matrices to eigenvalues and eigenangles in the gaussian ensembles, has nothing to do with the Jacobian between the space of transfer matrices and the space of radial coordinates in the present case. Its appearance in the DMPK equation has historical reasons, as the authors tried to mimic the well-known gaussian ensembles. This choice of coordinates makes the DMPK equation asymmetric and ultimately hard to solve. The relationship between the Jacobian and the radial part of the Laplace–Beltrami operator. The DMPK equation can be rewritten in a more tractable form by making the change of coordinates i = sinh2 xi :
(8.51)
This introduces the “right” Jacobian for the transformation to eigenvalue space in the transfer matrix ensemble. This Jacobian turns out to be the function K2 ({xi }) ≡ K2 (x) given by ! ! J ({xi }) = K2 (x) = |sinh2 xj − sinh2 xi | |sinh 2xk | : (8.52) i¡j
k
Let us introduce a new operator B de4ned as N 9 2 9 −2 B= K (x) K (x) : 9xk 9xk
(8.53)
k=1
It is easy to see, by direct substitution, that B is related to the DMPK operator by 1 D= B : 2E Thus we can rewrite the DMPK equation as 1 9P = BP : 9s 2E
(8.54)
(8.55)
By comparing Eq. (8.53) with Eq. (6.26) of Section 6.2 we see that the operator B is related to the radial part of the Laplace–Beltrami operator DB on the negative curvature symmetric spaces associated to the transfer matrix ensembles by DB = J −1 BJ . This is the 4rst indication of a general, important relation which we will discuss in more detail in Section 8.5 below. Similar Fokker–Planck equations can be derived for all the symmetry classes. In [50] the equation was derived in conjunction with a tight-binding model for the cases in which the transfer matrix is in the standard ensembles and the Hamiltonian in the chiral ensembles. In [76] it was derived for systems with Hamiltonians of the Bogoliubov–de Gennes (BdG) type (see the next section). The latter apply to quasiparticle transport at the Fermi level in a disordered superconducting wire. In this case charge is not conserved, but one can study transport of heat and spin. In standard systems, localization occurs when the wave function of an electron undergoes a transition from a Bloch wave extending throughout the sample to a localized exponentially decaying form
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(r) ∼ e−r=K , where K is the localization length. As a result, in the localized regime the conductivity is dominated by the lowest eigenvalue and decreases exponentially with the length of the wire. In the context of disordered wires, the seven universality classes pertaining to chiral and BdG Hamiltonians are more interesting than the standard ones in that they can display a departure from exponential localization [76]. 8.3.6. BdG and p-wave ensembles 8.3.6.1. Physical applications of the BdG ensembles. The so-called NS ensembles are examples of Bogoliubov–de Gennes, or BdG, ensembles. They get their name from the physical structures they describe [3]. The symmetry classes of these new ensembles are realized in normal metal–superconductor (NS) heterostructures. These are mesoscopic systems composed by a normal conductor in conjunction with a superconductor. Because of Andreev reQection at the NS interface, this system is diOerent from the conventional one. Just like the additional chiral symmetry of the Dirac operator and the symmetries of the transfer matrix of a normal mesoscopic conductor give rise to new ensembles, so the symmetries of the so called Bogoliubov–de Gennes (BdG) Hamiltonian give rise to four new symmetry classes depending on whether time-reversal (TR) and/or spin–rotation (SR) symmetry is present. The BdG Hamiltonian is a 4rst-quantized version of the BCS Hamiltonian in a mean-4eld approximation. It has an additional particle–hole grading which is absent in the Hamiltonian for a normal metal. This gives rise to a discrete particle–hole symmetry. 8.3.6.2. The symmetric spaces associated to the BdG ensembles. In [3] the authors show that the BdG Hamiltonian H (actually iH) belongs to one of four symmetry classes depending on which symmetries are present. Since the presentation in [3] is excellent, we will just summarize the results here. The space to which iH belongs is an algebra or an algebra subspace and is either SO(4N) (no TR and no SR), USp(2N) (only SR), SO(4N)/U(2N) (only TR), or USp(2N)/U(N) (both TR and SR). All these algebra subspaces can be considered to be symmetric spaces of zero curvature, by the construction we have repeatedly discussed. By comparing with Table 1 we see that they can be identi4ed with Cartan classes D, C, DIII and CI, respectively. The Jacobian for the transformation by the stability subgroups to radial coordinates is [3] ! ! Jr; s ({qi }) ˙ |qi2 − qj2 |r |qk |s ; (8.56) i¡j
k
where the pair (r; s) takes the values (2; 0), (2; 2), (4; 1), and (1; 1) for SO(4N), USp(2N), SO(4N)/ U(2N), and USp(2N)/U(N), respectively. For a review of NS junctions and the BdG equation, see [26]. Scattering matrices of the BdG type. The scattering matrix for an NS-type heterostructure is obtained by exponentiation of the Hamiltonian: S = eiH . Since iH is in the algebras or tangent spaces listed above for the respective symmetry classes, the scattering matrix is in the corresponding symmetric spaces of positive curvature. Further information on these ensembles can be found in [26]. 8.3.6.3. The symmetric spaces associated to the p-wave ensembles. Recently, Ivanov [7] found realizations of Cartan classes B and DIII-odd in the algebra SO(2N + 1) and the algebra subspace SO(4N + 2)=U(2N + 1) in p-wave superconductors. The corresponding ensembles are called p-wave
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ensembles and are characterized by the presence of a zero mode. We recall that I = |p − q| zero modes are also present for the chiral ensembles for p = q (for notation see Table 1). These two classes were obtained for disordered vortices in p-wave superconductors with or without timereversal symmetry, using the same Bogoliubov–de Gennes Hamiltonian as in [3]. As in the case of BdG ensembles, the identi4cation of the p-wave ensembles with algebra (sub)spaces maps them onto Qat symmetric spaces. For more comments on these ensembles see [7]. 8.3.6.4. Vicious walks and BdG ensembles. Recently a new and completely independent realization of the gaussian BdG ensembles has been proposed in the context of the two-matrix model description of the vicious walk in presence of a wall (see [77] and references therein). In this case the peculiar symmetries of the BdG ensemble are a direct consequence of the boundary conditions imposed by the presence of the wall in the model. 8.3.7. S-matrix ensembles 8.3.7.1. Physical applications of the S-matrix ensembles. The important class of matrix models referred to as S-matrix ensembles was introduced a few years ago in [78,79] to describe the behavior of ballistic chaotic quantum dots. These are microstructures in which impurity scattering can be neglected, and only boundary scattering is considered. 8.3.7.2. De
1 − Ti Ti
(8.57)
one obtains a representation suitable for studying the transport properties through a quantum dot. The resulting distribution P ({ i }) takes the form of a Gibbs distribution. A direct calculation [78] shows that P ({ i }) ∼ e−(
i
V ( i )−
2− V ( i ) = N + 2
i¡j
ln | i − j |)
;
ln(1 + i ) :
(8.58)
8.3.7.3. The symmetric space associated to the S-matrix ensemble for = 2. It appears that the S-matrix ensemble for = 2 corresponds to the compact symmetric space SU (2N )=SU (N ) × SU (N ) × U (1) (for p = q in SU (p + q)=SU (p) × SU (q) × U (1), which corresponds to the multiplicity of the short roots ms = 2(p − q) = 0, i.e. a root lattice of type BCn that reduces to Cn for p = q). Interestingly, as we will see in Section 8.4.1, the two other ensembles (for = 1; 4)
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do not 4nd correspondence in the Cartan classi4cation [5]. This issue will be discussed further in Section 10. 8.4. Identi
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ensembles. Algebraically, the curvature of the spaces is reQected in the explicit form of the Jacobian of the transformation to spherical coordinates. This can be understood by comparing with Eq. (6.42) which we reproduce here for comparison: ! J (0) (q) = (q )m ; ∈ R+
J (−) (q) =
!
(a−1 sinh(aq ))m ;
∈ R+
J (+) (q) =
!
(a−1 sin(aq ))m :
(8.59)
∈ R+
Note that the root multiplicities are the same for the spaces of positive and negative curvature in the triplet corresponding to the same symmetric subgroup. Even though we have not de4ned a restricted root system for the zero-curvature spaces arising from non-semisimple groups, the same multiplicities characterize also the Jacobian pertaining to the zero-curvature member of the triplet. This should be understood as explained in the remark following Eq. (6.30) in Section 6.2. 8.4.1. Discussion of the Jacobians of various types of matrix ensembles Let us now come back to the classi4cation scheme. The root multiplicities are listed in Table 1, to which we systematically refer in the following. This means that a few integers and the sign of the curvature (+; −; 0) are enough to completely characterize a matrix ensemble. 8.4.1.1. The gaussian ensembles. As we saw in Section 8.3.2, the standard gaussian ensembles are labelled by the Dyson index . It is noteworthy that this index is exactly equal to the multiplicity mo of ordinary roots of the restricted root lattice characterizing a triplet of symmetric spaces (see comments in the previous subsection concerning the “root multiplicities” characterizing a non-semisimple algebra G0 ). The Jacobian in the gaussian case is given by Eq. (8.28): ! J (0) ({ i }) ∼ | i − j |mo : (8.60) i¡j
This case corresponds to the An−1 root lattices with only ordinary roots. The Jacobian in (8.60) has exactly the form given in Eq. (8.59) for zero curvature spaces with a root lattice of type An−1 (ml = ms = 0). The positive roots in this case are = ei − ej (i ¿ j) and the expressions entering in the Jacobian are the radial coordinates ≡ · = i − j (see Section 6.2). The absolute value in Eq. (8.60) corresponds to a choice of Weyl chamber (cf. Section 2.6) 1 ¿ 2 ¿ · · · ¿ N . 8.4.1.2. The circular ensembles. The Jacobian of the circular ensembles of unitary scattering matrices was given in (8.17): ! |ei!i − ei!j | : (8.61) J ({!i }) ˙ i¡j
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It is not hard to see that this Jacobian can be rewritten in the form ! !i − !j m0 (+) J ({!i }) ∼ 2 sin 2
123
(8.62)
i¡j
which is exactly the form given by Eq. (8.59) for the Jacobian pertaining to the transformation to spherical coordinates on a symmetric space of type An−1 with only ordinary roots and of positive curvature, if we choose the radius a = 1=2. This allows us to identify these ensembles with the positive curvature symmetric spaces of Cartan classes A, AI and AII, in agreement with what we had already concluded from the corresponding integration manifolds. 8.4.1.3. The chiral ensembles. The Jacobian given for the chiral ensembles in Eq. (8.37) ! ! JI ({ i }) ˙ | i2 − j2 | | k |(I+1)−1 (8.63) i¡j
k
corresponds in the most general case to a root lattice with all types of roots. The chiral ensembles are algebra subspaces, and like for the gaussian ensembles we associate to it the restricted root lattice of the curved symmetric spaces in the same triplet. The restricted root lattices for the chiral ensembles are of type Bq or Dq for = 1 and BCq or Cq for = 2; 4. In case the root lattice is not of type An−1 , not only is ei − ej a positive root but also ei + ej . Thus the positive roots are as follows: for Bq {ei ; ei ± ej }, for Dq {ei ± ej }, for BCq {ei ; ei ± ej ; 2ei }, for Cq {ei ± ej ; 2ei } (i = j always). Using the root multiplicities mo = , ml = − 1, ms = |p − q| ≡ I we see that the Jacobian is of type (8.59). It can be rewritten as ! ! J (0) ({ i }) ∼ | i2 − j2 | | k | ; ≡ mo ; ≡ ms + ml : (8.64) i¡j
k
From the Jacobian it is evident that in addition to the usual repulsion between diOerent eigenvalues, i also repels its mirror image − i , and the eigenvalues are no longer translationally invariant. This kind of ensembles are therefore called boundary random matrix theories. The boundary random matrix theories include chiral and NS ensembles. 8.4.1.4. The transfer matrix ensembles. The “orthogonal” and “symplectic” transfer matrix ensembles correspond to the same classes of symmetric spaces as the BdG ensembles (see below), but with negative curvature. The “unitary” transfer matrix ensemble also corresponds to a symmetric space of negative curvature, but it is of the chiral, not BdG type (see Table 1). The relevant root systems are all of type Cn with long and ordinary roots. The multiplicities of these can be read from Table 1. ml is always equal to 1 and mo = . Therefore the Jacobian given in Eq. (8.52), ! ! J ({ i }) ˙ |sinh2 i − sinh2 j | sinh(2 k ) (8.65) i¡j
k
is of the general form J (−) ({ i }) if we note that the positive roots are {ei ± ej ; 2ei } and use the following identity for hyperbolic functions: sinh( i − j ) sinh( i + j ) = sinh2 i − sinh2 j
(8.66)
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8.4.1.5. The BdG ensembles. In exactly the same way as above, using Table 1 and the trigonometric identity similar to Eq. (8.66) sin( i − j ) sin( i + j ) = sin2 i − sin2 j ;
(8.67)
we check that the Jacobian for the BdG ensembles of scattering matrices ! ! Jr; s ({ i }) ˙ |sin2 i − sin2 j |r sins (2 k ) i¡j
(8.68)
k
is the one corresponding to positive curvature spaces (J (+) ({ i })) and determined by the corresponding restricted root systems of type Dn (only ordinary roots) or Cn (long and ordinary roots) according to Eq. (8.59). This Jacobian corresponds to scattering matrices of the BdG type, S = eiH . For the corresponding algebra subspaces to which iH belongs, the Jacobian was given in Eq. (8.56). Obviously, its structure is the one given by the same restricted root lattices according to Eq. (8.59), but for zero curvature spaces (J (0) ({ i })). 8.4.1.6. The S-matrix ensembles. In terms of the transmission eigenvalues Tn the probability distribution of Eq. (8.58) (which is the same as the Jacobian for circular ensembles) can be rewritten as ! ! P({Ti }) dTi ≡ J ({Ti }) dTi ∼ |Ti − Tj | |Tk |(−2)=2 dTk : (8.69) i¡j
k 2
Changing variables and setting Ti = sin i we 4nd ! ! J ({ i }) d i ∼ |sin2 i − sin2 j | |sin k |−2 |sin 2 k | d i¡j
k
:
(8.70)
k
As we have by now understood, this is the typical Jacobian for a positive curvature space, described by a root lattices of the BCn type with root multiplicities mo = ; ms = − 2; ml = 1, where the Dyson index as usual can take the three values = 1; 2; 4. However, in this case we see a rather unexpected feature: it seems that only one of the three ensembles can be mapped onto a symmetric space. It is the unitary ensemble ( = 2), which is described by a Cn root lattice, since for = 2, ms = 0. The other two cases correspond to choices of the root multiplicities which lie outside the classi4cation of Table 1. We will come back to this problem in Section 10.1, where we will see that the ensembles characterized by Dyson index =1 and 4 can be understood in terms of a non-Cartan parametrization of standard symmetric spaces. 8.5. Fokker–Planck equation and the Coulomb gas analogy An important tool in the theory of random matrices is the so called “Coulomb gas analogy” due to Dyson (see [11, Chapter 8] for a review). The content of this analogy is the following. The probability distribution P( 1 ; : : : ; N ) ˙ e−W ({ i }) (8.71) of random matrix eigenvalues is identical to the probability density of the positions { i } of N unit charges in one dimension in a stationary potential that is given by N 1 2 − ln| i − j | (8.72) W ({ i }) = 2 i=1 i i¡j
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if the random matrix potential is of the gaussian type. The temperature of this system is kT = −1 . The force exerted by the potential W is of the Coulomb type. If in addition to this force one considers the Brownian motion of the unit charges in the presence of a time-dependent random Quctuating force and a frictional force, one can derive a Fokker–Planck type equation describing the evolution of the Coulomb gas with time t [11]. This analogy allows us to write the probability distribution of eigenvalues in the gaussian and circular ensembles as the asymptotic limit of a Brownian motion process. Similar equations can be written also for all the other ensembles of positive and zero curvature listed in the previous section. The relevant feature of these equations for the purpose of the present review is that the diOerential operator which describes the Brownian motion in the case of the positive curvature ensembles can be exactly related to the radial part of the Laplace–Beltrami operator on the corresponding symmetric space. Remarkably enough, if one performs a similar mapping for the negative curvature symmetric spaces (i.e. the transfer matrix ensembles) one exactly 4nds the DMPK equation. Below we will discuss all this in more detail, but let us 4rst summarize the results: • The DMPK equation is the exact analog for the negative curvature symmetric spaces of the Coulomb gas equation introduced by Dyson for the positive and zero curvature ensembles. • All these equations describe the free diOusion—this is the ultimate meaning of the Laplace– Beltrami operator—of the eigenvalues on the corresponding symmetric spaces. • For the positive curvature ensembles, in the large time asymptotic limit the eigenvalues reach the distribution dictated by the Jacobian. In fact, if we had the whole Laplace–Beltrami operator (not just its radial part) we would expect a uniform distribution on the symmetric space, which is exactly the probability distribution of the random matrices in the circular ensembles. Since the Dyson operator is the radial projection of the Laplace–Beltrami operator, the 4nal result is that the eigenvalues are distributed according to the Jacobian of the transformation to radial coordinates. • For the negative curvature ensembles, in the asymptotic limit the eigenvalues depart from each other exponentially. This behavior has a deep physical meaning. Indeed, it is well known that for quasi-one-dimensional wires, as the length of the wire (which corresponds to the time in the Brownian motion analogy) increases, the wire eventually reaches the insulating regime. This perfectly agrees with the Brownian motion picture. As the time increases the eigenvalues depart arbitrarily far from each other. As a result the conductivity is dominated by the lowest eigenvalue and decreases exponentially, in agreement with the fact that we have reached the insulating regime. Let us now discuss these results in detail. 8.5.1. The Coulomb gas analogy Let us concentrate on the gaussian case for de4niteness. As we mentioned above, the probability density P ({ i }) of Eqs. (8.29) and (8.71) can also be interpreted as the probability density for the positions of N unit charges constrained to move on a line and interacting through the potential (8.72). The integral # ! Z() = d i e−W ({ i }) (8.73) i
for the probability can thus be interpreted as a standard thermodynamic partition function with playing the role of inverse temperature. A nice feature of this representation is that we may obtain
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the distribution P ({ i }) as the asymptotic limit of a Brownian motion type process. To this end the following steps are needed: First we must add a 4ctitious time dependence in P . Then we must assume that the out-ofequilibrium dynamics of the Coulomb gas is of the Brownian motion type (free diOusion). As we implied above, this involves introducing, in addition to the Coulomb force, a time-dependent rapidly Quctuating force giving rise to the Brownian motion, and a frictional force. As a consequence the time dependence of P ({ i }; t) must be described by the following Fokker–Planck equation [11]: & ' N 9 9P 1 92 P − [E( j )P] (8.74) f = 9t 9 j2 9 j j=1 where f is the friction coeVcient setting the time scale for the diOusion process, and 1 9W ({ i }) E( j ) ≡ − = − j + 9 j j − i
(8.75)
i =j
is the electric Coulomb-type force experienced by the unit charge at j . Standard manipulations show that the Fokker–Planck equation can be equivalently rewritten as N
9P 9 ˜ 9 ˜ −1 f J J P ; = 9t 9 j 9 j j=1
(8.76)
where J˜ exactly coincides with the joint probability density for the eigenvalues of Eq. (8.29) N 2 ! | i − j | e− j=1 j =2 (8.77) J˜ ({ i }) ≡ P ({ i }) = C i¡j
with C an undetermined normalization constant. We have lim P ({ i }; t) = P ({ i }) :
t →∞
(8.78)
One can write down similar Fokker–Planck equations for all the zero and positive curvature spaces. All these Fokker–Planck equations describe the approach to equilibrium of the eigenvalue distribution. As mentioned above, the equation for the negative curvature spaces coincides with the DMPK equation. 8.5.2. Connection with the Laplace–Beltrami operator Comparing the Fokker–Planck equation (8.76) in the case of positive curvature ensembles (in this case the potential is absent and J˜ ({ i }) = J (+) ({ i }), that is, the joint probability distribution of the eigenvalues is identi4ed with the Jacobian of the transformation to radial coordinates) with Eq. (6.26) we see that the Fokker–Planck operator F≡
N 9 ˜ 9 ˜ −1 J J 9 j 9 j j=1
(8.79)
is related to the radial part of the Laplace–Beltrami operator,
r 1 9 (+) 9 J ; DB = (+) J 9q 9q =1
(8.80)
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127
where q are the radial coordinates identi4ed with the eigenvalues j . More precisely, the two operators are related by DB = J −1 FJ :
(8.81)
Similarly, in case of negative curvature spaces the operator B of Eqs. (8.53), (8.54), that is proportional to the DMPK operator B=
n 9 2 9 −2 K (x) K (x) = 2ED 9xk 9xk
(E ≡ N + 2 − )
(8.82)
k=1
is mapped to DB by the following relation: DB = K−2 (x)BK2 (x)
(8.83)
" " after identifying K2 (x) with the Jacobian J ({xi }) = i¡j |sinh2 xj − sinh2 xi | i |sinh 2xi | of the transformation to radial coordinates (Eq. (8.52)) and the radial coordinates themselves with the eigenvalues xi related to i by (8.51): i = sinh2 xi . Eq. (8.83), that relates the DMPK operator to the free diOusion on a symmetric space, is one of the main results of the mapping between random matrix ensembles and symmetric spaces. As we shall see it also represents the starting point for the exact solution of the DMPK equation for the transfer matrix ensembles. 8.5.3. Random matrix theory description of parametric correlations The original motivation behind the Coulomb gas approach to random matrix theory was to obtain the equilibrium probability distribution of the eigenvalues in a diOerent and more physical way. In this respect the only interesting regime of the corresponding Fokker–Planck equations was the asymptotic t → ∞ limit. However, it was later realized that this equation contains interesting physical information for all values of t. In particular it could be used for describing the adiabatic response to an external perturbation of the energy spectrum of a mesoscopic system [80]. In this approach the role of the 4ctitious time is played by the perturbation parameter. An interesting application appears if the role of the external perturbation is played by an external magnetic 4eld. In a random matrix description of a disordered conductor, there is a smooth transition between ensembles characterized by various values of as a function of the magnetic 4eld B around B ∼ 0 (for a review see [26]). When such a transition is completed, the level distribution becomes independent of the magnetic 4eld. The random Quctuation of individual energy levels is still present, however, as a function of B. Such random Quctuations are described by Fokker–Planck type equations similar to the DMPK equation, but containing the Laplace–Beltrami operator on symmetric spaces of positive curvature [26]. 8.6. A dictionary between random matrix ensembles and symmetric spaces Probably the simplest way to summarize what we have said in this section is to write down the equivalences between random matrix and symmetric space concepts in a table. This is what we have done in Table 2.
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Table 2 The correspondence between random matrix ensembles and symmetric spaces Random matrix theories (RMT)
Symmetric spaces (SS)
Circular or scattering ensembles Gaussian or hamiltonian ensembles Transfer matrix ensembles Random matrix eigenvalues Probability distribution of eigenvalues Fokker–Planck equation Coulomb gas analogy Ensemble indices Dyson index Boundary index = (I + 1) − 1 Translationally invariant ensembles Boundary matrix ensembles Pair interaction between eigenvalues
Positive curvature spaces Zero curvature spaces Negative curvature spaces Radial coordinates Jacobian of transformation to radial coordinates Radial Laplace–Beltrami equation Brownian motion on the symmetric space Root multiplicities Multiplicity of ordinary roots ( = mo ) Multiplicity of short and long roots ( = ms + ml ) SS with root lattice of type An SS with root lattices of type Bn , Cn , Dn or BCn Ordinary roots
9. On the use of symmetric spaces in random matrix theory In this section we discuss some of the applications of the mapping between random matrix ensembles and irreducible symmetric spaces outlined in Section 8. The main application, which is a natural consequence of the Cartan classi4cation of symmetric spaces, is a tentative classi4cation of the random matrix ensembles. We will discuss this important issue in Section 9.1, while in Section 9.2 we discuss how the symmetries of the spaces are reQected in the random matrix ensembles. A second natural application is related to the orthogonal polynomial approach to the construction of eigenvalue correlation functions of random matrix ensembles [11]. As it turns out, the orthogonal polynomials associated to random matrix ensembles whose integration manifolds are symmetric spaces, are all of the classical type and can be directly constructed from the knowledge of the curvature and root multiplicities of the underlying symmetric space. We will discuss this issue in Section 9.3. Finally in Section 9.4 we discuss the applications of symmetric spaces to the transfer matrix ensembles which appear in the description of quantum transport. This is probably the 4eld in which the knowledge of the mathematical structure of the underlying symmetric spaces is most helpful. 9.1. Towards a classi
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applicable, and it becomes much more diVcult to extract meaningful information or predictions for the random matrix theory. 16 At the same time it appears that most of the matrix ensembles which have physically interesting realizations belong to the Cartan framework. It is possible that this is merely a consequence of the fact that, in trying to describe a physical problem, one prefers to use ensembles that are simple to deal with, even though this may imply stronger approximations. Let us mention an example which illustrates the issue at hand. Recently various generalizations of the DMPK transfer matrix ensembles discussed in this review have been proposed [81] as an attempt to avoid the “quasi-one-dimensional” approximation involved in the standard DMPK equation. The resulting equations cannot be mapped to a symmetric space, and thus cannot be solved exactly or even asymptotically. In spite of this, important information on the expected behavior of the eigenvalues can all the same be obtained by means of suitable perturbative expansions [81]. In Table 3 we have included the random matrix ensembles discussed in Section 8 in the Cartan classi4cation of irreducible symmetric spaces. This classi4cation was presented in Table 1. The scattering matrix for an NS-type heterostructure is obtained by exponentiation of the Hamiltonian S = eiH . Since iH is in the algebras or tangent spaces for the respective symmetry classes, the scattering matrix is in the corresponding symmetric spaces of positive curvature. The scattering matrix ensembles of NS systems have been listed in Table 3 with the notation Bm+o ;ml ;ms and are of the same type as the circular and S-matrix ensembles. We list the ensembles using this kind of notation, which is more consistent than some of the traditional names given to ensembles in the past with abuse of language (for example, an ensemble was called “unitary” if = 2, even though the stability subgroup was not unitary). The notation used here is as follows. An ensemble is labelled by a letter indicating the type of ensemble and alluding to the traditional name. Let C stand for circular ensembles, G for gaussian ensembles, P for the p-wave ensembles of Ivanov (these are of the BdG type, but have a zero mode), B for Bogoliubov–de Gennes ensembles, T for transfer matrix ensembles, S for S-matrix ensembles, and Q for chiral ensembles, a superscript taking the values +, 0, − indicating the curvature of the space, and three subscripts mo ; ml ; ms . Since the pair {; } can be the same for some pairs of distinct ensembles, it is better to keep all three root multiplicities in the label. For example, B2;+ 0; 0 indicates the BdG (NS) ensemble corresponding to a symmetric space of positive curvature with root multiplicities {mo ; ml ; ms } = {2; 0; 0}. In this way each ensemble is uniquely labelled. Let us stress that the empty spaces of Table 3 do not mean that a corresponding random matrix ensemble does not exist. Following the previous discussion, it is easy to see that for each symmetric space (of arbitrary curvature) one can construct a perfectly consistent matrix ensemble. The empty boxes simply mean that such an ensemble still has not found a relevant physical application or realization (or, possibly, that we are not aware of such a realization). It is likely that with time all the empty boxes in Table 3 will be occupied with physically interesting applications. Note also that the type of restricted root system changes within the same Cartan class for the ensembles labelled by two integers {p; q}, depending on whether p ¿ q or =q. The S-matrix and transfer matrix ensembles have p = q (I = 0) and have been written on the corresponding line, while the chiral ensembles may have p ¿ q or =q. 16
See the discussion in Section 10.1 below for a remarkable exception to this fact.
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Table 3 Irreducible symmetric spaces and some of their random matrix theory realizations Restricted root space Cartan class G=K (G)
G ∗ =K (G C =G)
m o m l ms X +
AN −1
A
SU (N )
SL(N; C) SU (N )
2
0
0
C2;+0; 0 G2;0 0; 0 T2;−0; 0
AN −1
AI
SU (N ) SO(N )
SL(N; R) SO(N )
1
0
0
C1;+0; 0 G1;0 0; 0 T1;−0; 0
AN −1
AII
SU (2N ) USp(2N )
SU ∗ (2N ) USp(2N )
4
0
0
C4;+0; 0 G4;0 0; 0 T4;−0; 0
AIII
SU (p; q) SU (p + q) 2 SU (p) × SU (q) × U (1) SU (p) × SU (q) × U (1)
1
2I S2;+1; 0 Q2;0 1; 2I T2;−1; 0
BN
B
SO(2N + 1)
SO(2N + 1; C) SO(2N + 1)
2
0
2
P2;0 0; 2
CN
C
USp(2N )
Sp(2N; C) USp(2N )
2
2
0
B2;+ 2; 0 B2;0 2; 0 T2;−2; 0
CN
CI
USp(2N ) SU (N ) × U (1)
Sp(2N; R) SU (N ) × U (1)
1
1
0
B1;+ 1; 0 B1;0 1; 0 T1;−1; 0
CII
USp(2p + 2q) USp(2p) × USp(2q)
USp(2p; 2q) USp(2p) × USp(2q)
4
3
4I
DN
D
SO(2N )
SO(2N; C) SO(2N )
2
0
0
B2;+ 0; 0 B2;0 0; 0 T2;−0; 0
CN
DIII
SO(4N ) SU (2N ) × U (1)
SO∗ (4N ) SU (2N ) × U (1)
4
1
0
B4;+ 1; 0 B4;0 1; 0 T4;−1; 0
BCN
DIII
SO(4N + 2) SU (2N + 1) × U (1)
SO∗ (4N + 2) SU (2N + 1) × U (1)
4
1
4
P4;0 1; 4
BDI
SO(p + q) SO(p) × SO(q)
SO(p; q) SO(p) × SO(q)
1
0
I
Q1;0 0; I
BCq (p ¿ q) Cq (p = q)
BCq (p ¿ q) Cq (p = q)
Bq (p ¿ q) Dq (p = q)
X0
X−
Q4;0 3; 4I T4;−3; 0
T1;−0; 0
The random matrix ensembles with known physical applications are listed in the columns labelled X + , X 0 and X − and correspond to symmetric spaces of positive, zero and negative curvature, respectively. Extending the notation used in the applications of chiral random matrices in QCD, where I is the winding number, we set I ≡ p − q. The notation is C for circular, G for gaussian, Q for chiral, B for Bogoliubov–de Gennes, P for p-wave, T for transfer matrix and S for S-matrix ensembles. The upper indices indicate the curvature, while the lower indices correspond to the multiplicities of the restricted roots characterizing the spaces with nonzero curvature. To the euclidean type spaces X 0 ∼ G 0 =K, where the nonsemisimple group G 0 is the semidirect product K ⊗ P, we associate the root multiplicities of the algebra G = K ⊕ P.
It is important to note that the symmetric space associated to a random matrix theory can be given either for an ensemble of random Hamiltonians H, for an ensemble of random transfer matrices M, or for an ensemble of random scattering matrices S. Thus, the symmetric space associated to the transfer matrix group of a given system (for example a quantum wire) is diHerent from the symmetric space associated to the Hamiltonian H or the scattering matrix S of the same physical system. A table of correspondences between the M and H descriptions was given in [6].
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9.2. Symmetries of random matrix ensembles Some known symmetries of the random matrix ensembles can be understood in terms of the symmetries of the associated restricted root lattice. In particular, ensembles of An type are characterized by translational invariance of the eigenvalues. This translational symmetry is seen to originate in the root lattice: all the restricted roots of the An lattice are of the form (ei − ej ). The Wigner–Dyson (circular and gaussian) ensembles are of the translation invariant type. For all the other types of restricted root lattices (Bn , Cn , Dn and BCn ) this invariance is broken and substituted by a new Z2 symmetry giving rise to the reQection symmetry of the eigenvalues that we discussed in the context of the Jacobians. Since these ensembles are characterized by the presence of a boundary (not always immediately evident) with respect to which they are reQection invariant, they are called boundary random matrix theories (BRMT in the following). They include all the remaining ensembles that are not of the circular or gaussian type. 9.3. Orthogonal polynomials An important role in the study of matrix ensembles is played by the set of polynomials orthogonal with respect to the random matrix theory integration measure. These polynomials come into the picture when rewriting the Jacobian for the transformation to eigenvalue space in terms of a product of Vandermonde determinants. By adding linear combinations of the rows, the determinant can be written as a determinant of monic polynomials (a polynomial Pn (x) is called monic if Pn (x) = xn + O(xn−1 )), for example N ! i¡j
(xi − xj ) ∼
det
16i; j 6N
xji−1 =
det
16i; j 6N
Pi−1 (xj ) :
(9.1)
If these polynomials are then chosen orthogonal with respect to the measure 17 w(x) d x = e−NV (x) d x, where V (x) is the random matrix potential # e−NV (x) Pm (x)Pn (x) d x = hn 6mn (9.2) I
(here I is some interval on the real axis and hn is a normalization factor), the eigenvalue correlation functions can be expressed in terms of a ChristoOel–Darboux kernel [11]. An arbitrary k-point correlation function is de4ned as # ∞ ! N 7(x1 ; : : : ; xk ) = d xj P(x1 ; : : : ; xN ) ; (9.3) −∞ j=k+1
where P(x1 ; : : : ; xN ) is the joint eigenvalue distribution of the random matrix model. The quantity 7(x1 ; : : : ; xk )d x1 : : : d xk equals the probability of 4nding one eigenvalue in each of the intervals between xj and xj + d xj (j = 1; : : : ; k). 17
The factor N in the exponent is common in QCD related applications. If the weight function is odd, the so-called pseudo-orthogonal polynomials may be used. We will not discuss them here, since our goal is just to remind the reader of the general mechanisms that make orthogonal polynomials useful.
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The general formula for the k-point function turns out to be 7(x1 ; : : : ; xk ) =
N k (N − k)! det KN (xi ; xj ) : 16i; j 6N N!
(9.4)
In particular, the spectral density is simply 7(x) = KN (x; x) :
(9.5)
In Eqs. (9.4) and (9.5), KN (xi ; xj ) is the ChristoOel–Darboux kernel de4ned in terms of orthogonal polynomials 2
2
KN (xi ; xj ) = N −1 eN=2(V (xi )+V (xj ))
−1 N
1 h− k Pk (xi )Pk (xj ) :
(9.6)
k=0
It turns out that for all the matrix ensembles related to symmetric spaces, the associated orthogonal polynomials belong to the set of the so-called classical orthogonal polynomials. With the term classical one usually denotes three families of orthogonal polynomials: the Jacobi, Laguerre and Hermite polynomials, whose unifying feature is the so-called Rodriguez formula 18 which allows to construct the polynomials once the weight function p(x) and the function X (x) (which speci4es the domain of support of the polynomials) are given: 1 1 dn Pn (x) = {p(x)X n (x)} : (9.7) An p(x) d xn Here An is a normalization constant which can be obtained explicitly, but is irrelevant for our purposes. The domain-specifying function X (x) is a polynomial in x of degree 6 2. The three possibilities are: • if X (x) is simply a constant, then the polynomials are de4ned on the whole real line, the weight 2 function must be p(x) = e−x , and we 4nd the Hermite polynomials. • if X (x) is a polynomial of 4rst degree, one can always shift the origin so as to obtain X (x) = x. In this case the orthogonal polynomials are de4ned on the positive real axis, the weight function must be p(x) = x e−x , and we 4nd the Laguerre polynomials. • if X (x) is a polynomial of second degree, then one can always normalize it so as to obtain X (x) = (1 − x2 ). In this case the orthogonal polynomials are de4ned on the interval [ − 1; 1], the weight function must be p(x) = (1 − x): (1 + x)7 , and we 4nd the Jacobi polynomials. The well known Gegenbauer, Chebyshev and Legendre polynomials are only special cases of Jacobi polynomials. Each one of these polynomial families is in one-to-one correspondence with a particular random matrix ensemble of Table 3. The mapping is complete, i.e. all the ensembles of Table 3 are covered. In particular the Hermite polynomials are related to the gaussian Wigner–Dyson ensembles (i.e., those corresponding to symmetric spaces de4ned by an An root lattice). Simple changes of variables allow one to show that the Laguerre polynomials are related to the gaussian BRMT’s (these are 18
Actually this formula is a generalization due to Tricomi of the original formula obtained by Rodriguez for the case of Legendre polynomials.
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the chiral, BdG and p-wave ensembles of zero curvature spaces) and the Jacobi polynomials to the circular BRMT’s (i.e., ensembles related to symmetric spaces of positive curvature not of the An type, like for instance the BdG scattering ensembles). A thorough discussion of the correlation functions for these ensembles using the orthogonal polynomials listed above can be found for instance in [56] (see also [83]). These results are well-known (for instance, in the early papers on chiral matrix ensembles, these ensembles were named Laguerre ensembles) and do not require any particular reference to the symmetric space description of random matrix theories. What is interesting in our framework is that the parameters which de4ne the polynomials can be explicitly related to the multiplicities of short and long roots of the underlying symmetric space and thus, by the identi4cation in Table 2, with the boundary universality indices of the BRMT. The relation is the following: Laguerre polynomials: L( ) (x) =
x− ex d n n+ −x (x e ) n! d xn
(x ¿ 0) ;
ms + ml − 1 : 2 Jacobi polynomials: (1 + x)n+7 (−1)n (1 − x)−: d n (7; :) (x) = n P 2 n! (1 + x)7 d xn (1 − x)−n−: ≡
(9.8)
(−1 6 x 6 1) ;
ml − 1 ms + ml − 1 ; :≡ : (9.9) 2 2 We see that and 7 have the same expression in terms of ms and ml . Thus the BRMT’s corresponding to Laguerre and Jacobi polynomials with the same = 7 indices belong to the same triplet in the classi4cation of Table 3. They are, respectively, the zero curvature (Laguerre) and positive curvature (Jacobi) elements of the triplet. This explains the so called “weak universality” which was observed a few years ago for the boundary critical indices of these ensembles [56], i.e. the fact that the form (near the boundary) of scaled k-level correlators is the same for Laguerre and Jacobi ensembles if the symmetry parameter is the same (see [56] for a discussion of this point). The weak universality turns out to be simply a consequence of the organization in triplets of the symmetric spaces! 7≡
9.4. Use of symmetric spaces in quantum transport One of the most interesting applications of symmetric spaces is in the transformation of the Fokker–Planck equations in random matrix theory into SchrNodinger equations in imaginary time (where, as before, the time coordinate in transfer matrix ensembles is identi4ed with the dimensionless length s of a quantum wire). As a consequence of this transformation, in the case = 2 the degrees of freedom in the SchrNodinger equation decouple and it can be solved exactly. This result traces back to the original work by Dyson [84] and was later extended to boundary random matrix theories by various authors. We mentioned this important result already at the end of Section 7. In Section 9.4.1 we will study this mapping in detail in a case which is particularly relevant from a physical point of view, namely for the DMPK equation of transfer matrix ensembles.
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Another important consequence of the mapping is that in the “interacting cases” (for = 1 and 4) one can use the results discussed in Sections 6.3 and 6.4 on the zonal spherical functions to obtain important information on the asymptotic behavior of the solutions to the DMPK equation. We will discuss this issue in Section 9.4.2. In Section 9.4.3 we will see an example of a change of symmetry class (hence also of the underlying symmetric space description) induced in a quantum wire by switching on an external magnetic 4eld. Finally in Section 9.4.4 we shall discuss a general scheme (based on the DMPK equation) for constructing the scaling equations for the density of states of a quantum wire that covers all the Cartan symmetry classes. 9.4.1. Exact solvability of the DMPK equation in the = 2 case The exact solution of the DMPK equation in the = 2 case was 4rst obtained in a remarkable paper [45] by Beenakker and Rejaei. Here we review their derivation in a slightly diOerent language, trying to stress the symmetric space origin of their result. The starting point is the mapping discussed in Section 8.3.5 which we brieQy recall here. By setting n ≡ sinh2 x n (cf. Eq. (8.51)) the DMPK equation can be rewritten as (see Eqs. (8.52) and (8.83)) 9P 1 = [K(x)]2 DB [K(x)]−2 P ; 9s 2E where K({xi }) =
!
|sinh2 xi − sinh2 xj |=2
i¡j
(9.10) !
|sinh 2xk |1=2
(9.11)
k
and DB is the radial part of the Laplace–Beltrami operator on the underlying symmetric space. At this point one can follow two equivalent ways. In the 4rst one (which was the one followed in [45]) one makes use of the results discussed in Section 7, in particular Eq. (7.5), to map the DMPK equation into a SchrNodinger equation. By comparing with Eq. (7.5) we see that this simply requires the substitution P({x n }; s) = K({x n })R({x n }; s) :
(9.12)
A straightforward calculation shows that the DMPK equation then takes the form of a SchrNodinger equation in imaginary time: −
9R = (H − U )R ; 9s
where U is a constant and H is a Hamiltonian of the form ( − 2) sinh2 (2xi ) + sinh2 (2xj ) 1 92 −2 + sinh (2x ) + : H=− i 2E i 2E (cosh(2xi ) − cosh(2xj ))2 9xi2 i¡j
(9.13)
(9.14)
At this point the main goal has already been reached: it is easy to see that if = 2 the equation decouples and an exact solution can be obtained [45]. Before going into the details of this solution, let us remark that the above equation can be recast in a slightly diOerent form, thus completing the chain of identi4cations DMPK equation—radial part of the Laplace–Beltrami operator—
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Calogero–Sutherland model. By using simple identities for hyperbolic functions, this Hamiltonian becomes [40] 1 92 go2 gl2 go2 − + + + +c ; (9.15) EH = 2 9xi2 sinh2 (2xi ) sinh2 (xi − xj ) sinh2 (xi + xj ) i i¡j where gl2 ≡ −1=2, go2 ≡ ( − 2)=4 and c is an irrelevant constant. This Hamiltonian, apart from an overall factor 1=E and the constant c, exactly coincides with the Calogero–Sutherland Hamiltonian (7.4) discussed in Section 7, corresponding to a root lattice R = {±2xi ; ±xi ± xj ; i = j} of type Cn with root multiplicities mo = , ml = 1. The values of the coupling constants go , gl are exactly the root values given in Eq. (7.8) g 2 =
m (m − 2)| |2 8
(9.16)
of Section 7, for which the transformation from H into DB is possible. Let us now come back to the exact solution of the DMPK equation following Beenakker and Rejaei. As we have seen, for = 2 H is reduced to a sum of single-particle Hamiltonians H0 , H0 = −
1 92 1 − : 2 2E 9x 2E sinh2 2x
(9.17)
At this point, to solve the DMPK equation one simply has to construct the Green’s function G0 of the single-particle Hamiltonian H0 . This requires solving the eigenvalue equation H0 (x) = S (x) ;
(9.18)
A standard analysis shows that the spectrum of H0 is continuous, with positive eigenvalues S= 14 k 2 =N and that the eigenfunctions k (x) are real functions given by )1=2 ( /k sinh 2x P1=2(ik −1) (cosh 2x) ; (9.19) k (x) = /k tanh 2 where PI (z) denotes the Legendre functions of the 4rst kind. From this we obtain the spectral representation of the single-particle Green’s function G0 2 # ∞ k s −1 d k exp − G0 (x; s | y) = (2/) k (x) k (y) 4N 0 2 # ∞ /k 1 k s 1=2 k tanh = (sinh 2x sinh 2y) d k exp − 2 4N 2 0 ×P1=2(ik −1) (cosh 2x)P1=2(ik −1) (cosh 2y) :
(9.20)
The N -particle Green’s function G is related to the single-particle Green’s function G0 through a Slater determinant, and the probability distribution of eigenvalues is related to G through a similarity transformation by the antisymmetrized eigenstate R0 (x) = K(x) ( = 2) of the N -fermion Hamiltonian (for details see [45]). From the expression (9.20), imposing so called ballistic initial conditions (which essentially amount to requiring that all the eigenvalues are concentrated at the origin for
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s = 0), one 4nally obtains the probability distribution P({x n }; s) for the eigenvalues ! ! P({x n }; s) = C(s) (sinh2 xj − sinh2 xi ) (sinh 2xk ) i¡j
(#
×Det
0
k
∞
2 ) /k k s tanh k 2m−1 P1=2(ik −1) (cosh 2x n ) : d k exp − 4N 2
(9.21)
This is the exact solution of the DMPK equation for = 2. 19 The second approach relies more heavily on the underlying symmetric space structure. The starting point is again the identi4cation made in Eq. (9.10) between the DMPK operator and the radial part of the Laplace–Beltrami operator on the underlying symmetric space. As a consequence of this identi4cation, if Tk (x) (x = {x1 ; : : : ; xN }, k = {k1 ; : : : ; kN }) is an eigenfunction of DB with eigenvalue k 2 , then K(x)2 Tk (x) will be an eigenfunction of the DMPK operator with eigenvalue k 2 =(2E). The eigenfunctions of the DB operator (which are known in the literature as zonal spherical functions) have been widely discussed in Sections 6.3 and 6.4. As we have seen, by means of the zonal spherical functions one can de4ne the analog of the Fourier transform on symmetric spaces: # dk W f(x) = f(k)T (9.22) k (x) |c(k)|2 (here we have neglected an irrelevant multiplicative constant). In particular, for the three symmetric spaces which are of interest for us one 4nds: 2 ! F(i kj =2) |c(k)|2 = |D(k)|2 (9.23) 1 F 2 + i kj =2 j
with
2 ! F(i(km − kj )=2)F(i(km + kj )=2) 2 ;
|D(k)| = m¡j F 2 + i(km − kj )=2 F 2 + i (km + kj )=2
(9.24)
where F denotes the Euler gamma function. This is a completely general result which we will use also in the next section. The problem is that in general the explicit form of the zonal spherical functions involved in Eq. (9.22) is not known. A remarkable exception to this situation is represented exactly by the = 2 case for which we have [9,85] (see Eq. (A.4) in the appendix): det[Qml ] Tk (x) = " ; (9.25) 2 2 2 2 i¡j [(ki − kj )(sinh xi − sinh xj )] where the matrix elements of Q are 1 1 l 2 Qm = F (1 + ikm ); (1 − ikm ); 1; −sinh xl 2 2 and F(a; b; c; z) is the hypergeometric function.
(9.26)
19 We remark that in [50], using the same technique as in [45], the Fokker–Planck equation for the probability distribution of eigenvalues in systems with a chiral Hamiltonian was solved exactly in the case = 2, and in [76], the equation corresponding to a system with BdG Hamiltonian was solved in the presence of time-reversal symmetry (for two of the four BdG symmetry classes).
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Eqs. (9.10), (9.22) and (9.24) allow us to write the s-evolution of P({x n }; s) from given initial conditions (described by the function fW0 (k)) as follows: # dk 2 : (9.27) P({x n }; s) = [K(x)]2 fW0 (k)e−(k =2E)s Tk (x) |c(k)|2 By inserting the explicit expression of |c(k)|2 and by using the identity F( 1 + i(k=2)) 2 k /k 2 ; = tanh F(i(k=2)) 2 2
(9.28)
we end up with the following general expression for P({x n }; s) with ballistic initial conditions (which, due to the normalization of Tk (x), simply amount to choosing fW0 (k) equal to a constant): # ! /kj 2 −(k 2 =2E)s Tk (x) : (9.29) dk e kj tanh P({x n }; s) = [K(x)] |D(k)|2 j 2 Inserting the explicit expression for Tk (x) from Eqs. (9.25), (9.26) into (9.29) and using the identity PI (z) = F(−I; I + 1; 1; (1 − z)=2) ;
(9.30)
we exactly obtain, as expected, the solution (9.21) found by Beenakker and Rejaei. This is a remarkable and nontrivial consistency check of the correctness of this solution. 9.4.2. Asymptotic solutions in the = 1; 4 cases The power of the description in terms of symmetric spaces becomes evident in the = 1 and 4 cases, in which the interaction between the eigenvalues does not vanish and the 4rst approach discussed in the previous subsection does not apply. On the contrary, the description in terms of zonal spherical functions (i.e., Eq. (9.27)) also holds in these two cases. Even though for = 2 one does not know the explicit form of the zonal spherical functions, one can use the powerful asymptotic expansion (6.63) discussed at the end of Section 6.4 to get asymptotic solutions. In our context this expansion reads: 1 Tk (x) ∼ (9.31) c(rk)ei(rk; x) ; K(x) r ∈W where rk is the vector obtained acting with r ∈ W on k (W denotes the Weyl group of the symmetric space). The important feature of Eq. (9.31) is that it is valid for all values of k, thus it can be used both in the metallic (k1) and in the insulating (k1) regimes. This leads to expressions for the probability distribution of the eigenvalues and for the conductance, which can then be compared to other theoretical results using numerical simulations or experiments (see [40] for a detailed discussion). 9.4.3. Magnetic dependence of the conductance In [86], the theory of symmetric spaces was applied to give a possible explanation for the discrepancy between the random matrix theory and nonlinear sigma model analysis of the magnetoconductance in the weakly insulating, localized regime close to the Anderson transition of a disordered wire. The Anderson transition is a disorder-induced transition from the conducting to the insulating regime. For a review of the sigma model approach see [12].
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Magnetoconductance is the change in the conductance of the wire due to the presence of a magnetic 4eld (for a review see [26]). More precisely, it is the suppression of weak localization (a quantum eOect due to time-reversal symmetry) which appears when a magnetic 4eld destroys time-reversal invariance. The disagreement between the random matrix theory approach and the nonlinear sigma model approach was evident in the prediction for the magnetoconductance in the presence of strong spin–orbit scattering: the sigma model approach gave a zero magnetoconductance, while the random matrix approach gave a strong negative value due to suppression of the localization length. A negative magnetoconductance has been observed in experiments. However, as we will review below, also within the random matrix approach a zero magnetoconductance can be expected, depending on whether Kramers degeneracy of the eigenvalues is conserved or not. The conductance in the insulating regime is related to the localization length K by G = G0 exp(−2L=K), where L is the length of the sample. In the standard random matrix approach, K is proportional to . Thus the transition (due to the switching on of a magnetic 4eld) between the ensemble characterized by = 4 and the ensemble characterized by = 2 means a negative contribution to G. The key observation made in [86] was that the matrix ensemble to which the transfer matrix belongs corresponds, for = 4, to a symmetric space SO∗ (4N )=U (2N ) with a root lattice of type CN . This root lattice is characterized by two types of roots, long and ordinary, and therefore by two indices ≡ mo = 4 and ; ≡ ml = 1. In the ensemble labelled by = 4 the eigenvalues are twofold degenerate. Therefore, if there are N degenerate scattering channels for = 4 (the so-called Kramers degeneracy), for = 2 there are 2N channels. Using the mapping of the DMPK equation onto a Calogero–Sutherland model (see Section 7), a generalized DMPK equation was derived to take into account the new index ;. As a consequence, it was shown that the localization length was not aOected if one performs the simultaneous change = 4 → = 2 and N → 2N , while keeping ; 4xed. This corresponds to the transition between ensembles SO∗ (4N )=U (2N ) → SU (2N; 2N )=(SU (2N ) × SU (2N ) × U (1)) (the latter ensemble is characterized by = 2, ; = 1) and gives a zero magnetoconductance. EOectively, the level statistics depends on and N , but the localization length depends more generally on , ;, and N . Assuming instead that Kramers degeneracy is conserved (N → N ) we have a transition SO∗ (4N )= U (2N ) → USp(2N; 2N )=(USp(2N )×USp(2N )) (the latter ensemble is characterized by =4, ;=3) with the result that K is smaller in the presence of a magnetic 4eld. In this case we have a negative magnetoconductance. Thus we see that both possibilities, zero and negative magnetoconductance, are feasible also within the framework of random matrix theory. 9.4.4. Density of states in disordered quantum wires Another interesting application of the DMPK equation is the construction of the scaling equation for the density of states of a disordered quantum wire which was recently discussed by Titov et al. in [6,87]. Computing the density of states 7(j) in the thermodynamic limit N → ∞ in a quantum wire of in4nite length at energy j turns out to be intimately related to the solution of the DMPK equation in presence of absorption [6,87]. Absorption is described by the addition of a spatially uniform imaginary potential i!, ! ¿ 0 to the Hamiltonian. A DMPK-like equation can be obtained also in this case. It is very similar to the standard one (see Eqs. (8.53) and (8.55)). The only change is an additional term proportional to the absorption
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constant
N 9Pi! 1 l! 9 sinh 2xj Pi! ; = B+ 9s 2E vF j=1 9xj
(9.32)
where l is the mean free path and vF the Fermi velocity. It is not diVcult to see that this equation has the following stationary (i.e. L independent) solution: N
Pi! ({xi }) =
|J | ! −a cosh 2xj e ; Z(a) j=1
(9.33)
where a=El!=vF is the adimensional absorption constant and Z(a) is a normalization constant which is needed to ensure that Pi! is normalized to one. The key (and non-trivial) point is that, once Z(a) is known it is possible to show (by making an analytical continuation from i! to j) that the density of states 7(j) is obtained from Z(a) as follows: ( ) 9 d 9 a log Z(a) (9.34) 7(j) = − Re lim /vF a→−iElj=vF 9a 9a (see [6] for a detailed derivation). It is interesting to observe that this procedure works for all the Cartan classes. In the nonstandard symmetry classes of the Hamiltonian, the density of states is singular at the band center (for chiral Hamiltonians) or Fermi energy (for BdG Hamiltonians). This point corresponds to extra symmetries of the Hamiltonian, an issue that was discussed in the context of chiral ensembles. The precise form of this singularity depends on the symmetry class and, for chiral Hamiltonians, on the parity of N (the number of channels). Apparently, there is a connection between the anomalous behavior of the density of states and the divergence of the localization length (signaling criticality) at the singular point [87]. 10. Beyond symmetric spaces In the previous section we have discussed some of the possible applications of the theory of symmetric spaces in random matrix models. In this last section we discuss three issues which in one way or another go beyond the theory of symmetric spaces developed up to now. We will see that the power of the group theoretical methods developed in the 4rst part of this review allows us to obtain interesting results also in some cases in which symmetric spaces appear not to be useful. At the same time the topics to be discussed represent new open directions of research which we hope will lead to future interesting results. In Section 10.1 we will discuss the nonCartan parametrization of symmetric spaces. We will see how it is possible to map exotic random matrix ensembles to nonstandard (in a sense that will be clear) symmetric spaces. In Section 10.2 we will discuss how a wide set of nonisotropic solutions to the DMPK equation can be constructed by simply resorting to the exact integrability of the associated Calogero–Sutherland models. This new tool might be useful in overcoming the quasi-1D constraint of the DMPK description of quantum wires. Finally, in Section 10.3 we discuss the triplicity of (in a certain sense) the most general potential in a Calogero–Sutherland model, the Weierstrass potential.
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We will see how this potential in various limits reproduces the components of an arbitrary triplet of symmetric spaces. 10.1. NonCartan parametrization of symmetric spaces and S-matrix ensembles As we have seen in Sections 8.3.7 and 8.4.1, apparently only the S-matrix ensembles labelled by = 2 (these were introduced in Refs. [78,79]) can be associated to a symmetric space. Let us brieQy recall the problem. The S-matrix ensembles correspond to root lattices of BCn type with the following root multiplicities: mo = ; ms = − 2; ml = 1, where takes values in the set {1; 2; 4}. Only the = 2 case corresponds to a set of root multiplicities associated to a symmetric space. In this case the restricted root lattice degenerates into Cn , since the multiplicity of short roots ms = 0. It is instructive to re-obtain this symmetric space description from the integration manifold of the random matrix ensemble. In the unitary case the scattering matrix is parametrized by (cf. Eq. (8.14) in Section 8.3.4) r t S= ; (10.1) t r where r; t; r ; t are N × N matrices. One is interested in the eigenvalues {Ti } of the product tt † , since these determine a variety of transport properties. This implies that there is a hidden symmetry: v3 0 S→S (10.2) 0 v4 with v3 ; v4 ∈ U (N ). Indeed, under the transformation (10.2), t → tv3 and the product tt † is invariant for unitary v3 . This additional symmetry de4nes the space in which the matrices S live. It is not SU (2N ) but the coset SU (2N )=S(U (N ) × U (N )), which is exactly the symmetric space described by the CN root lattice with multiplicities mo = 2 and ml = 1 mentioned above. We can write the coordinates {Ti } explicitly by using the fact that any 2N × 2N unitary matrix can be decomposed as [78,79] (cf. Eq. (8.15)) √ √ = − 1−= v1 0 v3 0 S= √ √ 0 v2 0 v4 = 1−= √ √ v1 = 0 v −v1 1 − = v3 0 3 ≡ S ; (10.3) = √ √ 0 v 0 v4 v2 1 − = v2 = 4 where v1 ; v2 ; v3 ; v4 ∈ U (N ) (in the presence of time-reversal symmetry v1 ; v3 and v2 ; v4 are related to each other by transposition) and = is a N × N diagonal matrix which collects the coordinates {Ti }. As was obvious to begin with, they are not the radial coordinates corresponding to a Cartan subalgebra in the symmetric space of the scattering matrix (cf. Section 8.3.4). The same reasoning can be followed in the = 1 and 4 cases. In these cases the cosets which one obtains by imposing the gauge symmetry discussed above turn out not to be symmetric spaces (Section 8.4.1). We can look at these random matrix ensembles as plain circular ensembles in which, for wellde4ned physical reasons, we have chosen a set of parameters (the eigenvalues of the matrix tt † )
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diOerent from the eigenvalues of the matrix S (i.e., the radial coordinates of the symmetric spaces associated to the Dyson circular ensembles). Since the standard radial coordinates are related to the Cartan generators, we can consider this non-standard choice of “radial” coordinates as a nonCartan parametrization of the symmetric space 20 of the scattering matrix. As we will see in a moment, the important point is that the construction of this nonCartan parametrization follows the same lines as the standard parametrization (identi4cation of an involutive automorphism, separation of the generators in even and odd, and so on). As a consequence of this, several properties of the standard Cartan parametrization are conserved, and allow us to treat these ensembles in a way which is essentially the same as for those which correspond to standard Cartan parametrization of symmetric spaces (see the results of [78,79]). Below we will see in detail how this nonCartan parametrization is constructed. To help the reader we will 4rst brieQy review the general method (a detailed discussion can be found in Appendix A of [82]) and then look at an example. As we will see, the procedure has many similarities to the procedure we used in constructing the restricted root lattice in Section 5.2, and like in Section 5.3, we will use two successive involutions. 10.1.1. NonCartan parametrization of SU (N )=SO(N ) Let us begin with a compact 21 symmetric space G=K and denote the corresponding algebra subspace G=K. Suppose : is the involutive automorphism that splits the algebra G into K ⊕ P. Let us now operate on the subspaces K and P = G=K with a second involution = exactly like in Section 5.3. Then the subspaces K and P are split into even and odd parts that were named K1 , K2 , P1 , P2 in Section 5.3. (In Ref. [82] these are the subspaces Ke , Ko , Pe , Po .) Let A denote a maximal abelian subalgebra contained in P2 . That means its elements are odd under the transformations : and =: :A: = −A, =A= = −A for A ∈ A (i.e. the elements of A anticommute with the involutions). Let M be the subspace of elements of K1 that commute with all the A ∈ A. The subspace of G that is invariant under = is K1 ⊕ P1 ≡ G1 . The central ingredient in obtaining the nonCartan parametrization is the bijective mapping ! of the manifold G1 =M × A+ into the symmetric space G=K such that ! : (gM; a) → gaK, where elements in the coset space are denoted gK (cf. Section 2.3). The nonCartan “radial coordinates” are encoded in the matrix a belonging to a connected open subset A+ of A = eA . In the tangent space this corresponds to a mapping Ta between algebra subspaces [82] Ta : G1 =M × A → G=K ; Ta (Z; H ) = H + aZa−1 |P ;
(10.4)
where the notation |P means the restriction to the subspace P. From the linear mapping Ta we will obtain the Jacobian as a determinant of the matrix expressing the diOerential of ! with respect to the new basis. Keeping in mind the commutation relations for symmetric subalgebras (Eq. (3.2) of Section 3.1) and the fact that A anticommutes with the two involutions : and =, we easily see that [A; P1 ] ⊂ K2 ; 20
[A; K1 =M] ⊂ P2 =A;
[A; K2 ] ⊂ P1 ;
[A; P2 =A] ⊂ K1 =M :
(10.5)
We thank Martin Zirnbauer for suggesting this possibility to us. The fact that we choose a compact space for de4niteness is not important and we can equally well choose a noncompact one. 21
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This means that the mapping ad(ln a) from the algebra G to itself maps the four eigensubspaces of := into each other as follows: ad(ln a) : K1 → P2 ; ad(ln a) : P2 → K1 ; ad(ln a) : K2 → P1 ; ad(ln a) : P1 → K2 :
(10.6)
Denoting the adjoint action aZa−1 in (10.4) with Ad(a)Z and using Ad(a) = exp ad(ln a) = cosh ad(ln a) + sinh ad(ln a), we see from Eq. (10.5) that if Z = X + Y is the decomposition of Z ∈ G1 =M into parts belonging to the subspaces P1 and K1 =M, respectively, Ta (Z; H ) = H + cosh ad(ln a)X − sinh ad(ln a)Y
(10.7)
(this follows from Eqs. (10.6) because cosh x is an even and sinh x an odd function of x, and keeping in mind that in Ta we take the projection on P). The Jacobian corresponding to the change of coordinates is the determinant [82] JNC (a) = det(cosh ad(ln a)|P1 →P1 ) det(sinh ad(ln a)|K1 =M→P2 =A )
(10.8)
and it is obtained as the product of the eigenvalues. The eigenvalues of the automorphism ad(ln a) are nothing but the restricted roots with respect to the abelian algebra A. We therefore obtain the general formula ! ! (j) JNC (a) = sinhm (ln a) coshm (ln a) (10.9) ∈ R+ 1
∈ R+ 2
where the subscript NC stands for “nonCartan” and the index + reminds us of the fact that the sum is over the positive roots only. (ln a) is nothing else than the projection q introduced in Eq. (6.27) + in Section 6.2. The positive roots have been divided into two subsets R+ 1 and R2 in an obvious notation. In case G=K is a compact space, the roots and in (10.9) are purely imaginary. If we then set = i and = i we obtain ! ! (+) JNC (a) = sinm (ln a) cosm (ln a) ; (10.10) ∈ R+ 1
∈ R+ 2
(− ) whereas for real , (noncompact space) we get hyperbolic functions in Eq. (10.9) for JNC (a) (cf. the similar situation in Eq. (6.43)). For comparison, recall that the Jacobian corresponding to the standard Cartan parametrization of the G=K space is given by Eq. (6.30), ! sinm (ln a) (10.11) J (+) (a) = ∈ R+
where the algebra corresponding to A in this case is a maximal abelian subgroup of the whole subspace P. Example. Let us now take as an example the = 1 S-matrix ensemble, whose parametrization in Section 8.4.1 falls outside of the Cartan classi4cation of symmetric spaces. We will identify it as a nonCartan parametrization of a standard symmetric space.
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The starting point in this case is the circular orthogonal ensemble of Dyson, i.e. G=K = SU (N )= SO(N ). Let us assume for completeness that we have a diOerent number of left and right scattering channels p = NL and q = NR . In this case the involution = is given by Ip; q = INL ;NR = diag(1NL ; −1NR ) (cf. Eq. (4.15)) with 4xed point set G1 = SU (NL ) × SU (NR ) and the rank of a maximal abelian subgroup A is r = min(NL ; NR ). We can choose A to be generated by the matrices Ak = i(Ek; NL +k + ENL +k;k )
(k = 1; : : : ; r) ;
where Eij is the matrix having a 1 in the ith row and jth column, and zeros elsewhere. After 4nding the corresponding roots like in the example in Section 5.2, the radial coordinates · x (leaving out the i) and the corresponding root multiplicities turn out to be R+ 1 :
(xk ± xl )
R+ 2
xk (m = |NL − NR |) (k = 1; : : : ; r) ; (xk ± xl ) (m = 1) (k ¿ l)
:
(m = 1)
2xk xk
(m = 1)
(k ¿ l)
(k = 1; : : : ; r)
(m = |NL − NR |)
(k = 1; : : : ; r)
(note that the total root multiplicities mo = 2; ml = 1; ms = 2(NL − NR ) are exactly the ones for a BCN type restricted root lattice; cf. Table 1). These are the radial coordinates in the Jacobian for the nonCartan parametrization, Eq. (10.10) ! ! ! (+) JNC (a) = sin(xi − xj )sin(xi + xj ) sinI xk cos(xl − xm )cos(xl + xm ) i¿j
! ! × cos(2x n ) cosI xq n
k
l¿m
(10.12)
q
where I ≡ |NL − NR |. By making now the variable substitution Tk = sin2 2xk in this Jacobian, we obtain the radial measure in the form ! ! (+) JNC (a) da = |Ti − Tj | |Tk |(|NL −NR |−1)=2 dTk ; (10.13) i¡j
k
which agrees with Eq. (8.69) for the special case NL = NR and = 1. As anticipated in Section 8.4.1, we conclude that Eqs. (8.69), (8.70) represent the Jacobian for a nonCartan parametrization (albeit expressed in diOerent variables) of a standard symmetric space, which for = 1 is the space SU (N )=SO(N ). 10.2. Clustered solutions of the DMPK equation As we mentioned previously, there are two major drawbacks in the DMPK approach to quantum wires. The 4rst is that the DMPK description only holds in the quasi-one-dimensional limit. The second is that the solution discussed in Sections 9.4.1 and 9.4.2 does not allow studying the intermediate cross-over region between the metallic and the insulating regimes, where no simplifying approximation is allowed. This is true even in the simplest case, for = 2.
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In the last few years various generalizations of the DMPK equation have been suggested [81] to avoid the quasi-one dimensional limit. However, as we discussed at the beginning of Section 9.1, in all these generalized equations most of the attractive properties of the DMPK equations are lost, mainly due to the fact that the description in terms of symmetric spaces is no longer valid. Only a few pieces of information on the expected joint probability density of the transmission eigenvalues can be obtained. Recently a diOerent strategy has been proposed in [88] where the DMPK equation is kept unchanged, but one looks for a set of special solutions (with nontrivial initial conditions) which break the isotropy ansatz. To this end one uses the exact integrability of the Calogero–Sutherland models: recall that the DMPK equation can be mapped into the evolution operator of a suitably chosen Calogero–Sutherland model (for a review see [9]). This is a highly nontrivial property which is more general than the underlying symmetric-space structure; in fact it holds also for generic integer values of the root multiplicities [9]. It is possible to show that as a consequence of their exact integrability, in these models—besides the well known symmetric solution—a wide class of nontrivial (but exact) solutions exists, in which the particles are grouped into clusters. Once the mapping to the DMPK equation is performed, the clusters of particles become clusters of eigenvalues. The exact integrability of the Calogero–Sutherland model ensures that this asymmetric distribution of eigenvalues survives in the asymptotic limit, and the remarkable properties of the underlying symmetric space allow us to explicitly write down such asymptotic expansions. Let us discuss these solutions in more detail. We assume the cluster to be composed of the 4rst N ¡ N eigenvalues. This means |xi − xj | ¡ ∞;
i; j = 1; : : : ; N
|xi − xj | → ∞;
i = 1; : : : ; N ;
(i = j) ; j = N + 1; : : : ; N
(i = j) :
(10.14)
In the symmetric space framework we can identify the cluster by selecting a subsystem of roots associated to the space. Let U be the system of simple roots associated to the symmetric space X , and U a subsystem of simple roots which satis4es the inequality (10.15) U = ∈ U | lim x ¡ ∞ ; |x|→∞
where x = (x; ). At this point there are two possibilities. Since U is a CN type lattice, U can be either of type CN (in this case it must also contain the long root, and the ordinary roots must be chosen so as to preserve the Z2 symmetry of the lattice), or it can be of type AN . In both cases one can construct, from the ordinary roots of U , the diOerences x = xi − xi+1 which correspond to the nearest neighbor distances between eigenvalues. It follows from the de4nition of U that these distances must remain 4nite in the asymptotic limit, so that U de4nes a cluster if it is connected, or a set of clusters otherwise. If the cluster is of type AN , there is no other constraint and the cluster can in principle Qow to an in4nite distance from the origin (while the eigenvalues inside the cluster are kept at a 4nite distance from each other). On the contrary, if the cluster is of type CN , the eigenvalues are bounded by the lattice structure of U itself and consequently remain within a 4nite distance from the origin. In the following we denote the radial coordinates outside the cluster by x˜ and the ones inside the cluster by x .
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The asymptotic expansion of the zonal spherical functions in the presence of such a cluster was obtained a few years ago by Olshanetsky in [89]. It turns out to be a rather natural generalization of the Harish-Chandra asymptotic expansion, Eq. (9.31): * x) ˜ * i(rk; Rk (x) ∼ cz (rk)e R(rk) (x ) ; (10.16) r ∈W=W
where W which appears in the coset W=W is the Weyl group associated to the cluster, and Rk (x) ≡ K(x)Tk (x) (cf. Eq. (9.31)). The function cz (k) is de4ned by ! c (k) ; (10.17) cz (k) = ∈R+ =R +
where R + is the set of positive roots associated to the cluster. In (10.16) (rk) denotes the projection * is its complement. The function K(x) is given by of the vector rk on the sublattice U and rk Eq. (7.9): ! ! K(x) = | sinh2 xj − sinh2 xi |=2 |sinh 2xi |1=2 : (10.18)
i¡j
i
In spite of its apparent simplicity, expression (10.16) is highly nontrivial. Notice for instance that the symmetrization with respect to the Weyl coset W=W acts not only on the part containing the coordinates x˜ but also on the momenta of the zonal spherical function describing the cluster coordinates T(rk) (x ). This means that the particles inside the cluster do not move independently in a section of the whole space but they “feel” the presence of the other particles and are subject to the symmetry group of the remaining space. We refer the reader to the recent paper [88] for some explicit examples of this type of solutions and for indications of how they could be used to address the two problems mentioned at the beginning of this subsection. We only mention here that the clustered solutions described in this section may 4nd another natural application if one tries to model systems in which the number of open channels is reduced by the structure of the wire itself (cf. the wide–narrow–wide geometry of [90]). In this case, one could consider a con4guration formed by a CN type cluster (bounded to the origin) made of N eigenvalues and let the remaining N − N eigenvalues Qow to in4nity. One can choose N and s (the dimensionless length of the wire) so as to keep the cluster in the metallic regime, while the other eigenvalues are in the insulating regime and do not contribute to the wire conductance. 10.3. Triplicity of the Weierstrass potential The Calogero–Sutherland potentials of type IV in Eq. (7.2), vIV (K) = P(K) ;
(10.19)
where P(K) is the Weierstrass P-function, is the most general type of potential. The Weierstrass P-function is de4ned as 1 1 1 ; (10.20) − P(z; !1 ; !2 ) = 2 + z (z − 2!1 m − 2!2 n)2 (2!1 m + 2!2 n)2 m; n where the prime on the sum means we are summing over all pairs (m; n) ∈ Z2 except (m; n) = (0; 0). The P-function is doubly periodic with periods 2!1 , 2!2 , which can be seen by rearranging the
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sum. The Weierstrass P-function is an elliptic function. An elliptic function is de4ned to be a single-valued, doubly-periodic analytic function, whose only singularities in the 4nite part of the complex plane are poles. We will show that as both, or just one, of the two periods goes to in4nity, we recover the potentials of type I, II and III, respectively. This fact was mentioned, but not proved, in [9], and a proof for the III type potential was brieQy indicated in Appendix A of [91]. Set ∞ 1 V (z) = ; (10.21) 2 4 sin ((z + im)=2) m=−∞ where z is a complex variable. It is immediately obvious that V (z) is doubly periodic with periods 2/ and i, has a double pole in each period-parallelogram at z = 2/k + im (k; m ∈ Z) and is analytic elsewhere. The function V (z) − 1=z 2 is analytic in z = 0 and in a neighborhood around this point. Expanding it around z = 0 we 4nd ) ( ∞ 1 1 1 − V (z) − 2 = : (10.22) 2 z z=0 12 m=1 2 sinh (m=2) Let wkm = 2/k + im denote the position of the double poles of V (z). Since V (z) is periodic, analytic except in the poles, and has double poles in wkm it must be of the form + , ∞ c− 2 c− 1 n + c0 + V (z) = : (10.23) + cn (z − wkm ) (z − wkm )2 (z − wkm ) n=1 k; m 2 (n) n From the condition that f(z) ≡ V (z) − 1=z is analytic and equal to ∞ n=0 f (0)z =n! in and near z = 0 we then get c−2 = 1, c−1 = 0 by matching terms. We now see that the two elliptic functions V (z) and P(z; /; i=2) have the same periods, poles, and principal parts at each pole. They then diOer by a constant k (see [92, Section 13.11]), so that cn = 0 for n ¿ 0 in (10.23). So we have proved that V (z) − 1=z 2 + k = P(z; /; i=2) − 1=z 2 where k is de4ned by Eq. (10.22), since both sides have to have the same value at z = 0: ( ) ( ) 1 1 1 1 V (z) − 2 − V (z) − 2 (10.24) = − z z z=0 (z − 2/k − im)2 (2/k + im)2 k; m
or in other words ∞
∞
1 1 1 + P(z; /; i=2) = − : 2 2 4 sin ((z + im)=2) 12 m=1 2 sinh (m=2) m=−∞ It is now a simple matter to show that in the limit → ∞, 1 V (z) → 2 4 sin (z=2)
(10.25)
(10.26)
1 that is, apart from a constant 12 , P(z; /; i=2) in the limit → ∞ becomes a potential of type III in Eq. (7.2). In a completely analogous way one shows that the hyperbolic potential ∞ 1 V˜ (z) = (10.27) 2 4 sinh ((z + m)=2) m=−∞
M. Caselle, U. Magnea / Physics Reports 394 (2004) 41 – 156
is related to the Weierstrass P-function by ∞ ∞ 1 1 1 + : − P(z; =2; i/) = 2 2 4 sinh ((z + m)=2) 12 m=1 2 sinh ( m=2) m=−∞
147
(10.28)
As → ∞, the potential V˜ (z) approaches 1 V˜ (z) → (10.29) 4 sinh2 (z=2) so that P(z; =2; i/) becomes a potential of type II in Eq. (7.2). Also, if both periods go to in4nity, we obtain a potential of type I: 1 (10.30) P(z; =2; i=2) → 2 as ; → ∞ : z The important consequence of this analysis is that we can see the triplicity of the symmetric spaces as a limiting procedure on the general Weierstrass potential. This gives us a framework for interpolating between spaces of diOerent curvature (at 4xed root lattice), and may have relevant applications in the context of random matrices. Indeed, in the last few years a substantial amount of nontrivial mathematical results has been accumulated concerning these P(z) type models [93]. The hope is that some of them will be useful for constructing nontrivial generalizations of the known random matrix theories, while preserving some of the attractive properties due to the underlying symmetric space structure. 11. Summary and conclusion In this review we have discussed the usefulness of viewing random matrix ensembles as symmetric spaces. Random matrix theory, that has evolved into an important branch of mathematical physics, is used in the description of physical systems with chaotic behavior, disorder, or a large number of degrees of freedom. The versatility of random matrix theories allows for a parameter-free description of an assortment of systems, ranging in size from nuclei to mesoscopic conductors. The unifying feature of these systems is chaotic behavior eOectively resulting in randomness, and the available information is the universal statistical behavior of spectra. On the other hand, symmetric spaces are well-understood mathematical objects that can be represented as coset spaces G=K of a Lie group G with respect to a symmetric subgroup K (or as Lie algebra subspaces). After a general introduction to Lie algebras and root spaces, we have seen how to construct symmetric spaces from semisimple Lie algebras using involutions, and how to identify them with the integration manifolds used in random matrix theory. In the process we gave concrete examples of all the mathematical concepts that were introduced. We discussed coordinate systems on symmetric spaces and identi4ed the spherical radial coordinates with the physical degrees of freedom in most matrix models (an exception is the transfer matrix ensembles where the physically interesting degrees of freedom, in the Landauer theory, are the transmission eigenvalues, which are not the same as the eigenvalues of the transfer matrix). We explained how the Dyson and boundary indices of the random matrix ensemble are related to the multiplicities of the restricted roots associated to this symmetric space. We have seen that the metric on a Lie algebra, de4ned in terms of the adjoint representation, leads to the concept of curvature tensor on the symmetric space, and we have shown that the
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symmetric spaces classi4ed by Cartan appear in triplets of positive, zero, and negative curvature corresponding to a given (restricted) root lattice (and therefore to a given set of multiplicities of long, ordinary and short roots). Further we have discussed Casimir operators in a general context and their representations in local coordinates (the Laplace operators) and given a general formula for obtaining the radial part of the Laplace–Beltrami operator in terms of the Jacobian resulting from a transformation from random matrix to eigenvalue space. We devoted a subsection and the appendix to their eigenfunctions, the zonal spherical functions, that were later used in some of the applications. In Section 8 we gave a general introduction to some commonly used random matrix theories, explaining how universality of the correlation functions leads to general universal predictions concerning the statistics of the eigenvalues. The spectral properties that can be described in this way are exactly those that are independent of the detailed dynamics of the system. The only input is the global symmetries. The fact that many of the physical systems described by random matrices fall into the universality classes of Cartan’s classi4cation is a consequence of this fact. The eigenvalue density is determined by the matrix potential and by the Jacobian obtained in diagonalizing the random matrices on the symmetric manifold. The Jacobian is the origin of spectral correlations in the matrix models discussed here. It is completely determined by the restricted root system of the underlying symmetric space and by its curvature. In this review we also discussed the mapping of Calogero–Sutherland models onto (restricted) root systems of Lie algebras and symmetric spaces. The mapping is based on the fact that the Hamiltonian of these models for certain values of the coupling constants (determined by the root multiplicities) can be exactly transformed into the radial Laplace–Beltrami operator. This mapping allows to obtain several exact results for the zonal spherical functions of the corresponding symmetric spaces (see the discussion in the appendix). The Dorokhov–Mello–Pereyra–Kumar equation is the diOerential equation determining the joint probability distribution of the transfer matrix as a function of an external parameter. The operator appearing in this equation is mapped (by a transformation involving the Jacobian) onto the radial part of the Laplace–Beltrami operator representing free diOusion on the symmetric space underlying the random matrix ensemble. This can be used to solve the Fokker–Planck equation exactly for Dyson index = 2, and to obtain approximate solutions for the probability distribution and the conductance in other cases using asymptotic expansions of the zonal spherical functions. As further examples of applications we reviewed how the connection from random matrices to symmetric spaces can explain a discrepancy between random matrix theory and the nonlinear sigma model regarding the magnetoconductance, and further how the orthogonal polynomials and the symmetries of the matrix models can be traced directly to the root lattice. In the last section we discussed some applications going beyond the Cartan classi4cation, and we showed that the Weierstrass potential of Calogero–Sutherland models in its various limits reQects the possible signs of the curvature of the symmetric space. An interesting project that might be worth exploring in the same spirit as above concerns nonhermitean matrix models, i.e. random matrices having complex eigenvalues. Recently there has been a boost of activity in this 4eld. Since nonhermitean random matrices are useful in a number of contexts, but not much is known about them (see however [94–96]), a thorough study of them would be extremely important. A classi4cation according to their symmetries was attempted in [15], where 43 spaces were enumerated. However, a study of the resulting spaces in the spirit of this work has not yet been performed. Some nonhermitean ensembles have only recently been applied to physical systems.
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Exploring what is known about the corresponding manifolds, and how this knowledge could be applied in the above mentioned contexts, would appear to be a worthwhile eOort. We are not aware of the extent to which the properties of such manifolds are known in mathematics, and how useful this knowledge might be in the problems involving nonhermitean random matrix theories. Let us mention a few physical problems in which nonhermitean random matrices are present: (a) Nonhermitean matrices are important in schematic random matrix models of the QCD vacuum [97] at nonzero temperature and/or large baryon density (i.e. 4nite chemical potential). The spontaneous breaking of chiral symmetry is one of the most important dynamical properties of QCD, as it shapes the hadronic spectrum. The study of the chirally symmetric phase of QCD is one of the primary objectives of heavy ion colliders. One of the main problems with including a chemical potential in the QCD action is that the fermion determinant becomes complex, which makes lattice simulations extremely diVcult. By using schematic random matrix models, one can study qualitative features of QCD at 4nite density and/or temperature. (b) In classical statistical mechanics, non-equilibrium processes can be studied as the time-evolution of nonhermitean Hamiltonians. As an example we mention non-hermitean spin chains related to a Kardar–Parisi–Zhang equation. Such an equation can describe for example interface growth, problems in Quid dynamics, a driven lattice gas, or directed polymers or quantum particles in a random environment [98]. (c) Nonhermitean eOective Hamiltonians appear in the S-matrix description of open systems connected to reservoirs. The S-matrix characterizes scattering in an open chaotic system like for example a ballistic microstructure pierced by a magnetic Qux [99]. (d) Neural networks are described by continuous local variables related through a non-linear gain function to a local 4eld that could represent the membrane potential of a nerve cell. The dynamics of the network is described by a large number of coupled 4rst order diOerential equations featuring (random) nonhermitean matrices. These contain the parameters coupling the output of the j’th neuron to the input of the i’th neuron. In the context of neurobiology, the study of chaos in neural networks is relevant to the understanding of several features of neural assemblies [100]. (e) Anderson localization in a conductor denotes the phenomenon when the wave function of an electron becomes localized and exponentially decaying. As a result the conductor becomes an insulator. In conventional conductors localization is known to happen in dimensions d ¿ 2 (d = 2 is the critical dimension). New interesting phenomena may occur in systems with nonhermitean quantum mechanics. In [101], the authors considered particles described by a random SchrNodinger equation with an imaginary vector potential. The model was motivated by Qux-line pinning in superconductors. They found that delocalization transitions arise in both one and two dimensions. Such models were discussed further in [102–104]. Even if the suggested direction of research concerning nonhermitean matrix models turns out not to be feasible, there are likely to be many more applications for hermitean matrix models that can be pursued within the framework of symmetric spaces.
Acknowledgements This work was partially supported by the European Commission TMR program HPRN-CT2002-00325 (EUCLID). U.M. wishes to thank Prof. S. Salamon for discussions.
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Appendix A. Zonal spherical functions A remarkable feature of the theory of zonal spherical functions (ZSF) discussed in Section 6.3 is that in some special cases exact formulae exist for these functions. This happens in particular for all the symmetric spaces with multiplicity of ordinary roots mo = 2 in the classi4cation of Table 1 or Table 3. The ultimate reason for this becomes clear by inspection of the mapping discussed in Section 7. The radial part of the Laplace–Beltrami operator can be transformed into a suitable Hamiltonian of n interacting particles which decouple exactly for mo = 2. Thus we may expect to be able to write the ZSF’s in these particular cases as a suitable determinant of single-particle eigenfunctions. In the simplest case in which the multiplicity of both the short and long roots is zero one can write ! ˙ [K(q)]−1 det sei(s ; q) (A.1) s ∈W
where W denotes the Weyl group (which in this case is simply the permutation group, while det s is simply the sign of the permutation s), labels the ZSF and is related to its eigenvalue (see Eq. (6.45)), K(q) is given by Eq. (7.9) and ( ; q) denotes the vector product. In the above expression we recognize, as anticipated, the determinant det e i qj written as an alternating sum over the permutation group W ≡ Sn . Notice, as a side remark, that if the curvature of the symmetric space is positive, then is “quantized” and must belong to the dual weight lattice. This expression 4xes the q-dependence of the ZSF. However, it does not completely 4x the -dependence, which is hidden in the proportionality constant. Fixing this constant turns out to be a highly nontrivial task. In the case of the An type spaces it can be obtained by direct integration of Eq. (6.44) [9]. The 4nal result (in which we have introduced explicitly the parameter a as in Section 6.3 so as to describe the three possible curvatures in one single equation) is 1!2! : : : (n − 1)! s∈W det sei(s ; q) (− ) " ! (aq) = " : (A.2) j¡k ( j − k )=a j¡k sinh[a(qj − qk )] The extension to other root lattices is slightly more involved. Due to the presence of the long and/or short roots or to the additional reQection symmetry of the lattice, the corresponding “single particle” SchrNodinger equations are more complicated and the solutions are no longer simple plane waves. The problem was solved in 1958 by Berezin and Karpelevick [85]. The solution they obtained is valid for a generic BCn lattice and, remarkably enough, holds for any value of ms and ml , not only for those related to the symmetric spaces (while obviously mo = 2 is mandatory). In this case there is no compact way to deal with the three possible curvatures in a single formula, so let us look at the three cases separately: Positive curvature spaces: !(+) (q) = C+
det [;l !F( + 1)=F(;l + + 1)P; ;l (cos 2qj )] ; " " 2 2 2 2 2 2 j¡k ( j − k ) j¡k (sin qj − sin qk )
(A.3)
where the P; ;l are Jacobi polynomials with = (ms + ml + 1)=2, = (ml − 1)=2 and ;l = (2 l − ms − 2ml )=4 (recall that in this case is quantized, thus ;l turns out to be an integer number). The normalization C does not contain any further dependence on and q. Its value is "n−1 constant n− C = 2n(n−1) l=1 l!(ms + l) l .
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Negative curvature spaces: ! (−) (q) = C− "
det[F(al ; bl ; c; −sinh2 qj )] " 2 2 2 2 2 2 j¡k ( j − k ) j¡k (sinh qj − sinh qk )
(A.4)
where this time F(al ; bl ; c; −sinh2 qj ) denotes the hypergeometric function and the three parameters are: al = (ms + 2ml + 2i l )=4, bl = (ms + 2ml − 2i l )=4 and c = (ms + ml + 1)=2. The normalization constant is the same as in the positive curvature case. This is the formula that we used in Section 9.4.1 to solve the DMPK equation in the = 2 case. Zero curvature spaces: In this case the expression is slightly simpler: " !(0) (q) = C0
det [( l qj )−E JE ( l qj )] " 2 2 2 2 j¡k ( j − k ) j¡k (qj − qk )
(A.5)
where E = (ms + ml − 1)=2 and C0 =
(−1)n(n−1)=2 2n(n+mˆ −3=2) F(mˆ + 1=2) : : : F(mˆ + n − 1=2) 1!2!3! : : : (n − 1)!
(A.6)
with mˆ = (ms + ml )=2. A.1. The Itzykson–Zuber–Harish–Chandra integral A remarkable feature of the zero curvature case discussed above is that not only the zonal spherical functions themselves are simpler, but also the integral representation from which they are obtained drastically simpli4es. In fact, in this case (recall that mo = 2) the symmetric spaces coincide with the classical Lie algebras and Eq. (6.44) can be written as a simple integral over the group manifold. For instance in the unitary case, i.e. for systems of type An , the integral representation becomes # † (0) (A.7) ! (q) = DU eTr(OUQU ) ; where U ∈ SU (n), O and Q are diagonal matrices: O = diag( 1 ; : : : ; n ) and Q = diag(q1 ; : : : ; qn ). On the other hand, if we inspect the a limit of Eq. (A.2) we 4nd 1!2! : : : (n − 1)! s∈W det sei(s ; q) (0) " " ! (q) = : (A.8) j¡k ( j − k ) j¡k (qj − qk ) In this last expression we recognize the det e i qj written as an alternating sum over the symmetric group W ≡ Sn , while the two products in the denominator can be interpreted as Vandermonde determinants ! D( ) = ( i − j ) : (A.9) i¡j
Moreover, we easily see that Eq. (A.7) holds unchanged if O and Q are two generic hermitean matrices with eigenvalues { i } and {qi }, respectively. At this point, equating these two expressions
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for !(0) we 4nd the remarkable relation: # 1!2! : : : (n − 1)!det e i qj † DU etr(OUQU ) = D( )D(q)
(A.10)
which is the well-known Itzykson–Zuber–Harish–Chandra (IZHC in the following) integral. This result was originally obtained by Harish-Chandra in [105]. It was later rediscovered in the context of random matrix models by Itzykson and Zuber [106] and fully exploited by Mehta in [107]. It plays a major role in several physical applications of random matrix ensembles, ranging from the exact solution of two-dimensional QCD to the study of parametric correlations in random matrix theories. It is not diVcult, using Eq. (A.5), to generalize Eq. (A.10) to other Lie groups, i.e. to other root lattices. This general result was already present in the original paper by Harish-Chandra [105]. A recent review with detailed expression for SO(n) (both for even and odd n) and Sp(n) can be found for instance in Appendix A of [108]. A.2. The Duistermaat–Heckman theorem The remarkable elegance and simplicity of the IZHC integral suggests that there should be some deep mathematical principle underlying this result. Indeed, in the last few years it has been realized that the IZHC integral is a particular example of a wide class of integrals which may be solved exactly with the saddle point method provided one sums over all the critical points (and not only over the maxima). In the IZHC case this means that one has to sum over all the elements of the Weyl group (i.e over all the permutations in the unitary case) and not only over the ones with positive signature (as one would do in a standard saddle point approximation). The rationale behind this remarkable result is the Duistermaat–Heckman theorem [109]. It states that the saddle point is exact (provided one sums over all the critical points) if the integration is performed over an orbit with a symplectic structure. As a matter of fact, it was later recognized that this is only an instance of a wider class of results which are known as localization theorems. Their common feature is that they can be used to reduce integrals over suitably chosen manifolds to sums over sets of critical points. A more detailed discussion of this beautiful branch of modern mathematics goes beyond the scopes of the present review. To the interested reader we suggest Ref. [110], where a comprehensive and readable review on the localization formulae and some their physical applications can be found. References [1] F. Dyson, Comm. Math. Phys. 19 (1970) 235. [2] A. HNuOmann, J. Phys. A 23 (1990) 5733. [3] A. Altland, M.R. Zirnbauer, Phys. Rev. Lett. 76 (1996) 3420, cond-mat/9508026; A. Altland, M.R. Zirnbauer, Phys. Rev. B 55 (1997) 1142, cond-mat/9602137. [4] M. Zirnbauer, J. Math. Phys. 37 (1996) 4986, math-ph/9808012. [5] M. Caselle, A new classi4cation scheme for random matrix theories, cond-mat/9610017. [6] M. Titov, P.W. Brouwer, A. Furusaki, C. Mudry, Phys. Rev. B 63 (2001) 235318, cond-mat/0011146. [7] D.A. Ivanov, in: Proceedings of the workshop on Vortices in Unconventional Superconductors and SuperQuids— Microscopic Structure and Dynamics, Dresden, March 2000, cond-mat/0103089.
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Available online at www.sciencedirect.com
Physics Reports 394 (2004) 157 – 313 www.elsevier.com/locate/physrep
Atomic negative ions: structure, dynamics and collisions T. Andersen Department of Physics and Astronomy, University of Aarhus, DK-8000 Aarhus C, Denmark Accepted 12 January 2004 editor: J. Eichler
Abstract This paper reviews the knowledge of the structure, dynamics and collisions of atomic negative ions, as accumulated at the end of 2003, and describes how the research exploring these ions developed during the last decade. New experimental information has mainly been obtained from photon–negative-ion interactions using lasers and more recently also synchrotron radiation as the photon source. Additional insights have been gained from the use of new experimental techniques like heavy-ion storage rings, which made long-time observations of negative ions possible and promoted the study of electron–negative ion-interactions. Substantial progress has also appeared on the theoretical side with computational methods leading to reliable predictions for many of the lighter negative ions. c 2004 Elsevier B.V. All rights reserved. PACS: 32.10.Hq; 32.80.Gc; 39.30.−w; 31.25.Jf Keywords: Negative ions; Photodetachment; Electron correlation; Electron a7nities; Multiple ionization; Autoionization; Ion-atom collisions
Contents 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1. Historical background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2. Previous review articles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3. Plan for the review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1. Production of negative ions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2. Photon sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3. Single-photon studies using <xed frequency lasers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4. Single-photon studies using tunable lasers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.1. Laser photodetachment threshold (LPT) spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.2. Laser photodetachment microscopy (LPM) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.3. Resonance structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . c 2004 Elsevier B.V. All rights reserved. 0370-1573/$ - see front matter doi:10.1016/j.physrep.2004.01.001
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2.4.4. Photodetachment cross sections and angular distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5. Multi-photon studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.1. Resonant ionization spectroscopy (RIS) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.2. Non-resonant multi-photon detachment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.3. Resonant multi-photon detachment: resonances above the detachment limit . . . . . . . . . . . . . . . . . . . . . . 2.5.4. Resonance multi-photon detachment: resonances below the detachment limit . . . . . . . . . . . . . . . . . . . . . 2.6. Electron collisions with atoms or negative atomic ions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7. Negative-ion–atom collisions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.8. Storage rings and ion traps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.9. Accelerator mass spectroscopy (AMS) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3. Theoretical approaches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1. Computational methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2. The hyperspherical-coordinate approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4. Review of speci
175 176 176 179 182 185 188 190 192 195 196 196 200 202 202 203 208 209 215 216 217 224 228 229 230 230 231 237 239 241 243 243 249 250 256 257 260 260 263 265 265 266 268 270 271 271 272 274 274 277 278
T. Andersen / Physics Reports 394 (2004) 157 – 313 4.10.1. Fluorine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.10.2. Chlorine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.10.3. Bromine and iodine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.11. Transition elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.11.1. Lanthanum and osmium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.11.2. Technetium and rhenium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.11.3. Iridium and platinum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.11.4. Copper, silver and gold . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.12. Lanthanides . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.12.1. Cerium and praseodymium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.12.2. Thulium, ytterbium, and lutetium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5. Electron-impact on negative ions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6. Negative ions in external electric and magnetic <elds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1. Electric <elds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2. Magnetic <elds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
159 279 281 282 283 284 285 286 286 287 288 289 290 294 294 297 298 298
1. Introduction Negative ions represent a special niche of atomic physics and oEer excellent opportunities for studies of atomic structure, dynamics and interactions in systems characterized by the binding of an extra electron to a neutral atom in a short-range potential. While the electrons in neutral atoms and positively charged ions are bound in the long-range Coulomb potential, which is proportional to r −1 , r being the separation between the electron and the nucleus, the excess electron in negative ions is bound in a short-range potential, of the order of r −4 , with the H− ion as the only exception. The short-range potential gives rise to a number of exotic properties of negative ions unlike those of neutral atoms or positive ions. Whereas nearly 90% of atoms are able to form stable negative ions in the gas phase by attaching an electron to the ground state electron con
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Another well-understood result of the short-range potential is the variation of the photodetachment cross section with energy. The cross section is zero at threshold and rises in a manner that only depends on the angular momentum of the detached electron [2,3], unlike the photoionization cross section of neutral atoms, which is non-zero at threshold. This threshold behaviour is vital for experimental methods designed for accurate binding energy measurements (see Section 2.4), e.g. the laser photodetachment method utilizing resonance ionization detection of the excited neutral atom produced by the detachment process [4–6]. The zero cross section value at threshold also facilitated the
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of the atomic negative hydrogen ion in the Solar photosphere was put forward in 1939 [28] and later proved to be correct. Examples like these clearly illustrated that negative atomic ions were not only a curiosity of academic interest, but could play an important role in various branches of physics and chemistry as evident today. The existence of H− as a bound system had already been proposed theoretically by Bethe [29] in 1929, whereas predictions based on simple perturbational or variational methods had failed, even though these methods were well suited to predict the gross properties of the isoelectronic systems such as He, Li+ , Be2+ , etc. (for a review of the history of the H− ion see [30]). A combination of experimental, theoretical, and empirical methods had already in 1950 [27] established a reasonable set of electron a7nity (EA) values for the atoms belonging to the
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passed since Massey’s monograph [38] appeared it is still a useful source of information. The progress obtained during the 1980s has been described by Bates [40], who treated the atomic structure and spectra, by Esaulov [41] who focussed on the collisional aspects, and by Buckman and Clark [42], who in a very comprehensive review entitled “Atomic negative-ion resonances” updated the famous article by Schulz [37] from the early 1970s. Classi
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information has mainly been obtained from studies of photon–negative ion interactions using lasers and more recently also synchrotron radiation as the photon source. The development during the 1990s of new and often sophisticated linear and non-linear laser techniques, sometimes combined with very sensitive detection techniques [4,68] have yielded signi
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(2) direct extraction of negative ions from a Penning-type or a plasma source, (3) sputtering from a caesium contaminated surface. The need for negative-ion beams in connection with tandem accelerators entailed already in the 1970s [72,73] a signi
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sputtering projectile. As the caesium on the surface will be sputtered mostly as positive ions, a large fraction of the sputtered caesium will be driven back to the surface by the cathode potential drop. This mechanism tends to stabilize the caesium coverage and limits the caesium consumption of the source [73]. The design of the sputtering cathode, which usually is placed in a discharge opposite the outlet of the ion source, can be very important to ensure a highly e7cient beam formation by focusing the negatively charged, sputtered ions onto the outlet opening. The negative ions formed by the sputter technique will usually have to undergo a mass analysis before being injected into the experimental section of a negative-ion equipment. 2.2. Photon sources The photon sources used for negative-ion studies can be divided into two groups: (1) Lasers for low-energy photons (below 15 eV), (2) Synchrotron radiation for photons above 15 eV. Studies of photon–negative ion interactions can only be performed with low-density targets. Thus, the development and use of photon sources able to deliver high photon Mux (at narrow bandwidth) have had an important inMuence on the history of negative ions. During the last three to four decades we have witnessed a development from crossed ion-photon beam geometry with <xed frequency pulsed or cw lasers, via the use of tunable dye lasers to the more recent use of collinear geometry and the appearance of new laser sources, particularly in the infrared region e.g. the titanium-sapphire laser or the generation of intense coherent, Raman-shifted, infrared radiation produced from tunable dye laser radiation [31,48,49]. High-resolution studies of negative ions have bene
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experimental investigation of the H− ion Bryant and co-workers [81] introduced in 1977 a technique, which for two decades has provided a remarkable amount of spectroscopic information about this ion (for a review see [50]). The energetic photons were obtained using standard lasers in the visible or UV range and the frequency was up-shifted by as much as a factor of 3.4 by employing the large Doppler-shift associated with the high-velocity H− beam provided by the Los Alamos Meson Physics Facility. The 800 MeV H− beam energy was <xed, as was the photon energy of the lasers applied; tunability was achieved by adjusting the angle of interaction between the laser and ion beams. A very large photon energy range could be covered by this technique and used for studying the gross behaviour of the cross section, as well as detailed studies of resonances near the various H(n) thresholds. The same principle was also used in connection with the two-photon absorption study [82] of resonances possessing the same parity as the ground state. The resolution of the one-photon experiments was about 8 meV, mainly limited by the momentum spread of the relativistic ion beam and the angular divergence of the crossed-beam geometry. The main disadvantage of the technique described above to study the H− ion is the limited resolution, which prohibits a study of the much narrower H− resonances, located near the H(n = 2) threshold, which would be stringent tests of the many theoretical predictions available for the positions of the resonances [83–92] and their widths [83–85,88–90,92–95]. An alternative approach of Doppler-tuned spectroscopy was utilized by Balling et al. [96–99], who applied collinear ion–photon beams combined with a strongly reduced energy spread of the H− ions. Fixed frequency laser light at 118 nm (close to 11 eV) was produced as the 9th harmonic of the fundamental 1064 nm output from a pulsed, seeded Nd:YAG laser. Non-linear crystals were used to generated the 3’harmonic, which was further frequency shifted a factor of three in Xe gas using four-wave mixing. The tunability of this laser source was obtained by applying small changes in energy to the stored H− (or D− ) beam. The experimental resolution was improved a factor of 40 [99], compared with the experiments performed by Bryant et al., making it possible to explore some of the narrow resonances located near the H(n = 2) thresholds, see Section 4.1. The photo-induced processes in negative atomic ions by means of various laser techniques are all concerned with outer-shell electrons, whereas inner-shell excitation processes are outside the present range of conventional lasers and no inner-shell electron excitation or detachment in a free negative atomic ion had been reported before 2001. The reason was probably a combination of the very low densities which can be obtained for negative ions, the limiting photon Mux available from bending magnets at synchrotron radiation facilities and the relatively small cross section for photodetachment. However, during the second half of the 1990s synchrotron radiation beam lines started to be equipped with insertion devices called undulators, which could improve the photon Mux at selected photon ranges by several orders of magnitude. During 2001–2002 the
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Mux/meV generated by the 9th harmonics from a Nd:YAG laser at 118 nm and applied to study resonances in the H− ion [98] with the photon Mux/meV applied in the
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Fig. 1. Photoelectron kinetic energy spectrum for photodetachment of Ce− using 514:5 nm light [105].
research following the original discovery of the Ca− ion will be reviewed illustrating the important development in the study of negative ions, which took place both experimentally and theoretically in the 1990s. The LPES method has been revitalized at the end of the last decade by Thompson and co-workers in connection with studies of negative ions of lanthanides [102–105]. These ions can be expected to have a rather complex structure, possessing up to several excited states located below the energy of the ground state of the parent neutral atom. In order to establish the gross structure of these ions it is attractive to use the LPES technique to obtain the EA values even though the accuracy will be limited to approximately 25 meV. When the gross structure is established it is easier to perform EA measurements using LPT methods and thereby reducing the uncertainties. Fig. 1 shows a photoelectron kinetic energy spectrum for photodetaching Ce− using an argon ion laser (514:5 nm). The energy scale for the Ce− photoelectron spectra in the laboratory frame was transformed into the ion rest frame using the Cu− photoelectron spectra as a reference and interpreted using spectroscopic data for the neutral atom [106]. The electrons photodetached in the interaction region were energy analysed using a spherical sector, 160◦ electrostatic kinetic energy analyser, which operated in a <xed pass-energy mode. The structure labelled 3 is the most intense structure, and since there are no discernible higher energy structure, it was assumed to be associated with the ground state to ground state transition and studied in more detail at higher resolution, see Section 4.13. Single-photon studies using <xed frequency lasers were also applied, as described above in Section 2.2, to explore the structure of the H− ion, but tunability of the laser light was then obtained using Doppler-shift methods. 2.4. Single-photon studies using tunable lasers Single-photon and multi-photon studies using tunable lasers have dominated the experimental negative-ion studies during the 1990s.
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2.4.1. Laser photodetachment threshold (LPT) spectroscopy By measuring the threshold photon frequency of the transition from the ground state of the negative ion to the ground state of the neutral atom plus a threshold electron the binding energy of a negative ion can be established. Up to year 2000 [31] the binding energies measured covered the energy range from approximately 25 meV (Ca− ) to 3:61 eV (Cl− ). If the rather few negative ions with binding energies below 150 meV were excluded, it should be possible to determine the binding energies of the remaining stable negative ions, more than 60, utilizing single-photon absorption from tunable lasers combined with recording the rate of formation of the detachment products, the neutral atoms, the emitted electrons, or both, as a function of the frequency. Tunable laser photodetachment threshold (LPT) spectroscopy is well suited for such studies, utilizing UV, visible or IR tunable laser light. LPT spectroscopy using visible laser light was already established in 1970 [107] in combination with crossed ion-laser beam geometry, and later improved utilizing coaxial beams [108,109]. It is, however mainly within the last decade that tunable IR radiation has been applied to the study of negative ions. Binding energies of many negative stable ions are of the order of 1 eV or below [31]; tunable IR laser light is therefore advantageous for photodetachment studies, especially for detachment studies to the ground state of the neutral atoms. In a series of experiments Haugen and coworkers from the McMaster University group introduced and applied tunable IR laser photodetachment threshold (LPT) spectroscopy in a crossed beam set-up to measure a large amount of precise EA values [31] covering elements like B [110], C [111], Al [112], Si [111], Cr [113], Co [114], Ni [114], Cu [113], Ge [111], Mo [113], Ru [115], Rh [114], Pd [114] and even some heavier ones [31]. Nd:YAG pumped dye laser light was used to generate light with wavelenghts ranging from 400 to 980 nm with typical pulse energies up to 50 mJ. The tunable IR radiation covering the range up to 5:2 m was obtained using Raman conversion of the dye laser light in a high-pressure (22 bar) molecular hydrogen cell [31,64]. The IR radiation technique was originally developed by Bischel and Dyer [116]. Under the experimental conditions applied by the McMaster group, the
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only in the energy region close to threshold. The energy range over which the relationship given by Wigner’s law is valid to describe the threshold behaviour is not known from theory. It may therefore be necessary to apply correction terms associated with the long-range interaction between the electron and the neutral atom. If the dipole polarizability of the
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Fig. 2. Photodetachment yield vs. photon energy for the B− ion [110]. The measurements are performed by tuneable infrared spectroscopy. The solid line represents an analysis of the data using a Wigner s-wave
Crossed beam experiments are usually carried out choosing =90◦ to avoid the
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Fig. 3. Threshold photodetachment cross section for H− ions. The solid line represents a Wigner p-wave
Due to the experimental di7culties mentioned above with the H− ion in connection with threshold frequency determination (emission of p-wave) it has not been reasonable to select this ion as the reference standard for binding energies of negative atomic ions; the O− ion [49] has been adopted. The very precise binding energy reported by Neumark et al. [124] for the O− ion with a relative uncertainty of 5 × 10−7 , obtained from coaxial laser photodetachment threshold (LPT) spectroscopy, was for more than a decade considered to be the proper reference standard, since it was considered to be the most accurately known of all measured atomic electron a7nities [49]. The recent development of the laser photodetachment microscope (LPM) [125] and its use for determining the electron a7nity of the 16 O atom [126,127] have, however, caused some reconsiderations concerning selection of the O− ion as the reference standard [31], since the LPM and LPT results deviate by 0:032 cm−1 or 4 eV. This issue will be further addressed in connection with the negative oxygen ion, see Section 4.10. 2.4.2. Laser photodetachment microscopy (LPM) Photodetachment microscopy can directly image the spatial distribution of electrons photodetached from a negative ion in the presence of a uniform electric <eld [125,128]. Provided the electric <eld is small compared to the internal <eld of the atom, which is the usual situation when dealing with negative atomic ions in the laboratory, the electron is emitted from the ion in the form of a spherical wave of energy j, which will be folded back onto itself by the external <eld. The two halves of the wave will interfere and produce a pattern cylindrically symmetric around the electric <eld direction. Since the spectral resolution of the excitation scheme usually is of the order of 0:02 cm−1 and the spectral resolution experimentalists can achieve with electron detectors never is much better than
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0:1 mm, the ring pattern can be observed provided the following conditions are present: The kinetic energy at which the electron is brought above the detachment threshold is in the range 0.1–3 cm−1 [129], the electric <eld remains in the 102 –104 V m range, and a high spatial resolution electron detector is included in the experimental set-up [125,126]. The electron energy can be determined from the pattern radius since the radius increases as j1=2 . One can, however, also measure the accumulated phase or the number of rings, both of which have an j3=2 variation with energy, which makes them more sensitive to energy variation than the spot radius. Furthermore, the latter two are dimensionless quantities, which means that no absolute calibration of the electron image sizes will be needed. Counting the bright and dark rings immediately yields the number of rings. A more quantitative analysis is usually performed by
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Fig. 4. Scheme of experimental setup used in laser photodetachment microscopy (LPM), showing the expansion of the electron wavefunction from the photodetachment zone up to the spatial resolution detector. L is the laser, I the ion beam, P0 and P27 the two extreme plates, 28 in total number, which produce the uniform electric <eld F. The stack of micro-channel plates MCP is followed by a resistive anode encoder RAE, only partly drawn, with two of the four outputs shown, A and B, leading to the position analyser [126].
the population of a speci
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For neutral thresholds located very near to the two-electron escape thresholds it may be possible to combine single-photon excitation with state-selective electric <eld ionization to study energetic high lying resonance structures by selectively monitoring the neutral Rydberg atoms formed. By proper design of an electric <eld ioniser Petrunin et al. [19] could perform a photodetachment study of the He− (1s2s2p 4 P) ion in the vicinity of the two-electron escape threshold by selective monitoring the formation of He(1snl3 L) Rydberg states for n being 11–14, which are located only 50–100 meV below the He+ limit. The Rydberg atoms were reaction products from very weak reaction channels, representing only 10−6 –10−7 of the total detachment yield. Resonance structures have also been observed in connection with inner-shell excitation studies of negative ions using far more energetic photons than considered above. In the negative carbon ion [101] a pronounced shape resonance structure was observed just above 281 eV. The synchrotron radiation-induced excitation and detachment studies could bene
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and the cross section can be obtained from the shape of the curve. This technique has been in use for single as well as multi-photon detachment studies [152–154]. The saturation method has the advantage that it is not necessary to know the detection e7ciency of the neutral atoms nor the current of the negative ions, but information about the overlap between the laser and ion beams is needed, since the ion beam is not stationary and the laser Mux not constant over the beam spot. If the velocity of the ions and the inhomogeneity of the laser Mux have to be included then one has to consider explicitly the trajectory of the ions through the laser beam [154], but still it is not necessary to know the detection e7ciency, since the shape of the curve determines the cross section. Absolute cross sections using this method have been reported to be accurate within 10%. To avoid many of the complications related to photon–negative ion absolute cross section measurements relative measurements are often performed and made absolute by measuring the neutral atom production from the negative-ion beam of interest relative to that from an O− ion beam. Measurements of the cross section of O− have been reported long ago by Branscomb et al. [155] in the energy region from 1.5 and to 4 eV. This procedure has been applied to several photodetachment studies in the 1990s, e.g for the B− ion [139], with the absolute cross section data claimed to have an accuracy of better than 10%. Angular distributions of photoelectrons produced by single-photon detachment may be measured to give a valuable supplement to cross section data. The measurements are usually performed at a few selected wavelengths using a crossed laser-ion beam apparatus, see e.g. the measurements concerning the negative ions of Al, Si, and P [156] or the negative C ion (Section 4.7). 2.5. Multi-photon studies Multi-photon studies have had an important impact on negative-ion studies during the last 15 years. As neutral atoms, negative ions can undergo multi-photon excitation in su7ciently intense <elds. Even though the
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λ1,λ2 Ion Source t Chargeexchange Cell Na
~8ns
Faraday Cup Be
Be
λ1,λ2
Be+ +
Be
Power Meter
Fig. 5. Schematic diagram of the experimental setup used for laser photodetachment resonant ionization spectroscopy.
a second photon into a known state, often a Rydberg state, which subsequently is either photoionized or electric <eld ionized. The positive ions produced are subsequently detected. Photodetachment of a negative ion to an excited state of the neutral atom had been used for many years to obtain binding energies of negative ions, such as the alkalis, but the accuracy of the values obtained was lower than could be obtained with detachment to the ground state [49]. The reason was that changes in the photodetachment cross section, from the opening of a new detachment channel, would be superimposed on the much larger photodetachment cross section for the lower lying atomic states. This is particularly the case if the new detachment channel is associated with emission of a p-electron and the detection of the neutral atoms has to rely on the conventional methods for neutral particles [49]. A signi
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Be+ 1s22s 15d 3D
λ2
1s22p2 3P 1s22s 3p 3P
1s22p3 4S
λ1
1s22s 2p 3P 1s22s 2p2 4P
1s22s2 1S
Be--
Be
Fig. 6. Schematic energy diagram of Be− and Be states with the detachment and excitation channels indicated.
has the purpose to act, at higher voltages, as a state-selective electron-<eld ionized optimized for detection of speci
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Be+ YIELDS (arb. units)
150
100
50
0 39272
39274
39276
39278
39280
PHOTON ENERGY (cm-1)
Fig. 7. Be+ -yield following photodetachment of the Be− (4 P3=2 ) ion to the Be(2s3p 3 P) state, which subsequently is monitored by resonant ionization via the Be(2s15d 3 D) Rydberg state [21].
to obtain binding energies with high accuracy for several negative atomic ions such as the He− [13], Li− [68], K − [163], and the alkaline earths, Be− [21], Ca− [5], Sr − [6], and Ba− [4] ions. The RIS method has also been very important for the exploration of the resonance structures belonging to the He− (1s2s2p 4 P) ion and located in the energy region of the He(1snl3 L) thresholds with n = 3–5 [14–16]. Additional studies have covered the Li− ion in the energy region below the Li(6p) threshold [164–167], the Na− ion below the Na(5p) threshold [168], and the K − ion below the K(7p) threshold [169]. These studies have all stimulated theoretical studies of the relevant ions and the comparisons between the experimental results and the theoretical predictions will be discussed in Sections 4.2 and 4.3. Furthermore, the state-selective detection method also plays an important role in combination with non-linear laser techniques, see Section 2.5.3 [170,171]. 2.5.2. Non-resonant multi-photon detachment The access to strong laser <elds created in the beginning of the 1990s an interest for exploring the impact of strong laser <elds on negative atomic ions [43,54]. At that time strong laser <eld studies of neutral atoms had been performed for more than a decade, but due to the atomic structure of neutral atoms and the inMuence of the Coulomb potential on the outgoing electron non-resonant multi-photon detachment studies of negative atomic ions oEered possibilities for studying electron emission under quite diEerent and sometime much better conditions. First of all it would be possible to select negative ions, like the halogen ions, so the studies could performed without interfering eEects from excited states, secondly the emitted electrons would not be under inMuence of the Coulomb <eld. Thus, non-resonant multi-photon detachment provided conditions which, from a theoretical point of view, would be very attractive with a perfect non-resonant bound-free multi-photon transition. During the
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Fig. 8. Schematic of experimental cross-beam apparatus employed in the non-resonant excess photon detachment work on the Au− ion.
a clear distinction between existing theories [43]. It was more successful to study (1) Excess-photon absorption in negative ions. (2) Multi-photon detachment diEerential cross sections. (3) Multi-photon detachment with diEerent polarization of the laser light. Fig. 8 shows a set-up used for non-resonant photon detachment studies of negative ions [54]. The negative ions, supplied from a sputter ion source, are introduced to an ultrahigh vacuum chamber (UHV) after undergoing acceleration and mass and charge-state analyses. By deMecting the negative-ion beam about 15◦ within the UHV chamber, at a pressure of 10−8 –10−9 mbar, the collisionally produced neutral atoms are eliminated. At the centre of the UHV chamber the negative ions are crossed by a focussed laser beam and after subsequent electric <eld analysis the detached neutral atoms and positive ions are recorded by electron multipliers. The electrons emitted from the photon–ion detachment process can be analysed by a magnetically shielded time-of-Might (TOF) spectrometer, equipped with tandem-channel plates [54]. Three publications appeared within a few weeks in 1991 all reporting the observation of excess photon absorption in either negative F [158], Cl [159], or Au ions. [160]. In the case of the Au-ions (Fig. 9), two 1:165 eV photons from a Nd:YAG laser were su7cient to exceed the detachment threshold with about 20 meV. According to Wigner’s threshold law [2], the two-photon detachment cross section will be suppressed just above threshold and it was therefore easier to search for higher order excess-photon processes in negative ions than in neutral atoms [54]. In all three experiments mentioned above it was possible to reach at the same qualitative conclusion, that non-resonant excess photon absorption is an important process in a negative atomic ion at laser intensities of the order of TW=cm2 . In the negative Au ion experiment the
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Fig. 9. Energy level diagram of Au− . A mode-locked Nd:YAG laser was employed, such that the 2-photon absorption barely exceeds thresholds [54].
Fig. 10. Angular distribution of four-photon detached electrons from F− , at the wavelength of 1064 nm; three-photons are enough to detach the negative ion. The qualitative correspondence with the plane-wave approximation (dashed line) is a precise signature of excess-photon detachment [43].
(three photon absorption), and the second order excess-photon processes (four photon absorption) were estimated to represent 5% and 0.6%, respectively, of the total electron yield at an irradiation intensity of 3×1012 W=cm2 . In the F− experiment the angular distributions of excess-electron detached electrons were measured. Fig. 10 shows the angular distribution of four-photon detached electrons from F− ion at the wavelength of 1064 nm [158]. Three photons are su7cient to detach the F− ion at this wavelength, so the shape of the angular distribution gives a precise signature of excess-photon detachment. The
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160
140
Cs 7d
7s
Cs SIGNAL (arb. units)
120 +
100
80
60
40
6p1/2 7s
40
6p3/2 6p1/2
20 30 0 20
10
Cs 6s Cs 6s2
(a)
14940
(b)
14960
14980
Cs+ SIGNAL (arb. units)
6p3/2 7s
0
WAVENUMBER (cm-1)
Fig. 11. (a) Simpli<ed energy-level diagram of Cs− and Cs. (b) Cs-signal (upper part) and Cs+ -signal (lower part) [54].
2.5.3. Resonant multi-photon detachment: resonances above the detachment limit Resonant multi-photon detachment methods have been developed during the last decade and may be subdivided into two groups, (1) The resonant state is located above the detachment limit. (2) The resonant state is located below the detachment limit. The exploration of resonant multi-photon detachment as an experimental method to study negative ions was pioneered by Haugen and co-workers [54,173,174]. Resonant phenomena can occur in negative ions in several ways. The
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rise time of the laser pulse; in addition the doubly excited state is expected to favour the absorption of additional photons. As indicated on the
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ns 15s 3S1
λ excite λ probe J=2
ns np2 4P J=5/2 J=3/2 J=1/2
J=1 J=0 ns np 3P
λ detach
J=3/2 J=1/2
ns2 1S0 ns2 np 2P
Neg. Ion
Atom
Fig. 12. Schematic energy-level diagram of negative alkaline earths like the Ca− ion (n = 4) and the corresponding Ca atom, illustrating the principle of the state-selective depletion spectroscopic technique [170].
The Ca− (4s4 p2 4 P) state has played a very important role in the process leading to the present understanding of the Ca− ion. Before 1987 the existence of the negative Ca ion was considered to be due to the 4 P state, which was assumed to be long lived and metastable like the homologous state in the Be− . The discovery of the stable Ca− ion [10] dismissed this assumption, but subsequent studies [177] still claimed that a long lived, metastable component was present in negative Ca-beams and that this component most likely should be attributed to Ca− (4s4p2 4 P) ions. The state-selective depletion spectroscopy study by Kristensen et al. [170]
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Ca+ 6
4s13p 1P1
λ4 5
ENERGY (eV)
4p3 4S 4
4s4d 1D2
λ3
3
λ2
2
4s4p 1P1
λ4
λ1 5/2 3/2 1/2
1
λ2 4s4p2 4P
λ3 t
λ1 0
3/2 1/2
4s24p 2P
Ca--
4s2 1S0
Ca
Fig. 13. Schematic energy-level diagram of the Ca− ion and the corresponding Ca atom showing non-linear resonant multicolour absorption via the Ca− (4s4p2 4 P) autodetaching state, leading to population of the Ca− (4p3 4 S) state [80].
a schematic energy level diagram of the negative calcium ion and the corresponding parent atom used. The
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ION YIELD (arb. units)
186
75
25
439.05
439.10
439.15
439.20
λ2 (nm)
Fig. 14. Ca+ photo-ion yield, reMecting the population of the Ca(4s4p 1 P) level generated by the two-photon detachment process with the 2 laser (Fig. 13) being scanned in the region of the 4s4p2 4 P1=2 –4p3 4 S transition in the Ca− ion [80].
electronic con
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Fig. 15. Schematic energy-level diagram of Te− illustrating 2 + 1 photon detachment of Te− (2 P3=2 ) when the Raman condition is ful
“forbidden” transitions to study a range of heavier negative ions of elements like Sn [114], Sb [182], Te [180], Ir and Pt [181,183] to obtain information about the binding energies of excited, but bound negative-ion states including
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5 Sb (5p3 4S3/2)
6
5p4 1D2 2 3
4
1
5
J=0 5p4 3P
6
J=1 4 J=2
6
Sb--
Fig. 16. Schematic energy level diagram of Sb− . Arrows indicate diEerent photodetachment schemes: (1–3) single-photon detachment thresholds; (4–5) two-photon detachment via single-photon M1 resonance; (6) three-photon detachment via two-photon E1 resonance [182].
to the ground state con
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189
Whereas electron-impact ionization of atoms and positive ions has been studied since the early days of quantum mechanics, and much of our understanding of the structure of atoms has emerged from electron scattering experiments, the situation has been completely diEerent with respect to electron detachment from negative atomic ions. Only very few electron–negative atomic ion experiments have been conducted and only a few elements studied, before storage rings dedicated to atomic and molecular research became available (see Section 2.8 about storage rings). Tisone and Branscomb [185,186] and Dance et al. [187] studied the impact of 10–500 eV electrons on the H− ion, whereas the studies by Peart et al. [188–190] in addition to H− also included ions like C− , O− and F− . From an experimentalist’s point of view, negative ions may be di7cult to work with since the cross section for collisions with rest gases is large, leading to detachment of the negative ions. This results in a signi
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dismissed the previous claims [197–199] for the existence of doubly charged, negative atomic ions (see Section 5), produced by electron scattering from ground state negative atomic ions. 2.7. Negative-ion–atom collisions Heavy-particle collisions, using fast negative ion–atom or positive ion–solid interactions (solids in the form of thin carbon foils) as projectile–target combinations, have proved to yield valuable information about doubly excited states of simple negative ions, which otherwise are di7cult to study due to lack of selective excitation techniques. The negative ion–atom collision experiments were introduced by Edwards and co-workers in the 1970s, for a review see [70], and were performed by interacting momentum analysed and focussed negative-ion beams, having energies in the keV range, with atomic (He, Ne, etc.) or molecular (H2 ) gases. The experiments were carried out under conditions usually applied to heavy-ion collisions at that time, so ultra-high vacuum was not used. The electrons resulting from the diEerent collision processes were energy analysed using various types of energy analysers. Only electrons originating from excitation of either the projectile or the target will lead to structures in the yield of electrons as a function of energy. Doubly excited negative-ion states can be generated both in the projectile negative ion due to excitation, but also in the target due to charge exchange and excitation. By proper selection of target and collision energy it is possible, however, to eliminate the latter and only focus on the projectile; at low-energy it is also possible to control, to some extent, the doubly excited states created. The starting point for interpretation of inelastic processes in low-energy (1–10 keV) heavy-ion collisions is the diabatic molecular orbital-correlation diagram for the projectile–target pair. Fig. 17 shows schematically such a diagram for the F− –He collision, constructed from the Barat–Lichten rules [200]. Only the
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Fig. 17. Diabatic molecular orbital-correlation diagram for low-energy F− –He collisions.
Light sources based on heavy-ion collisions as the fast beam-foil/gas sources have been shown to e7ciently populate multiple excited states. The beam-foil technique relies on the passage of an energetic (keV–MeV) positively charged ion beam through a thin carbon foil with subsequent observation of the light emitted from the particles passing the foil. By varying the energy of the projectile ions it is possible to inMuence the charge distribution of the beam after its passage of the foil. The beam-foil technique has mainly been used to study: Atomic spectra of highly charged ions, multiply excited states in few electron systems (for a review of the three electron systems see [207]), and lifetimes of excited states decaying by photon emission. Since negative ions were not considered to possess excited states able to decay by photon emission it was never suggested before 1980 that optical spectra recorded from beam-foil experiments could contain spectral lines originating from negative ions. However, that year Bunge [208,209] predicted on the basis of very accurate calculations that an unidenti<ed spectral line in Li-beam foil spectra at 349:0 nm originated from a transition in the core excited Li− ion, which very quickly was con
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proved to be very valuable in connection with the characterization of the structural and dynamic properties of the metastable Be− (2s2p2 4 P) state [21,22]. The collisional aspects of keV negative ion–atom collisions were studied in the 1970s and 1980s and reviewed by Esaulov [41]. The dominant process was single-electron detachment, which theoretically could be accounted for within an “independent scattering model” in which a simple picture of the negative ions was used, consisting of a loosely bound electron and a core of the neutral atom. It was assumed that the electron and the core collided independently with the target [214–216]; a critical experimental test of this model was performed more recently [217] by comparing the orientation and alignment of the Li(2p2 P) states created in Li–He and Li− –He collisions, using polarised photon-scattered particle coincidence technique. Such complete scattering experiments have contributed signi
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Fig. 18. The storage ring ASTRID. Shown are some of the essential features for the experiments with negative ions. The 16 quadrupoles and correction dipoles are not shown.
intensity of the stored beam is typically monitored by Schottky pick-ups, but in order to greatly extend the sensitivity of beam monitoring, neutral atom detectors were installed at the corners of the ring. Whereas most of the negative-ion experiments were performed using nA or more intense beams, the sensitive detection technique allowed for experiments with pA beams, too. The count rate of neutral particles detected behind one of the magnets was recorded as a function of time after injection. This provides not only a method to follow the beam storage, but also allows for an easy method to measure the lifetimes of negative, metastable atomic ions that decay by electron emission, on a time scale considerably shorter than the storage time. By circulating a metastable negative-ion beam for periods of seconds it is possible by time-of-Might technique to make measurements of lifetimes in the range from about 10 s to 100 ms at the ASTRID storage ring [12,20,56,63]. The short time limit is due to the round-trip of the ion, the long time by negative ion-rest gas collisions leading to destruction of the ion. The advantages of using a storage ring rather than a single-pass beam [220] to study lifetimes of metastable negative ions are: data can be extracted over a much greater time range out to several lifetimes with a good signal-to-noise ratio; slit scattering is essentially eliminated in the ring, and ultrahigh vacuum conditions render collisional quenching entirely negligible. For photon- or electron-impact studies, the ions are accelerated to MeV energies before being exposed to VUV laser photons or merged with the electron beam from the electron cooler, respectively. The acceleration procedure can be performed within a few seconds, before the negative ions interact with 118 nm photons in the case of the H− ion (see Section 4.1) or are merged with mA electron beams with energies ranging from 50 eV to 2 keV. The electron beam is essentially uniform with a density of the order of 107 cm−3 [196,221]. It is important that the electron velocity distribution is known, particularly when threshold behaviour or resonances are studied. The electron cooler, which is mounted on one of the straight sections of ASTRID consists of an electron gun, a 1 m interaction region, and an electron collector. A 4–5 mA electron beam is emitted by a tungsten cathode and accelerated through a grid to about 450 V. The electron beam
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Fig. 19. Schematic diagram of the ELISA storage ring with injector. The negative ions are injected from the lower left side and the neutral particles detected by multi-channel plate detector (MCP) at the upper left side.
is guided in and out of the interaction region by solenoid magnetic <elds of 100–200 G connected with two deMection toroids. Since the electrons are continuously renewed, the electron cooler may be considered a heat exchanger in which the velocity spread of an ion beam with longitudinal velocity equal to that of the electron beam is reduced as the ions moving too fast (slow) are decelerated (accelerated) by collisional energy transfer [222]. By the cooling process an equilibrium situation with equal ion- and electron-beam temperatures is approached, implying a velocity spread of the ion beam which is reduced with respect to that of the electron beam by a factor (M=m)1=2 , with M and m representing the ion- and electron masses, respectively, provided Maxwellian velocity distributions can be assumed. The longitudinal electron temperature will be reduced by the acceleration process to approximately 1 meV [221], while the transverse temperature can be reduced from its original value, given by the cathode temperature, by adiabatically expanding the electron beam [223]. In the cooling of the H− ion [99] the transverse temperature was determined to be 26 meV, which represented a reduction by a factor of 4.3. From an atomic physics point of view the use of a magnetic storage ring for lifetime studies may have some disadvantages, since the steering quadrupoles magnets can cause mixing of magnetic substates originating from diEerent, but close-lying
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The electrostatic lattice con
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by combining AMS with laser photodetachment [238]. The electric <eld technique has also provided preliminary binding energies for negative ions of Tm and Dy [231] and Yb [239]. The latter was reported to have a binding energy of only 10(3) meV, which would have been the lowest binding energy known for a stable negative atomic ion. However, a reinvestigation [240] of the Yb− ion has dismissed the claim for the existence of a stable Yb− ion and also for a long-lived, metastable one. 3. Theoretical approaches The theoretical analysis and description of negative ions have followed two more or less independent, but interacting, directions one being predominantly computational, the other predominantly modelistic. Buckman and Clark [42] have given a very good survey of the various methods within both directions and their survey may be consulted for a more extensive description of the theoretical methods used most often during the last decade. 3.1. Computational methods Theoretical models of neutral atoms are generally based on the central-<eld approximation, according to which each electron moves in a spherical symmetric potential representing the interaction with the nucleus and the average eEect of the interaction with the remaining electrons. If the electronic motions are only weakly coupled, so the introduction of spherical symmetric potentials can account for the major part of the electron–electron interaction term of the Hamiltonian, then the remaining part can be treated as a perturbation The con
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197
atomic ions. In principle, energy levels can be calculated numerically to arbitrarily high accuracy by inclusion of an arbitrary number of con
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function of continuum states from which the resonance parameters are obtained via the analysis of scattering matrices or eigenphase shifts; (ii) direct calculation of complex energies of decaying states by treating the resonance as an eigenfunction of a non-Hermitian system, and (iii) computations that treat resonances like ordinary bound states and neglect interactions with the continuum as a
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199
indicates that the applied wave functions are suitable for the problem studied. At a given energy E, the wave function inside the reaction volume is then described as a linear combination of the eigenchannel wave functions generated. In the outer region all the long-range interactions are treated numerically by close-coupling procedures to obtain a basis set of multi-channel wave functions that can describe the outgoing electron and the residual neutral atom with the eEects due to the polarization of the residual atom included. Finally, the linear combinations of the multi-channel basis functions for the inner and outer regions are matched at the reaction surface to obtain the exact
200
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The complex-coordinate rotation method has been used rather extensively and with signi
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201
analogous to those arising in the Born–Oppenheimer approximation for molecules. For a given R the potential energy of the system can be represented by an eEective charge, which is a function of and 12 [266]. A resonant state is obtained in a situation where the two electrons are bound to the nucleus, i.e. they remain at
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4. Review of speci c results by elements 4.1. Hydrogen The H− ion occupies a special role in the study of negative atomic ions. It is the simplest ion of this type, its dipole potential can support an in
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4.1.1. Resonance structures Resonances of H− near H(n = 2): In the unique series of experiments performed by Bryant et al. before 1992 the gross behaviour of the photodetachment cross section as well as detailed studies of 1 P resonances near the n = 2–8 thresholds were explored. The initial observations of the lower lying resonances were in remarkably good agreement with the calculations by Broad and Reinhardt [274]. The detailed investigations of the higher lying resonances prompted theoretical investigations in the relevant energy regions [42,50] and again the agreement was good. More recently, the New-Mexico group has also explored the application of two-photon detachment technique to study a 1 D resonance located just below the n=2 threshold [82] and extended this work to include isotope shifts [275,276], see below. The speci
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+ Bryant and coworkers [50] observed the
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Fig. 20. Relative photodetachment cross section for D− vs eEective photon energy [97]. (a) The measured cross section has been normalized to theoretical results of Lindroth [86] (solid curve). The data of the 2 {0}− 3 resonance at 10:9277 eV exceeds the vertical scale of the plot. (b) Blow-up of the region near the 2 {0}− 3 resonance; the solid curve is a
The hydrogen experiments can be performed applying either H− or D− ions, which allows for an investigation of the isotope eEects. The very narrow A = − structures permit a study of even small shifts. Both the experimental and the theoretical eEorts have been centred towards determining the so-called speci
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Fig. 21. Relative photodetachment cross section vs. eEective photon energy in the vicinity of the 1 P 2 {0}− 3 resonance [99]. The solid curve represents the best
turning points and that it is the only resonance of its type observed in the H− photodetachment spectra. The energy of this resonance has been predicted in good agreement with the observed one of 10:971 eV, but the resonance width still causes some problems. The width has been calculated to be within the range of 16.9–18:6 meV [84–86,90,282], whereas the experimental results of 21(1) or 30(1) meV reported by the New-Mexico group [281] and the more recent value of 25(2) meV, obtained at the ASTRID storage ring [283], are indicating a somewhat larger width. In the latter experiment a new-positioning method was applied to prevent changes in the overlap between laser and ion beams over the large kinetic-energy range employed. In order to establish the width, the energy and asymmetry of the shape resonance a function is needed which coincides well with the experimental data. Traditionally, a Fano function [176] has been used to
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One-photon excitation of H− ions present in their ground state can only lead to population of P states, whereas two-photon excitation can lead to population of 1 D or 1 S states, but so far 1 S states have not been observed. In the energy region close to the H(n = 2) threshold triplet states are also present, but the selection rules for photon impact prohibit the exploration of such states, which previously have been observed in elastic electron-scattering experiments [42]. Theoretical calculations, which have been proved to yield excellent agreement with the experimental photon-impact data available, have been used to generate energies and widths of several of the resonance states which have not been observed so far. Several methods have been used for this purpose (see [98]), such as the saddle-point complex-rotation method used by Chen [87] or the numerically accurate Harris–Nesbet variational method applied by Gien [88]. Resonances of H− near H(n = 3–8): Several series of doubly excited H− resonances were observed by Harris et al. [144] in the energy region covering the H(n = 3–8) thresholds and these pioneering data and their identi
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Fig. 22. Detachment data for the e− + D− collision process obtained at the ASTRID storage ring [195] compared with previously reported results [199], the latter connected by means of a solid line.
and positive ions. The perturbed series result from interchannel coupling and the remaining electron correlation. 4.1.2. Search for H2− Doubly charged negative atomic ions have several times been reported to exist as stable ions, but none of these observations have been able to stand the test of time. The large repulsive energy of two electrons con
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using the Tokyo storage ring. The previously reported resonance structures were most likely experimental art eEects, as
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one- and two-electron ejection from H− as a function of the energy of the collision partner (photon, electron, etc.). Single photon–H− collisions leading to one-electron ejection have played an important role in the history of H− [30]. The abundant presence of both hydrogen atoms and low-energy electrons in the ionized atmosphere of the Sun and other stars argues for the formation of H− by electron attachment. At the same time, subsequent photodetachment back to hydrogen atoms and electrons for photon energies larger than the binding energy of H− points to the importance of the opacity of these atmospheres to the passage of electromagnetic radiation [30]. Besides photodetachment, other collision processes involving H− are also important in stellar atmospheres. Prominent among these are collisions with neutral hydrogen or with protons that are abundantly present [30]. The single-photon detachment cross section of H− has previously received a considerable amount of attention for low photon energies, see [30,50,89,310] and references therein, and the main collision processes between H− and heavier particles have been reviewed by Esaulov [41]. In this and the following sections only the more recent development will be covered for photon- or heavy particle impact, whereas collisions between electrons and H− ions will be treated in Section 5 of this review. Single-photon-impact collisions: Studies of single-photon two-electron ejection from H− at photon energies above the double escape threshold at 14:35 eV has been far more limited than those of the isoelectronic neutral He atom. The threshold behaviour for double photodetachment of H− has been explored both theoretically [311] and experimentally [312], but these studies have been restricted to just the
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important, giving rise to an E −1 high-energy behaviour for the double to single-electron ejection ratio. It should be noted that the asymptotic value for the branching ratio for H− (0.015) is lower than for helium (0.0167) although one might expect otherwise because of the stronger role of electron –electron correlation for smaller nuclear charge (Z). Several explanations for this behaviour have been proposed [314,315]. Multi-photon-impact collisions: During the last decade, intense lasers have made it possible to observe eEects of multi-photon-absorption (Section 2.5) by atoms and ions, including the negative hydrogen ion [30,50]. Among notable eEects are “above-threshold ionization” wherein more photons are absorbed than necessary to break up the system, the extra energy going to increase the kinetic energy of the ejected photo electron by multiples of the photon energy [30]. Experimental studies were
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Fig. 23. Left: Two-photon detachment cross section of H− using linearly polarized light as a function of the photoelectron kinetic energy jf . Nc is the number of coupled channels within each term level of the initial, intermediate, and
cross section from the partial waves (L = 0 and 2) exhibit very diEerent behaviour as a function of the photon energy. The cross section for detachment to
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Fig. 24. Angular integrated photoelectron energy spectrum showing the results of absorption of 2–5 photons in H− ; solid curve: theoretical results, dots: experimental data. Angular distributions at energies labelled a and b, respectively, are shown in Fig. 25. The inset shows the higher energy peaks in greater detail [331].
Nikolopoulos and Lambropoulos [327]. A few years later Reichle et al. [331] used intense IR light at 2:15 m (0:577 eV) from a femtosecond laser to observe three- and four-photon detachment of H− ; hereby they could overcome the saturation problems caused by the long interaction time using pulses from Nd:YAG lasers. Previously, Nicolaides et al. [334,335] had calculated the multi-photon detachment rates of H− for IR-laser photons with energies ranging from 0.136 to 0:326 eV and intensities in the range 2:5 × 1010 –1011 W=cm2 , covering three- to seven-photon detachment rates, by means of the many-electron, many-photon theory (MEMPT), and obtained cross sections indicating that it should indeed be possible to observe several of the higher order detachment channels. Reichle et al. [331,336] studied the angular distribution of the photoelectrons produced in two-, three-, and four-photon detachment of H− . The infrared light used required absorption of at least two photons to overcome the binding energy of H− . Fig. 24 shows the photoelectron spectrum, which reveals at least three prominent excess photondetachment channels. An analysis of the angular distribution of the photoelectrons revealed an unusual dependence of the angular distribution on the electron kinetic energy. This eEect was most pronounced in the two-photon detachment channel near the threshold of photodetachment. Fig. 25 shows the angular distributions of photoelectrons in the two-photon detachment channel at two diEerent laser intensities, 1:3 × 1011 W=cm2 for the upper part (a), and 6:5 × 1011 W=cm2 for the lower part (b). The corresponding kinetic energies are marked as a and b in Fig. 24. The distribution at the lower intensity (a) can be described by a superposition of s and d waves with their relative phase taken into account. At the higher intensity the distribution has a shape with a maximum pointing perpendicular to the laser polarization. Gribakin and Kuchiev [337] have described such a behaviour in terms of quantum interference of electron trajectories. An electron can tunnel into the continuum at diEerent times within a period of the <eld oscillation, and the corresponding continuum wavefunctions will carry the phase of the <eld at which the electron is emitted. The interference eEect results from the superpositions of the wavefunctions [331]. The two-photon detachment cross section of H− has also been extended to the energy regions below the H(n = 2) and H(n = 3) thresholds, where 1 S and 1 D resonances may appear. As described in the section about H− resonances, a 1 D resonance located below the H(n = 2) threshold was observed and characterized [82,275,276] in good agreement with several theoretical predictions.
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Fig. 25. Angular distributions of photoelectrons in the lowest two-photon detachment channel at two diEerent intensities: (a) I = 1:3 × 1011 W=cm2 , (b) I = 6:5 × 1011 W=cm2 [331]. Experimental data (dots) and theoretical curves (solid curves) are normalized to each other.
Resonant and non-resonant two-photon absorption at still larger photon energies (8 eV or above) have been calculated by several groups [338,339]. Du et al. [338] calculated the cross section for 1 resonant two-colour, two-photon detachment of H− . The
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elliptically polarised laser light have been reported for the negative halogen ions and reviewed by Blondel [43], but so far no experimental data are available dealing with H− . 4.1.4. Heavy particle-impact collisions Collisions between H− and an atom, molecule or ion have been studied quite extensively for several decades [30,41,342], but to a less extent during the last decade. The importance of collisions between H− and a positive ion has been recognized for long time within astrochemistry, planetary atmospheres and astrophysical processes, and for generation of intense neutral hydrogen beams employed in the heating of magnetically con
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electron free. The Penning detachment is shown to be enhanced when several decay channels are open and the target (Li or Ca) is left preferentially in the nearest excited state. Angular distributions of neutral hydrogen atoms following electron detachment from H− ions interacting with molecular hydrogen or nitrogen at 2–6 MeV have been studied experimentally and reported rather recently [361]. The measurements were, however, performed already in 1985, but the data were classi<ed for more than 10 years. The experimental distributions are found to be in good agreement with existing Born approximation theory for scattering processes [362]. Single- and double-electron detachment in collisions of two negative atomic ions has been studied experimentally [363,364] and theoretically [364,365] for H− . Since the single-detachment cross section reaches a magnitude of 10−14 cm2 at 10 keV centre-of-mass collision energy these collisions can play an important role in electromagnetic storage rings (see Section 2.8) and be responsible for loss of negative ion beams due to intrabeam scattering, an eEect noticed at the ELISA storage ring during the running-in period. Finally measurements of H− , H0 , and H+ yields produced by foil stripping of 800 MeV H− ions have been performed passing H− ions through foils of carbon or Al2 O3 [366–368]. The time of interaction is less than a femtosecond, when H− is experiencing a pulse of the “matter <eld” [30]. Since this is the appropriate orbital period of loosely bound electrons, complicated energy transfer and chaotic processes may occur. A signi
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ion has become an attractive negative ion to study. Storage ring and ion trap experiments have yielded accurate lifetimes of the metastable ion, interaction with lasers detailed information about the photodetachment cross section below the He+ limit from which the binding energy of the ion and the energies and widths of excited 1snln l 4 L resonances with n and n reaching 15 have been obtained, whereas interaction with synchrotron radiation provides information about the photodetachment cross section and resonances located above the He+ limit. Here the only other bound state of He− , 2p3 4 S, is located, see [371] and references therein. 4.2.1. Below the He+ limit The negative-ion spectrum of helium has probably been the subject of the most extensive experimental and theoretical investigation of any element in the periodic table [42] and the knowledge about the many 2 L resonances has been accumulated over a period of four decades resulting in energies and widths with accuracies about 1 meV for the former and 0:5 meV for the latter. The 1s(2s2 ) 2 S resonance is located at 19:365(1) eV above the He ground state with a width of 11:2(5) meV according to the most recent experimental study by Gopalan et al. [372]. They improved the accuracy of the energy and obtained a width in good agreement with the one reported by Kennerly et al. [373], but larger than claimed more recently by DubRe et al. [374]. The paper by Gopalan contains 19 references to previous experimental investigations of the 1s(2s2 ) 2 S resonance and 17 to theoretical studies. Gopalan et al. [372] also reported numerical calculations based on the R-matrix method with pseudo-state approach, as described by Bartschat et al. [375] and Bartschat [376], and obtained results for the energy and width of the He− (2 S) resonance agreeing with the experimental values within their error limits. Theoretical studies have also been reported by Chrysos et al. [377] with the aim to calculate the energies and partial widths of the He− 2 S two-electron ionization ladder resonances and by Bylicki [378], who focussed on the 2 P and 2 D Feshbach resonances, but none of the latter mentioned resonances have so far been identi<ed. The metastable He− (1s2s2p 4 P) ion has received considerable attention since its discovery by Hiby in 1939 [379]. Its binding energy,
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Fig. 26. Lifetime data recorded for He− (1s2s2p 4 P5=2 ) at the ASTRID storage ring [12]. The neutral He atom signal from a tandem-channel plate detector is shown as a function of time following beam injection.
and 2 P levels of the 1s2s2p con
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Fig. 27. He− (1s2s2p 4 P5=2 ) lifetime data recorded at eight diEerent temperatures of the ELISA storage ring [18]. Open circles: measured data, full circles: data corrected for blackbody radiation. Solid line: mean weighted value of the corrected data, whereas the dotted lines indicate the uncertainty limits.
storage ring was in agreement with the older value of 345(90) s [220], but the uncertainty had been improved a factor of six, allowing detailed tests of the many theoretical results. The new lifetime clearly eliminated a series of calculational approaches considered to be among the most reliable [381,382]. The storage ring lifetime stimulated new theoretical research [383]. Three factors were considered as being responsible for the fact that the previously calculated lifetimes were too long: the autodetachment mechanism, the electron correlation eEects in the initial state, and in the
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and correction for the blackbody-induced photodetachment leads to a consistent set of lifetimes in the range 360–370 s from which the
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Fig. 28. Partial cross sections for photodetachment via the He(1s3p 3 P) + e− (jp) channel [16]. The upper and lower spectra show the regions below the He(n = 4) and He(n = 5) thresholds, respectively. The insets show the regions near the thresholds in greater detail.
input’. Most likely some of these resonances could have 4 S or 4 D symmetry, since they did not appear in the calculations of 4 P resonances by Bylicki [389]. These and the previous observations clearly called for a comprehensive theoretical study covering all the resonance features as well as the partial cross sections for photodetachment of He− (1s2s2p 4 P) in the vicinity of He(1snl) thresholds with n = 3, 4 and 5.
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Fig. 29. Partial cross sections in the vicinity of He(n = 4) thresholds. (a) (3p), (b) (3p, 4 S), (c) (3p, 4 Pe ), (d) (3p, 4 De ). The experimental data points in (a) are from Kiyan et al. [16] and are compared with theoretical results by Liu and Starace [390]. The vertical dashed lines indicate the location of the He(1s4l 3 L) thresholds.
Liu and Starace [390] performed a comprehensive R-matrix study of the photodetachment of the He− (1s2s2p 4 P) ion in the vicinity of the three thresholds mentioned above. The calculated He (1s2s 3 S) and He (1s2p 3 P) partial cross section results were in excellent agreement with the relative measurements of KlinkmWuller et al. [14,15]. Near the He (1snl = 4l and 5l) thresholds Liu and Starace predicted the presence of about 30 quartet Feshbach resonances and four quartet shape resonances which had not been observed or predicted before; the calculated He (1s3p 3 P) partial cross section was also in excellent agreement with the measurements of Kiyan et al. [16]. Of the 12 observed resonances 11 were identi<ed. The measured (3p) partial cross section is a sum of three partial cross sections (3p, 4 L) with L being S, P or D. By decomposing the (3p) cross section in this way it was possible to accurately characterize the term value of the doubly excited state responsible for the observed resonance structure (see Fig. 29). The labels (a–f) used in Figs. 28 and 29 are the same. The upper part of Fig. 27 shows the excellent agreement reached between experiment and theory. The (3p; 4 P) partial cross section gives the largest contribution to the (3p) partial cross section and the 4 P resonances are the ones which dominate the (3p) spectrum, whereas a series of very narrow 4 S resonances exists situated on top of a nearly constant background. The calculations show that both the feature b and d should be assigned to resonances with 4 S symmetry, con
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of the resonances were calculated using the complex rotation method combined with the use of B splines in a spherical cavity to describe the negative ion and its decay channels. The calculations had con
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The data from the vicinity of two-electron escape threshold [19] exhibit the same tendency as seen at the lower lying He(1s3p 3 P) state by Kiyan et al. [16]; it is 4 P double Rydberg states, which are the dominating resonance structures populated in photodetachment of the He− (1s2s2p 4 P) ion. The energies and widths of a few intra-shell He− (4 P) resonances in the n = 13–15 manifolds were reported, but so far theoretical calculations for comparison are only available for smaller n-values [395]. The experimental data also revealed the presence of several broad and some very narrow resonance structures, the latter with widths down to 0:1 meV. Some of the resonances are strongly bound and obviously belong to channels diEerent from the lowest one within a hydrogenic manifold [46]. Concurrent with the experimental and computational studies attempting to locate, identify, and characterize the He− resonances additional insights to the He− resonances have been searched for using the hyperspherical method within the adiabatic approximation [396]. In this approach, bound and resonance states are not calculated directly. Instead the hyperspherical adiabatic potential curves are
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scattering experiments from the ground state of He and they occur in a region of the spectrum (57–71 eV) where the existence of overlapping, doubly excited autoionizing He states greatly complicates their observation and unambiguous classi
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resonances, the
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Fig. 30. Photodetachment cross section of He− (1s2s2p 4 P) in the 1s photoabsorption region producing He+ . The experimental data [100] are compared with the results of two diEerent calculations: solid line [419] and dotted lines [482]. The He thresholds and He− resonances are indicated by hatched lines and arrows, respectively.
He+ ions via a two-step process. Just above the
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4.3. The noble gases The heavier noble gases, Ne, Ar, Kr, and Xe, are often treated separately to He, as the spectrum of negative-ion resonances for these atoms reMects the signi
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Fig. 31. Schematic energy diagram of Ar and Ar − with the detachment and excitation channels indicated.
evaluation of the
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con
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atoms and positive ions since the negative ion is the lowest member of the sequence and the electron– nucleus interaction is weakest relative to the electron–electron interaction. Thus studies of doubly excited states in H− , as described in Section 4.1, and of doubly excited states in negative alkali metal ions can yield new insight to fundamental aspects of atomic physics. However, doubly excited states in the Li− ion had not been observed even at the time when the Buckman–Clark review was written [42], whereas the photodetachment cross section near the Li(2p) threshold had been studied. One of the main achievements in recent negative-ion research has been the detailed studies of the photodetachment spectra of the lighter alkali metal ions, Li− , Na− and K − , and the possibility to compare the photodetachment spectra of the H− and Li− ions with the purpose to explore to what extent the existing theories for two-electron systems are applicable to pseudo-two-electron systems having
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Fig. 32. Total photodetachment cross section for Li− in the region between the Li(3s) and Li(3p) thresholds [164]. The dots represent the experimental data, the solid line the theoretical result.
Li(3s) and Li(3p) thresholds. Fig. 32 shows the experimental data compared with the calculated photodetachment cross section. The experimental data exhibit three signi
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resulting in nearly equal admixtures of 3snp and 3pns con
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Fig. 33. Partial cross section for photodetachment of Li− via the Li(3sjp) decay channel. Top section: the region below the Li(4p) threshold; bottom section: the region below the Li(5p) threshold. Experimental data points are shown together with
allowing identi
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Fig. 34. Top part: Experimental Li(3s) + e− (jp) partial photodetachment cross section for Li− below the Li(6p) threshold. The solid line represents a
It is possible to make a comparison between the series observed in the photodetachment of the Li− and H− ions and obtain information about the inMuence of the core electrons on the photodetachment processes by comparing the spectra shown in Fig. 34 [167]. The H− spectrum is dominated by the two intense window resonances marked B and D (analogous to b and d in the Li− spectrum) and are members of the (4; 1)+ series. The tiny features marked A and E (the latter is hardly visible in the spectrum) are members of the (3; 0)− series. Thus the pattern in H− is broken, since even though photoexcitation of Li− to the (K = n − 2; T = 1)+ states still dominates like in H− the excitation to the (K = n − 3; T = 0)− states is signi
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In the case of H− , the motions of the electrons in a highly correlated, doubly excited state were shown to be similar to those in a Moppy three-body rotor, consisting of a proton situated between the two excited electrons [264,447]. In this model the “+” series corresponds to the ground state bending vibrational mode ( = 0), whereas the “−” series corresponds to an excited state ( = 1) of this motion. The latter type of excitation is present in the photodetachment spectra of Li− and thus represents a violation of the propensity rules established for H− , since this type of excitation is not observed in the experimental H− data [50]. Inner-shell excitation and detachment of Li− : Inner-shell excitation of Li− ions was
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Fig. 35. Total double photodetachment cross section of Li− giving rise to Li+ in the vicinity of the 1s threshold [8]. The solid curve represents an R-matrix calculation. The experimental data are normalized to the calculation at 62 eV. Note the deviation below 58 eV, which later was accounted for by post-collision eEects.
negative ion as consisting of a loosely bound electron and a neutral core atom, which independently of each other can interact with the rare gas. The experimental test [217] compared the population of the excited Li(2p 2 P)state for keV Li–He and Li− –He collisions. In the region of maximum excitation probability, a strong preference for population of the magnetic sublevel m = −1 was observed for both types of collisions; the shape of the excited electron cloud and its alignment with respect to the projectile axis were identical for the two collision processes, yielding support to the independent scattering model. The degree of polarization, however, showed that the electron correlation of the two outer electrons in Li− should be taken into account to account for all the properties related to the electron detachment process. 4.4.2. Sodium During the last decade the Na− ion has not attracted a similar great interest as the Li− ion. The binding energy of the Na− was already well established in the 1980s [49] and in contrast to the relative large number of studies of highly excited two-electron states in photodetachment spectra of H− and later Li− the Na− photodetachment spectrum and, more speci
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Fig. 36. Na(4s) + e− (jp) photodetachment for Na− in the photon region from the Na(5s) to the Na(5p) threshold. The data points [168] are normalized to the theoretical result [390] between the Na(5s) and Na(4d) thresholds. The vertical dashed lines indicate the location of the thresholds.
[448], who reported partial as well as total cross sections covering the energy region up to the Na(5p) threshold. The cross sections were calculated by the eigenchannel R-matrix method, which previously had been used successfully to predict the H− and Li− photodetachment spectra [444–446]. The photo electron angular distribution asymmetry parameter for photodetachment of Na− leading to Na(3p) + e− formation was also calculated over the same energy range. The analyses focussed on the identi
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intensity (1010 W=cm2 ) two- and three-photon detachment rates have been calculated at low frequencies, where multi-photon absorption is the lowest-order process, and at higher frequencies where twoand three-photon absorption are studied as above threshold processes. Similar theoretical studies have previously also been performed for the Li− ion [451], but so far there are no experimental data to compare with for either of the negative ions. Photodetachment of the Na− using XUV photons with energies in the range 30–51 eV has been studied [438] at the advanced light source (ALS) at the Lawrence Berkeley National Laboratory. At these photon energies, electron detachment leads prominently to the production of Na+ ions. The structures in the measured cross section are associated with correlated processes involving detachment or excitation of a 2p electron, processes that often are accompanied by the excitation of one or more valence electrons and thereby showing that multiple excitation, as previously observed for inner-shell excitation of Li− [7], is the dominant process in inner-shell excitation of a negative atomic ion. The most prominent feature in the Na− cross section is a strong resonance at 36:292 eV associated with excitation of a 2p electron from the core and a 3s valence electron leading to the Na− 2p5 3s4s 2 Po doubly excited state. So far the inner-shell photodetachment of Na− has not been studied theoretically. 4.4.3. Potassium The K − ion has recently attracted new interest following the success of the studies of the lighter negative alkali metal ions. Previous studies of the K − ion were mainly based on electron scattering experiments [42,435], but the lack of resolution sometimes prohibited unambiguous assignment of the observed structures. The use of photon-impact has made more detailed studies possible dealing with the photodetachment spectrum from threshold to nearly 4 eV above. The binding energy of K − was improved by Andersson et al. [163] using laser photodetachment threshold spectroscopy combined with selective detection of the excited K(4p) atoms generated. The binding energy value of 0.501 459(12) eV represented an improvement in accuracy of nearly an order of magnitude compared with the previous value [31]. The spectrum of 1 Po doubly excited states in the K − ion has been the object of an experimental photodetachment study by the Gothenburg group [169]. The energy region studied covered the K(5d), K(7s) and K(5f) thresholds, which are located between 4.2 and 4:3 eV above the ground state of K − . The experimental technique used was similar to the one applied to study the higher lying Li− resonances and included measurement of the partial photodetachment cross section for the K(5s) + e− (jp) channel. Seven resonances were observed and six of these could be assigned to two series, one consisting of two states (a–b) located below the K(5d) threshold the other of four states(d–g) located below the K(5f) threshold, see Fig. 37. The
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Fig. 37. Partial cross section for photodetachment of K − via the K(5s) + e− (jp) channel [169]. The upper and lower spectra show the region below the K(5d,7s) and K(5f,5g) thresholds, respectively. The solid lines represent sums of Shore pro
where C0 is a numerical constant (1.848), n is the principal quantum number for the outer electron, the dipole polarizability of the parent atomic state, and the integer part of the parameter nmax represents the maximum number of states that can be bound to the given atomic state. It should be noted that n refers to the quantum number of the outer electron in the model polarization potential and thus one should assign n=1 to the lowest state of a given series, such as resonance a converging on the K(5d) threshold and resonance d converging on K(5f). The widths of the resonances along a given series such as d − g are decreasing by nearly an order of magnitude from one member to the next. The resonances d − f have widths of 10(2), 1.5(2) and 0:10(8) meV, respectively [169]. Kiyan [453] has developed a semi-classical model to describe the two-electron dynamics of the process of autodetachment from which a formula can be derived for the partial width of a doubly excited state as a product of an amplitude and phase factors. The phase factor reMects phase matching between the semi-classical phase of the outer electron in the initial
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state and the phase of the continuum wave of the outgoing electron. This factor is nearly constant at small energies below the threshold, where the functional behaviour of widths can be described by the amplitude factor determined by the wave function normalization coe7cient. The semi-classical model describes reasonably well the experimental K − data and is also able to account for the width of the last resonance in the series d − g. It is located close to the threshold and has a width of 0:30(3) meV, which deviates considerably from the decreasing trend mentioned above. The model predicts that in the case where the outer electron has a high angular momentum, the width can have an anomalous behaviour near threshold. The experimental K − study [169] was quickly followed of a detailed theoretical study [454], which calculated the photodetachment cross section over the energy region from the K(5s) threshold to the K(7p) threshold using R-matrix technique; thus the entire region explored experimentally was covered. The calculated spectrum is in good agreement with the experimental observations by Kiyan et al. [169] and the 1 Po resonances could all be identi<ed with K − doubly excited states and characterized by their energies and widths. The calculations show that the resonance marked a in Fig. 37 is due to two overlapping resonances, but appears as a single window feature in the experimental data, which indicates that
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explained by the existence of a resonance in the 1 P channel just below the respective excited state thresholds. Resonances above the
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Dirac-R matrix method described by Thumm and Norcross [456]. For incident electron energies up to 2:8 eV kinetic energy, the resonance structures located between the continuum limit, the neutral atom ground state, and the
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Be− (2s2p2 4 P): The observation of the metastable 1s2 2s2p2 4 P state was quickly followed by calculations of its binding energy,
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Fig. 38. Autodetachment yield of Be atoms vs. time for Be− ions stored in the ASTRID storage ring at 50 keV. The straight line represents a
exhibited the expected linear dependence, allowing an extrapolation of the decay rate to zero beam energy and a determination of the J =3=2 lifetime. From the analysis of the inMuence of the magnetic <eld it was also possible to get some information about the short lived components (J = 5=2 and 1=2); taking into account that the J = 5=2 component will be the dominantly populated of these two its lifetime could be estimated to be 0:25(15) s [20]. The use of a magnetic storage ring to measure lifetimes of metastable, negative atomic ions with lifetimes longer than 10 s was a signi
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Fig. 39. Autodetachment decay of Be− (2s2p2 4 P3=2 ) ions. The neutral Be signal is shown as a function of time after the injection of the Be− ions into the ELISA storage ring [24]. The solid line represents a single exponential decay component plus a small background.
The decay rate measurements at the ELISA storage ring were independent of the kinetic energy of the Be− ions as expected as no magnetic <elds is present, and in addition the decay rate is insensitive to blackbody radiation from the surrounding walls since the binding energy is about 291 meV. A detailed statistical analysis of the data (see Fig. 39) [24], which could be
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ε s,d
3876Å
3876Å
1s2 2p 3s 3P
Be+ λ2
λ2
λ2
1s2 2p2 3P
εp
2653Å
1s2 2p3 4S
1s2 2p2 1D
λp
λ1
λ2 t
λp
λ1
1s2 2s 2p2 4P
1s2 2s 2p 3P
5/2 3/2 1/2
1s2 2s2 1S
Be--
Be
Fig. 40. Energy level diagram of Be− and Be. The probe lasers 1 and 2 perform selective photodetachment and ionization of the Be− (2s2p2 4 PJ ) levels, while the pump laser p redistributes ions from the J = 3=2 level to the complete multiplet. The inset indicates the time ordering of the light pulses [22].
the laser pulses is indicated on the
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with the calculations available [470,477], but the experimental accuracy is too low to distinguish between them. The outer-shell photodetachment process of the Be− (1s2 2s2p2 4 P) metastable state has been studied experimentally by Bae and Peterson [466] and Pegg et al. [479], who measured the total cross section and the angular distribution at selected photon energies between 1.5 and 2:5 eV. Theoretical studies have been reported by Sinanis et al. [480] and Ramsbottom and Bell [481] for photon energies ranging from threshold to 4:6 eV. No resonance structure was observed in this region. More recently, Xi and Froese Fischer [482] and Zeng et al. [483] have performed calculations ranging from threshold to 7:5 eV, whereas the study by Sanz-Vicario and Lindroth [484] covers the region up to about 10 eV. The former reports partial and total photodetachment cross sections, angular distribution parameters, and analysis of possible resonance structures in this energy region, which covers energies below the n = 4 threshold, where n is the principal quantum number of the outermost electron of the Be target. The only experimental value available for the angular asymmetry parameter [479] is in excellent agreement with the calculation, whereas there are signi
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Possessing only 5 electrons the Be− ion attracts a good deal of interest as test object for theoretical calculations, some of which are aimed at accurate binding energies, but obtained using simpler calculations [486,241] than the large-scale con
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to search for the optical decay from the excited 3p3 4 S state and estimate the lifetime from an analysis of the spectral line width. Photodetachment of the Mg− (3s3p2 4 P) ion has recently been treated by Zeng et al. [483] who predicted a rather smooth cross section below 3:5 eV; only a 4 P resonance is predicted in this energy region and assigned to the 3s4s4p 4 P state. The possibility for testing these calculations are limited due to the short lifetime of the target. Mg− (3p3 4 S): The Mg− (4 S) state is predicted to decay mainly by optical emission [422,474, 491,492] with a radiative lifetime of 1.4 ns. According to Beck’s calculation [492] the 4 S– 4 P transition would appear at 292:1(4) nm, whereas a more detailed investigation [474], predicted 289:5 nm. Froese Fischer [474] studied variational procedures for predicting energy diEerences for many-electron systems and included calculations of the transition wavelengths for the Li− and Be− ions, respectively, in addition to the unknown wavelength for the Mg− ion. The calculations were based on the energy diEerences between the parent states in the neutral atoms combined with the expected or known electron a7nities including contributions from the valence electrons and the core –valence interaction. For the Be− ion the calculated transition wavelength agreed with the observed one within 0:05 nm, for Li− within 0:2 nm. Since the two theoretical estimates for the optical transition in the negative Mg-ion were deviating by nearly 3 nm, an experimental search had to cover a wavelength region ranging from about 286 nm to 295 nm [493]. The search [493] was based on the use of beam-foil and beam-gas spectroscopy, the latter applied to study the step-wise formation of the possible Mg− (4 S) ion from the initial Mg+ ion via charge exchange processes, considering Mg(3p2 3 P) atoms as the intermediate state. The analysis of the optical spectra showed that the cross section for production of the Mg− (4 S) ion by this mechanism is very much lower, by a factor of 30, than for formation of the homologous negative Be ion. The beam-foil spectra reveal a large number of spectral lines in the investigated wavelength region, but none of these could be assigned to a negative ion. While the possibility for observing the 4 S– 4 P transition near 292:1 nm can be excluded, the region around 289:5 nm is more congested with a rather intense spectral line at 289:6 nm, which was shown to originate from the quintet spectrum of neutral Mg [493,494]. This Mg I line may perhaps cover a much weaker spectral line from the negative Mg ion. It should be noted, that if the cross section for producing the Be− (4 S) state was reduced by a factor of 30, it would have been extremely di7cult to observe the optical transition in this ion using beam-foil technique [178]. 4.5.3. Calcium Peterson stated in 1992 [495] that “the Ca− ion has proven to be a formidable and moving target for both experimental and theoretical attempts to determine its properties”, when he summarized the knowledge about this negative ion, which at that time already had attracted interest for nearly 10 years. Two electronic states of Ca− were considered to have been observed, the stable ground state, 4s2 4p 2 P, and a metastable 4s4p2 4 P state. This was also the situation when Buckman and Clark wrote their review [42], in which a detailed description of the studies performed before that time is given. The development since then has led to a signi
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Ca− (4s2 4p 2 P): Pegg et al. [10] and Froese Fischer et al. [11] gave the study of the Ca− ion a new start in 1987 when they experimentally and theoretically, respectively, proved that the Ca− ion existed as a stable negative ion with a binding energy less than 50 meV. Both studies reported the binding energy for the 4s2 4p ion to be close to 45 meV. In the following years these studies had a signi
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ION YIELD (arb. units)
1.00
0.75
0.50
0.25
0 31700
31720
31740
31760
PHOTON ENERGY (cm-1)
Fig. 41. Ca+ yield following photodetachment of the Ca− (4s2 4p 2 P1=2; 3=2 ) ground state levels to the Ca(4s5s 3 S1 ) level, which subsequently is monitored by resonant ionization via the Ca(4s15p 3 P2 ) Rydberg level. Arrows indicate location of the thresholds [4].
of this process as a function of the principal quantum number n can also yield information about electron a7nities of weakly bound atoms and molecules [511–513], which otherwise may be di7cult to obtain. For the Ca− ion the combination of photodetachment and resonant ionization spectroscopy (RIS) (see Section 2.5.1) of the excited Ca atom [5] has, however, proved to be superior to the methods mentioned above. For Ca− , a small part of the ground state ions was photodetached to the Ca(4s5s 3 S) state, which was monitored by RIS via the 4s15p 3 P2 Rydberg level. The data obtained are shown in Fig. 41, allowing an accurate determination of the binding energies of both
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binding energies in weakly bound ions. Among these calculations the ones being in best agreement with the experimental binding energies are performed by Salomonson et al. [518], who evaluated the many-body self-energy potential beyond the second order in perturbation theory and obtained 19 meV for the binding energy of the J = 1=2 level, and by Avgoustoglou and Beck [520] who reported 22 meV for the same level as the result of an all-order relativistic many-body calculation. More recent calculations like the many-body approach used by Veseth [423] or the con
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Fig. 42. Total photodetachment cross section for Ca− . The thick solid and dashed lines represent, respectively, the length and velocity forms of the core–valence correlation result, whereas the thin solid and dashed lines represent the same forms of a valency-only calculation [527]. The calculated cross sections are compared with experimental data [143,170,495,504].
between the calculations by Yuan [527] and the experimental data available; the agreement is rather good, particularly below 2 eV. The absolute cross section values calculated at 1.39 and 1:903 eV lie within the error bars of the experiments at these energies [143,170], and the resonance at 3 eV is also reproduced with shape and width of the peak in good agreement with the experiment [495,504]. The study of the photodetachment of the Ca− ion shows very clearly, that correct binding energies, position of Cooper minima or shape resonances, and the absolute magnitude of the total cross section, all rely on a proper description of the electron correlation. Without including the core–valence correlation or core polarization, the binding energies will be overestimated and the Cooper minimum appear at an incorrect photon energy. In addition, core–core and relativistic interactions may also have a distinct inMuence on the binding energies. Some of these eEects, such as the core–valence electron correlation, are also important for a proper description of low-energy electron-Ca atom scattering [528]. Ca− (4s4p2 4 P): From the early days of negative-ion research, the Ca− (4s4p2 4 P) ion has proved to be a most elusive object. For many years it was assumed that a long lived Ca− ion could exist only if it had this con
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Fig. 43. Fano-pro
does not exist for the J = 5=2 level of the He− ion. A few years later Froese Fischer et al. [529] predicted the binding energy to be about 40 meV below the experimental value reported. Looking back, it seems likely that the Ca− (4 P) ion never has been observed before 1997, when Kristensen et al. [170] developed state-selective depletion spectroscopy and applied it to the Ca− ion. The state-selective depletion experiment [170] was based on the use of a fast negative Ca-ion beam being overlapped by three pulsed laser ns beams within a 1 m long interaction zone. Fig. 12 in this review showed a schematic energy level diagram of the Ca− ion and the laser–ion interactions described in connection with the experimental technique used, see Section 2.5.3. By recording the positive ion signal, obtained by resonant excitation of the Ca(4s4p 3 P2 or 3 P0 ) levels, as a function of the laser wavelength used to induce the transition from the 2 P ground state to one of the 4 PJ levels, the positive ion signal will display a Fano pro
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the use of strong laser <elds, which will inMuence the spectroscopic data, excess-photon absorption via an autodetaching state is a suitable technique to gain spectroscopic information about the doubly excited Ca− (4p3 4 S) state. Combining resonant ionization with non-linear resonant multicolour spectroscopy Petrunin et al. [80] obtained information about the Ca− (4 S) state (see Section 2.5.3, where this technique is described with the present experiment as an example). The binding energy was determined to be 586:86(10) meV with respect to the parent Ca(4p2 3 P) state, deviating only 3 meV from the prediction (589:6 meV) by Hart [486], who used a B-spline con
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between the calculated [498] and measured cross sections may be considered fortuitous, since the former diEers considerably in the length and velocity forms. Theoretical data [486,533] are also available for the 5p3 4 S and 5s4d5p 4 F states, respectively, but none of these states have so far been observed experimentally. 4.5.5. Barium and radium The experimental proof for the existence of a long-lived metastable or stable Ba− ion was already available at the time, when Buckman and Clark published their review in 1994 [42], but the experiments [534,535] had not been able to distinguish between the two types of possible ions. Based on the observation of a stable Ca− ion, a similar Ba− ion could be expected to exist with a 6s2 6p ground state con
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Ba+ n*~19
6s6d 3D3 5d6p 1F3 5d6p 3F4
6s6p 3P2 5d6s 1D2 5d6s 3D3 5d6s6p 4F9/2
2
2
6s2 1S0
3/2 1/2
6s 6p P
Ba-
Ba
Fig. 44. Schematic energy-level diagram of Ba and Ba− with the detachment and detection channels indicated as dashed and solid lines, respectively [4].
energies of weakly bound, heavier negative ions. However, a more recent all-order many-body calculation [520] has yielded binding energies (145 and 96 meV for J = 1=2 and 3=2, respectively,) in good agreement with the experimental data. Storage ring experiments revealed that the Ba− beams contained at least two components with storage times of 1.75(10) and 10:5(2) ms, respectively. Petrunin et al. [4] attributed the two components to the J (3=2 and 1=2)-levels belonging to the stable 6s2 6p 2 P ion, whereas the minor component, Ba− (4 F9=2 ), could not be observed. The metastable Ba− (4 F9=2 ) ion could either be so long lived that its lifetime only would be limited by the blackbody radiation-induced photodetachment similar to the 2 P ground state levels and it would then decay with a lifetime equivalent to the storages times (ms) of the stable ion, or the lifetime was so short that the ion decayed during the
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Fig. 45. Ba+ yield following the photodetachment of the Ba− (2 P1=2 ) component to the Ba(5d6s 1 D2 ) state, see Fig. 44. The solid line represents a Wigner s-wave
to perform relativistic calculations, taking account of many-electron eEects, to obtain an appropriate description of the cross section. Ba− (5d6s6p 4 F9=2 ): By irradiating Ba− beams with the fundamental IR radiation from a Nd:YAG laser Petrunin et al. [4] observed a signi
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suggests that the short-lived component contains two independent decay components with slightly diEerent lifetimes. The two components represent the blackbody induced decay of the Ba− (2 P3=2 ) level and the autodetachment rate for the odd-mass isotopes of the metastable Ba− (4 F9=2 ) level. The lower limit of the metastable lifetime is therefore a few ms and the Ba− (4 F9=2 ) ion by far the longest-living metastable negative atomic ion observed so far. Ba− (6s6p2 2 P1=2 ): The excited Ba− (6s6p2 P1=2 ) level, which is located below the Ba(6s6p 3 P0 ) level, is connected to the ground state of the Ba− ion by an electric dipole transition. Due to the autodetachment selection rules it cannot decay to the ground state of the Ba atom via Coulomb autodetachment, whereas this is possible to the Ba(5d6s) states by emitting an jd-electron. The 2 P1=2 level was studied in connection with the development of spectroscopic techniques based on the use of excess-photon absorption via an autodetaching state [171] (see Section 2.5.3) leading to determination of its binding energy and lifetime. Ra− (7s2 7p 2 P): The Ra− ion has been observed in accelerator mass spectroscopy studies [231,232]. Its binding energy was
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appeared in 1981. Among the theoretical calculations reported [540–547] three of these were aiming at accuracies better than 10 meV. Noro et al. [543] performed a large-basis-set multi-reference single and doubly excited con
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Fig. 46. Photodetachment cross section of B− at the vicinity of the B(2s) threshold. Experimental data points: open circles [556], dark squares [139]. Theory 1: R-matrix calculation [555]; theory 2 and 3: Combined many-body method results [557], with the length and velocity forms shown, respectively.
np3 subshell, n = 3 for Si− , would create dramatic deviations from the independent-electron picture in the photodetachment cross section. The experimental Si− photodetachment cross section exhibited a signi
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Fig. 46. The results for the length and velocity forms diEer, however, because the method applied is not self-consistent; the results may, however, be considered as a
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Fig. 47. Photodetachment yield vs. photon energy for Al− [112]. The data are analysed using a Wigner s-wave
below 3585 cm−1 (or below 444:4 meV). The
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transitions involving the 3s subshell. The calculation indicates that the resonance may be classi<ed as a doubly excited negative-ion state having predominantly 3s2 4s4p 3 P character. It should be noted that this resonance in contrast to many other negative-ion resonances only interacts with a very small portion of the underlying continuum; thus the resonance pro
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4.7.1. Carbon C − (2s2 2p3 4 S and 2 D): The C− ion is easy to produce and its binding energy was established already in the 1970s [48,49] with Feldmanns value of 1:2629(3) eV [571] being considered to be the most precise for the 4 S state. It was obtained from his pioneering infrared laser photodetachment threshold measurements, and only slightly corrected by Scheer et al. [111] more than 20 years later to be 1.262 119(20) eV, using essentially the same technique. The excited, bound 2 D state was also observed by Feldmann and its binding energy determined to be 33(1) meV. C− (2 D) ions can be produced in signi
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to treat open-shell systems. Gribakin et al. [552] used the spin-polarized version of the random-phase approximation with exchange (SPRPAE) and calculated the cross section for the
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Fig. 48. Photodetachment cross section of C− [52]. Experimental data points [68,572]; calculated partial cross sections for 2p → jd(1), 2p → js(2), and 2s → jp(3) photodetachment; total cross section (4) calculated with interchannel interaction relaxation taken into account [578]. Total R-matrix calculated cross section [577] in length and velocity forms are shown as (5) and (6), respectively.
The negative ions of group IV elements have been studied with emphasis on collisional detachment by the interaction with He, Ne or Ar gases at relative velocities ranging from 0.2 to 2.2 a.u. [580,581]. It was observed that it is possible to scale all the cross sections into a single curve of total cross section for each noble gas. This observation is in agreement with previous studies of collisional detachment cross sections of H− and the alkali ions, Li− , Na− , and K − ions, which showed a similar multiplicative scaling, but for which the understanding may be simpler due to the ns2 con
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infrared laser photodetachment threshold spectroscopy (LPT) [114,588] and laser photodetachment microscopy (LPM) [131] resulting in much improved accuracy, changing the uncertainty for the binding energies from 5 meV [587] to 2:4 eV [131], or by more than three orders of magnitude. The binding energy value for the 28 Si− (4 S) state is reported to be 1.389 5220(24) eV [131], which makes it among the most accurate binding energies reported so far for a negative ion [31]. It should be noted that Blondel et al. [131] using LPM were able to improve the rather accurate binding energy value reported by Scheer et al. [114] using LPT by one order of magnitude, which clearly demonstrates the potential of the photodetachment microscope. The binding energy of the Si− ground state ion has also attracted interest from several theoretical groups in recent years [242,549,568,589] using very diEerent approaches and with de Oliveira et al. [242] ab initio calculation being far the most accurate, deviating only 1 meV from the experimental result. The lifetimes for the 2 P1=2 and the 2 D3=2 levels have been calculated to be 24 and 162 s, respectively, whereas the 2 D5=2 level should have a much longer lifetime, of the order of 27 h [590]. Si− (3s3p4 ): Strong and complicated intershell interactions can manifest themselves in the photodetachment of negative atomic ions with open subshells [52]. The main features of these ions, with Si− as an example, are related to the production of diEerent terms in the
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Fig. 49. Photodetachment cross section of Si− in the vicinity of the window resonance. The experimental data are indicated by crosses [141]. The solid curves (1) and (2) represent calculated cross sections applying various modi
Fig. 49 shows a comparison between the experimental data and two calculations both performed within the random phase approximation with exchange, using frozen-core wavefunctions or by taking static rearrangement eEects into account [52,552]. The location of the experimental data in between the two theoretical curves shows that it is necessary to take other many-electron processes into account to obtain a better agreement between experiment and theory, as done more recently for the photodetachment of the homologous C− ion [577,578]. 4.7.3. Germanium and tin The binding energies of the bound s2 p3 terms, 4 S and 2 D, and the 2 D
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experimental values of up to four orders of magnitude and provided the
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approximation. It should be noted that the observation of longer-lived N− species
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Fig. 50. M1 resonance in the two-photon detachment yield after excitation of the Sb− (5p4 3 P2 ) ground state level via the Sb− (5p4 1 D2 ) excited state to above the detachment limit. The solid line represents the simulated resonance pro
spectroscopy resulting in experimental data with far better accuracy, of the order of 100 times more accurate. Wijesundera and Parpia [598] performed multi-con
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account. The solid line in Fig. 50 represents a simulated resonance pro
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[31] it was, however, pointed out that the Doppler shifted thresholds reported for parallel and anti-parallel laser and ion beams extracted from the
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Fig. 51. Electron spectrum of 5 keV O− –He collisions [205]. The two peaks originate from autodetachment of the same doubly excited state, O− (2p3 (2 D)3s2 2 D).
and the two electron resonances did represent the branching between the dismissed autodetachment channels, only p electrons would be emitted. Dahl et al. [205] performed a combined experimental and theoretical investigation of the O− system with emphasis on the decay mechanisms proposed. The energy separation between the two intense resonance peaks was determined to be 1:97(1) eV, which deviates from the 2:01 eV previously reported [605], but is consistent with the energy separation between the 1 D and 3 P states in the oxygen ground state con
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16 RELATIVE INTENSITY
10eV
14
12eV
12 10 8 6
0
30
60
90 120 ANGLE (deg.)
150
180
Fig. 52. Angular distributions of autodetaching electrons from the doubly excited 2p3 (2 D)3s2 2 D state in O− . The curves represent least-squares
have also been investigated covering the electron energies from threshold to several keV [196,583]. The low-energy electron collisions will be treated in more detail in Section 5. The higher energy electron-negative-ion collisions will lead to single and double ionization of the negative ion and autoionizing, neutral states have been shown to play an important role in these processes [583]. However, Rost and Pattard [586] have shown that direct double photoionization still dominates the measured cross section for an ion like O− . Electron detachment and charge transfer collisions between O− ions and atomic hydrogen have been studied at relative collision energies ranging from 0.1 to 15 eV [606] with the purpose to obtain experimental cross sections. Associative attachment is considered to be an important mechanism for formation of OH in the interstellar space and this process displays no energy barrier. At the low collision energies used the experimental cross sections could be accounted for by simple models such as a Langevin orbiting or a curve-crossing model [606]. 4.9.2. Sulphur, selenium, and tellurium During the last decade the laser based structural studies of the negative ions of S, Se, and Te, have focussed on the properties of the ground state con
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in a Penning ion trap, leading to a determination of the dipole and quadropole hyper
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Fig. 53. Cross section for photodetachment of a 4d electron in Te− . Upper curve and left axis: Cross section for production of Te+ ; the narrow resonance represents excitation of a 4d to a 5p electron in the energy region below the 4d threshold, whereas the broad structure represents 4d → jf excitation resulting in a giant resonance located above the threshold. Lower curve and right axis: Cross section for Te2+ production [607].
by Hall et al. [157] to the recent laser photodetachment study of F− in a strong infrared laser pulse [613]. The binding energies of the negative ions are well established with high accuracies (2–27 eV) [31,121,131,614,615], and theoretical calculations for F− and Cl− have reached a level of accuracy of 1–2 meV [242]. 4.10.1. Fluorine F − (2p6 1 S): The binding energy of the F− ion has been studied by LPT [121] and LPM [131] techniques. The values obtained were in good agreement and the recommended binding energy is now 3.401 1895(25) eV [131]. The theoretical methods applied to calculate binding energies of strongly bound negative ions, like Muorine, [242,548,549] can reproduce the binding energy within 1 meV [242]. F − (2p4 )3s2 1 D): Due to the large energy gap between the ground state level of the F− ion and the
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are nearly identical to the energy splittings between the F 2p4 (3 P) 3p(2 P and 2 D) levels. Poulsen et al. [203] resolved this problem by use of electron–photon coincidence technique, which proved that all the autodetached electrons originated from the same upper level, F− 2p4 (1 D)3s2 1 D. In addition, calculations showed [203] that the decay process emitting low-energy electrons was at least as favourable as decay to the ground state of the F atom even though a change of core was involved. This result was in good agreement with the experimental data since the branching ratio was determined to be close to unity. Excitation of the F− (1 D) resonance state is far from being the dominant excitation process in low-energy F− –He or Ne collisions. Poulsen et al. [203] reported that the cross section for populating the F− 2p4 (1 D) 3s2 1 D state was one order of magnitude lower than the cross section for collisional population of F(1 D)nl states, which again only accounted for a minor part of the F atoms produced in the collisions. The neutral F atoms are mainly formed by detachment processes leaving the projectile and target atoms in their respective ground states. Photon and electron collisions: The interaction between strong light <elds and negative ions has been a major research topic over the last decade, both from an experimental and theoretical viewpoint. In a strong laser <eld, the probability of absorbing more photons than required becomes signi
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So far these predictions have not been tested experimentally for negative ions, but asymmetries in above threshold ionization electron angular distributions from rare gas atoms have been observed applying elliptically polarized light [622]. A few years ago Hart [623] applied the R-matrix Floquet approach to examine two- and three-photon detachment cross sections for electron-correlation eEects. Using diEerent expansions the inMuence of the overlap between the F and F− wavefunctions was found to be very important for determining cross sections at the onset of the open channels. By including only a limited number of interactions it was possible to get good agreement with experiment, but the author states that a full inclusion of all possible interactions is required to guarantee a proper result. It is, however, clear that correlation eEects have a signi
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responsible for the original, but incorrect interpretations. Thus, the collisional properties of the Cl− and F− ions are very similar with respect to population of doubly excited states. If molecular targets such as H2 were used it is also possible to populate the Cl− 3p4 (3 P)4s2 resonance state [42,629], whereas the 3p4 (1 S)4s2 resonance still remains unobserved. Photon, electron and atomic collisions: Like F− , the Cl− ion has often been an object in multi-photon detachment studies both experimentally and theoretically, for reviews see [43,632] and references therein. The experimental studies have aimed at absolute cross sections and angular parameters measured at selected photon energies. The number of photons needed to reach the detachment threshold could be expected to increase with increasing wavelength, and the intensity to saturate the photodetachment process should behave correspondingly. Surprisingly, Crance [633] reported calculations indicating that multi-photon detachment saturation intensities could decrease when the number of photons required to detach increased, and Davidson et al. [634] have claimed that they have observed such an eEect in multi-photon detachment of Cl− . Blondel [43] has, however, pointed out that this interpretation of the multi-photon detachment signal as a function of intensity may not be possible, since intricate eEects such as inhomogeneity of the laser beam and the
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ring applying the merged beam technique with energetic photons (40–160 eV) from a synchrotron source. The purpose was to study the contraction of the 4f wavefunction along the isoelectronic Xe sequence by comparing 4d photoexcitation of the ions and neutral atoms ranging from I− to Ba2+ . The assumption that a gradual contraction of the 4f wavefunction occurs along the isoelectronic sequence was supported, whereas the previously claimed abrupt collapse could be dismissed. Exposing I− ions to energetic photons will cause excitation and detachment of electrons from the 4d 10 electron shell and result in the 4d → jnf resonance phenomenon. Since the photoexcitation process to a high degree of accuracy can be described as a one-electron process, the single-electron detachment process is expected to be dominant. In the present case, however, the
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(Sections 2.4.1 and 2.5.4). These studies have been performed by the McMaster group [1,31,111,113– 115,649] and cover 12 of the negative transition element ions, which all can be characterized by possessing
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Fig. 54. Left: Photodetachment spectrum of Os− [1]. Right: Energy level diagram of low lying states of Os− and Os. The states labelled a and b give rise to the two strong resonance features in the spectrum. A second photon, not shown on the right
A recent laser spectroscopy experiment revealed, however, the presence of two unexpected states in the Os− ion [1]. Whereas the 5d 7 6s2 4 F9=2 ground state level of the Os− ion has a binding energy above 1 eV (1:077 80(12) eV) [31]) and the remaining three
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from nd m−1 (n + 1)s2 con
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con
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the binding energies of Ce− [105], Pr − [104], Tm− [103], and Lu− [654] by laser photodetachment electron spectroscopy (LPES). Semi-empirical extrapolations established already in the 1970s and 1980s [49] that is should be possible to add 5d or 4f electrons to the rare earth elements leading to stable negative ions with binding energies in the range 0.2–0:8 eV. Computational studies [655–658] performed more recently indicated, however, that it is be more likely that the extra electron will be a 6p electron leading to lower electron a7nities than previously assumed. The experimental proofs for the existence of stable or long-lived metastable negative ions of the rare earth elements came from accelerator mass spectroscopy (AMS) groups [231–233], reporting that at least 11 of the 14 elements could form negative ions detectable with AMS technique. The only exceptions were holmium and erbium, whereas the radioactive promethium (Pm) had not been investigated. Nadeau et al. [231] reported lower limits for the electron a7nities of a number of rare earths elements by assuming that the probability for producing negative ions by sputtering, given by the ratio of the negative ions produced to the sputtered neutral atoms, could be expressed by a rather simple function, which included the work function of the sputtered surface, the electron a7nity of the sputtered species, and the mass of the sputtered particles. The negative ions were expected to be generated in the ground state, if not, the lower limit of the binding energy would deviate considerably from the real one. Large sputtering yield was only obtained for Ce indicating that this ion had a large binding energy or a great part of the negative ions was formed in excited, but bound states. For Tm− ; Dy− , and Yb− ions, the sputtering yields were low and the binding energies expected to be below 100 meV; more precise values could be obtained applying electric dissociation technique in combination with AMS. Subsequent studies of Tm− [103] and Yb− [240] have, however, disproved the AMS results, since the Tm− ion has a rather large binding energy, whereas the Yb− ion does not exist as a stable or long-lived negative ion. 4.12.1. Cerium and praseodymium The complex atomic structure of the lanthanides makes it di7cult to perform reliable calculations of the negative ions. The recent calculation performed for the Ce− ion by O’Malley and Beck [659] illustrates this. They performed a relativistic con
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It should be noted that the AMS study by Nadeau et al. [231] estimated the binding energy for Ce− to be larger than 0:5 eV, whereas an AMS study combined with laser excitation [661] indicated that the binding energy may be approximately 0:7 eV, but both results are somewhat away from the 0:955 eV obtained by Davis and Thompson [105], which illustrates that a reasonable high resolution is needed in this type of studies to obtain accurate results. For praseodymium an analogous situation exists. A theoretical prediction by Dinov and Beck [658] yielded an electron a7nity of 0:128 eV; Nadeau et al. [231] reported a lower limit of 0:1 eV, whereas a LPES study by Davis and Thompson [104] resulted in an electron a7nity value of 0:962(24) eV, with respect to the Pr(4f 3 6s2 4 I9=2 ) ground state level. The ground state con
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Andersen et al. [240] performed a detailed investigation of the possible existence of a stable Yb− ion using the ASTRID storage ring and a single-pass beam facility in combination with tunable laser photodetachment spectroscopy. After a careful search using diEerent experimental approaches to generate and observe a possible long-lived Yb− ion it was concluded that if a stable Yb− ion do exist, its binding energy would be lower than 3 meV, a limit given by the electric <elds present. This conclusion was supported by the later
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Fig. 55. Electron-impact detachment cross section of D− as a function of energy. The model calculation is compared with the experimental data for three diEerent values of the radius of the reaction zone [195].
usually cover the range from threshold to about 30 eV above with the single detachment cross section being nearly two orders of magnitude larger than the double detachment cross section, which can be observed at the higher energies for ions like B− [559]. Electron energies up to 95 eV have been applied to study e− –Cl− collisions [637] with the aim to study single-, double- and triple-detachment cross sections. The single detachment data, see Fig. 55, presented a challenge to theory since previous work dealing with electron detachment from negative ions, see references in [671], appeared to be of limited applicability in the near-threshold region. The electron-impact detachment of weakly bound negative ions exempli<es one of the more important processes in physics: the breaking of a target by a projectile giving at least three distinct bodies in the
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well known from nuclear and molecular calculations and this classical “reaction model” reproduces the general behaviour of the experimental data rather well. The classical calculations were quickly followed by a number of semi-classical techniques [671–674] in which the projectile electron was treated as a classical particle, whereas the electron that is attached to the atom was treated quantum mechanically. A theoretical description of the detachment process may be based on a rather simple <eld-ionization concepts. It is, however, important to account for quantum tunnelling eEects at large separations as well as for saturation eEects that appear when the incident electron comes close to the negative ion. Quantum tunnelling and classical over-barrier models were applied by Ostrovsky and Taulbjerg [671] to describe the detachment process and similar simple models based on a non-stationary wavepacket approach were investigated by Kazinsky and Taulbjerg [672]. These semi-classical models did not aim at a complete description of the detachment process and their shortcomings are also noticeable [193]. It is not possible to include electron exchange in these methods nor is it possible to make successive improvements in the main approximation of a classical incident electron. Robicheaux [193] has also pointed out that it is di7cult correctly to describe the
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Fig. 56. Total electron impact detachment cross section of H− =D− and B− vs. energy of the incident electron. The experimental data points [195,559] are compared with theoretical curves for T -matrix calculations [193] for diEerent limits of partial waves.
time-dependent close-coupling theory. The total cross sections obtained are in very good agreement with the experimental storage ring data [195], while the diEerential cross sections con
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6. Negative ions in external electric and magnetic elds Negative-ion states can be inMuenced by external electric and magnetic <elds leading to the well-known Stark and Zeeman eEects, which cause splitting of degenerate
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Fig. 57. Photodetachment of H− just above the detachment threshold at 0:75 eV (dashed line) and in the presence of a strong electric <eld F (solid line). Note that the detachment sets in below 0:75 eV in the presence of the electric <eld and that <eld-induced modulations appear at higher photon energies. The arrow marks the data point that was normalized to the theoretical result of Rau and Wong [30,684].
Du [687] and Kondratovich and Ostrovsky [694] have pointed out, that the ripple structure can be related to the interference pattern observed in photodetachment microscopy (see Fig. 4). When the photon energy is increased, the microscope picture will either exhibit a bright spot inMated in the middle of the pattern or a dark, destructive interference spot; the former can be related to the maximum slope in the ripple structure, the latter to the zero slope. More recently, Kramer et al. [695,696] have given an explicit derivation of the total photodetachment current in the framework of a Green function calculation; they commented the staircase look of the total current as the remaining imprint of a remarkable interference pattern distribution. The same authors have also shown [696] how similar the expression of the total photoelectron current is to the total current density at the very centre of the interference pattern, which illustrates the close relationship between both quantities (same Airy function, same argument). The oscillations mentioned above can be manipulated by adding a static magnetic <eld [681,687, 697,698]; alternatively, at <xed excitation frequency these quantum interference eEects may be controlled by changing the duration of the laser pulse or by using two or more short laser pulses [697–699]. With the advent of intense lasers, there has been an interest in multi-photon detachment and non-perturbative phenomena due to the dynamic electric <eld of the detaching laser, in particular since Gao and Starace [700] reinvestigated the problem through an exact solution for the outgoing electron in combined static and dynamic electric <elds. When applied to the H− ion, they claimed that a cross term between the two <elds leads to a somewhat diEerent result from previous studies even in the weak-laser <eld limit, the cross section near the zero-static <eld detachment threshold being lowered [693]. What exactly is going on very near threshold created a strong disagreement between theoretical groups [66]. Contrary to Gao and Starace [700], subsequent calculations by Rangan and Rau [693] and Zhao et al. [701] did not support the claims of lowered cross sections arising from a cross term between static and dynamic <elds, but the
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Bryant [66] has recently performed a comparison between the original experimental data, obtained at 64, 80, and 143 KV=cm, and four of the theoretical calculations [686,693,700,704] covering the critical region of high <elds near threshold. The comparison shows that the recent calculation by Manakov et al. [704], in which rescattering is added to the original contributions given by Gao and Starace [700], is in best agreement with the experimental H− data. A few years before Bao et al. [705] had pointed out, that rescattering eEects, originally neglected by Gao and Starace [700], are small for one-photon detachment, but important in describing two-photon detachment. However, Bryant points out [66] that it seems unlikely that any of the presently available theories, which recently were reviewed by Manakov et al. [704], are right since a realistic model of the residual atom has been omitted. In all the theoretical work, the hydrogen potential is taken in the zero-range approximation, where the potential is a delta function. New experiments with better experimental resolution accompanied by more realistic theoretical calculations, including work at much larger <elds, are recommended by Bryant [66], who adds that electron correlations must be taken into account. Experimental studies of photodetachment in static electric <elds were also performed with the negative ions of Cl and S [609,706], as representative for s-wave photodetachment, and with Au− [707] (p-wave detachment). Detachment below threshold and oscillations on the cross sections above threshold were observed near the S− and Cl− thresholds. The phase of the oscillations in the S− and Cl− data was in good agreement with predictions for s-wave photodetachment in a static electric <eld, whereas the amplitude of the oscillations was observed to be slightly reduced, but not for Au− . This diEerence between s- and p-wave detachment was attributed to rescattering, since this eEect would be most pronounced for s-wave detachment. Near-threshold photodetachment of the S− ion was recently studied both theoretically and experimentally [708] for combined parallel magnetic and electric <elds. The theory predicts both a loss of contrast in the detachment rate and a shift of the photocurrent maxima towards higher laser frequencies as a consequence of adding an electric <eld parallel to a magnetic <eld. The experimental results are supporting the theoretical predictions. Experiments investigating the eEects of static electric
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In the photon energy region near the H(n = 4) threshold the experimental data indicate that mixing of 1 Po states with 1 Se and 1 De states becomes possible at increasing <eld strengths. 6.2. Magnetic 2elds Magnetic <elds can also inMuence the lifetimes of negative ions, as seen in connection with the determination of the lifetimes of the metastable He− (1s2s2p 4 PJ ) levels [12]; the magnetic <elds in the ASTRID storage ring caused Zeeman-mixing of magnetic sublevel populations originating from diEerent
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1 T. Since the number of bound, magnetically induced ionic states depends on the product of mass and polarizability of the neutral system and the applied magnetic <eld the authors [721] hope that negative ions, which do not exist in free-<eld space, like Xe− , may be detected in the presence of strong magnetic <elds. Acknowledgements Aarhus Centre for Atomic Physics (ACAP), which has been funded by the Danish National Research Foundation in the period 1994–2003, has supported the negative ion research carried out at University of Aarhus including the preparation of this review. The author thanks C. Blondel for critical reading of the manuscript and suggested modi
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Physics Reports 394 (2004) 315 – 356 www.elsevier.com/locate/physrep
Neutrino masses and oscillations: triumphs and challenges R.D. McKeown∗ , P. Vogel W.K. Kellogg Radiation Laboratory, California Institute of Technology, Pasadena, CA 91125, USA Accepted 5 January 2004 editor: G.E. Brown
Abstract The recent progress in establishing the existence of 3nite neutrino masses and mixing between generations of neutrinos has been remarkable, if not astounding. The combined results from studies of atmospheric neutrinos, solar neutrinos, and reactor antineutrinos paint an intriguing picture for theorists and provide clear motivation for future experimental studies. In this review, we summarize the status of experimental and theoretical work in this 3eld and explore the future opportunities that emerge in light of recent discoveries. c 2004 Published by Elsevier B.V. PACS: 12.15.Pf; 14.60.Pq; 14.60.Lm; 23.40.Bw Keywords: Neutrino mass; Neutrino mixing; Neutrino oscillations
Contents 1. Introduction and historical perspective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Theoretical framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1. Neutrinos in the standard electroweak model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2. Neutrino mass terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3. Neutrino oscillation in vacuum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4. Neutrino oscillations in matter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5. Tests of CP, T and CPT invariance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6. Violation of the total lepton number conservation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7. Direct measurement of neutrino mass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3. Experimental results and interpretation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1. Neutrino oscillation results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.1. Atmospheric neutrinos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.2. Solar neutrinos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ∗
Corresponding author. Tel.: +1-626-395-4316; fax: +1-626-564-8708. E-mail address:
[email protected] (R.D. McKeown).
c 2004 Published by Elsevier B.V. 0370-1573/$ - see front matter doi:10.1016/j.physrep.2004.01.003
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3.1.3. SNO and KamLAND . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.4. LSND . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2. Direct mass measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3. Double beta decay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4. Cosmological constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4. Near-term future . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1. MiniBooNE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2. Determination of Dm232 and 23 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3. Studies of 13 , neutrino mass hierarchy, and CP violation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.1. High-precision reactor neutrino experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.2. Long baseline accelerator experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4. Future direct mass measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5. Plans for double beta decay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6. Cosmological input . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5. Longer-term outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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1. Introduction and historical perspective For almost 70 years, neutrinos have played a pivotal role in the quest to understand elementary particles and their interactions. The neutrino was actually the 3rst particle proposed by a theorist to guarantee the validity of a symmetry principle: Pauli boldly suggested [1] that neutrinos (invisible then and remaining so for another twenty 3ve years) are emitted together with electrons in nuclear beta decay to salvage both energy and angular momentum conservation in the beta decay process. Experimental con3rmation of Pauli’s hypothesis required many decades of experimental eJort, but the discovery of the antineutrino in the 3fties by Reines and Cowan [2] and subsequent experiments in the early sixties ultimately led to the Nobel Prizes for Reines in 1995 and Lederman et al. in 1988. More recently, the 2002 Nobel prize was awarded to Davis and Koshiba for their seminal roles in the development of neutrino astrophysics (along with Giaconni for X-ray astrophysics). The standard model of electroweak interactions, developed in the late 1960s, provided a theoretical framework to incorporate the neutrinos as left-handed partners to the charged leptons, organized in two generations along with the quarks. The subsequent discovery of charmed quarks and the third generation of quarks and leptons completed the modern view of the standard model of electroweak interactions. This version possessed additional richness to incorporate CP violation, and further eJorts to unify the strong interaction led to the development of grand uni3ed theories (GUTs). These GUTs provided a natural framework for nucleon decay and for neutrino masses, and motivated many experiments in the 3eld. Following on the successes of big-bang nucleosynthesis (BBN) and the discovery of the cosmic microwave background, it also became clear that neutrinos were potentially major players in the history of the early universe. The increasing evidence for the existence (and then dominance) of dark matter in the universe then led to the economical and seductive hypothesis that neutrinos, with small but 3nite mass, could provide the mass to explain the dark matter. This set the stage for a major experimental assault on the issue of neutrino mass and its role in cosmology, and provided
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substantial impetus to a worldwide program of experiments addressing the issues of 3nite neutrino mass and the possibility of mixing between generations. Although it now appears that neutrinos are not likely the source of dark matter in the universe, the experimental evidence obtained in the last decade for 3nite neutrino masses and mixing between generations is strong and irrefutable. The pattern of masses and mixing angles emerging from the experiments provides an intriguing glimpse into the fundamental source of particle mass and the role of Mavor in the scheme of particles and their interactions. The scale of neutrino mass diJerences motivates new experimental searches for double beta decay and end-point anomalies in beta decay, as well as new studies of oscillation phenomena using accelerators, nuclear reactors, and astrophysical sources of neutrinos. In this review, we attempt to synthesize these themes, present a concise and coherent view of the experimental and theoretical developments leading to the current picture, and motivate the future explorations necessary to resolve the remaining issues. 1 We begin with a summary of the theoretical motivation for studying neutrino masses and mixing, along with a development of the phenomenological framework for interpreting the experiments. This includes neutrino oscillations in vacuum as well as in matter, CP violation, ordinary beta decay, and double beta decay. We then review the recent experimental data that contribute to our present knowledge of these neutrino properties. We discuss the successful neutrino oscillation measurements, including the key contributions from various solar neutrino experiments, cosmic ray-induced atmospheric neutrino studies, and the recent dramatic results from the Sudbury Neutrino Observatory and the KamLAND reactor neutrino experiment. Important additional information is obtained from the experimental attempts that have thus far only yielded limits on neutrino masses such as double beta decay and tritium beta decay. We also will brieMy discuss the role of massive neutrinos in cosmology, and the corresponding constraints. The discussion of the future experimental program is separated into two diJerent time scales. In the near-term future, the planned experiments will focus on a better determination of the mixing matrix parameters, in particular 13 , and resolution of the LSND puzzle (right or wrong). Further studies in the longer-term will of course depend on the outcome of these measurements, but there is substantial interest in pursuing the search for CP violation, the possibility that neutrinos are Majorana particles and other issues related to the mechanism(s) responsible for the observed phenomena. 2. Theoretical framework It is natural to expect that neutrinos have nonvanishing masses, since they are not required to be massless by gauge invariance or other symmetry principles. Moreover, all other known fermions, quarks and charged leptons, are massive. Nevertheless, neutrinos are very light, much lighter than the other fermions, and this striking qualitative feature needs to be understood, even though we do not know why e.g. the electron mass is what it is known to be, and why muons and taus are heavier than electrons. The most popular explanation of the small neutrino masses is the “see-saw mechanism” [3] in which the neutrino masses are inversely proportional to some large mass scale MH . Simply put, if 1
The reference list is, for reasons of brevity, far from complete. We apologize to those whose work is not quoted, and refer to numerous reviews and monographs to provide a more complete reference list.
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neutrinos are Majorana particles, i.e., indistiguishable from their own antiparticles, they have only two states, corresponding to the two possible spin orientations. On the other hand, Dirac particles, distinct from their antiparticles, have four states, two spin orientations for the particle and antiparticle. In the “see-saw mechanism” there are still four states for each neutrino family, but they are widely split by the “Majorana mass term” (explained later) into the two-component light neutrinos with masses ML and the very heavy (sterile) two-component neutrinos with the mass MH , in such a way that ML MH ∼ = MD2 . Further, if we assume that the “Dirac mass” MD is of the same order of magnitude as the Dirac fermion masses (masses of quarks and charged leptons), we can understand why ML is so small, provided MH is very large. If that explanation of the small neutrino mass is true, then the experimentally observed neutrinos are Majorana particles, and hence the total lepton number is not conserved. Observation of the violation of the total lepton number conservation (we explain further why it is diRcult to observe it) would be a signal that neutrinos are indeed Majorana particles. Neutrinos interact with other particles only by weak interactions (at least as far as we know); they do not have electromagnetic or color charges. That further distinguishes them from the other fermions which also interact electromagnetically (charged leptons and quarks) and strongly (quarks). It is well known that the weak charged currents of quarks do not couple to a de3nite Mavor state (or mass state), but to linear combinations of quark states. This phenomenon is described by the Cabibbo-Kobayashi-Maskawa (CKM) matrix, which by convention describes linear combinations of the charge −e=3 quark mass eigenstates d; s; b. The mixed states d ; s ; b , form weak doublets with the u; c; t quark mass eigenstates. Consequently, the three generations of quark doublets are not independently preserved in charged-current weak processes. An analogous situation is encountered with neutrinos. There, however, one can directly observe only the ‘weak eigenstates’, i.e., neutrinos forming weak charged currents with electrons, muons or tau. 2 Information on the neutrino mixing matrix is best obtained by the study of neutrino oscillation, a quantum mechanical interference eJect resulting from the mixing. As a consequence, like in quarks, the Mavor (or family) is not conserved and, for example, a beam of electron neutrinos could, after travelling a certain distance, acquire a component of muon or tau Mavor neutrinos. We describe the physics and formalism of the oscillations next. The formalism has been covered in great detail in numerous books and reviews. Thus, we restrict ourselves only to the most essential points. The interested reader can 3nd more details in the monographs [4–10] and recent reviews [11–18]. 2.1. Neutrinos in the standard electroweak model In the standard model individual lepton charges (Le = 1 for e− and e and Le = −1 for e+ , and Te and analogously for L and L ) are conserved. Thus, processes such as + → e+ + , or KL → e± + ∓ are forbidden. Indeed, such processes have not been observed so far, and small upper limits for their branching ratios have been established. 2 The analogy is not perfect. For quarks, the ‘mass eigenstates’ are also ‘Mavor eigenstates’, with labels d; s; b. Since the CKM matrix is nearly diagonal, the labels d ; s ; b can be used for the ‘weak eigenstates’. In neutrinos, on the other hand, only the ‘weak eigenstates’ have special names, e ; ; while the mass eigenstates are labelled 1 ; 2 ; 3 and cannot be associated with any particular lepton Mavor.
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Based on these empirical facts, the standard model places the left-handed components of the charged lepton and neutrino 3elds into the doublets of the group SU (2)L , ‘L ; ‘ = e; ; ; (1) ‘L = ‘L while the right-handed components of the charged lepton 3elds are singlets. The right-handed components of the neutrino 3elds are absent in the standard electroweak model by de3nition. As a consequence of this assignment, neutrinos are deemed to be massless, and the individual lepton numbers, as well as the total one, are strictly conserved. Note that with these assignments neutrino masses do not arise even from loop corrections. Thus, observation of neutrino oscillations, leading to the neutrino Mavor nonconservation, signals deviations from this simple picture, and to the ‘physics beyond the standard model’. Note also that studies of e+ e− annihilation at the Z-resonance peak have determined the invisible width of the Z boson, caused by its decay into unobservable channels. Interpreting this width as a measure of the number of neutrino active Mavors, one obtains N = 2:984 ± 0:008 from the four LEP experiments [20]. We can, therefore, quite con3dently conclude that there are just three active neutrinos with masses of less than MZ =2. (The relation of this 3nding to the fact that there are also three Mavors of quarks is suggestive, but so far not really understood.) Besides these three active neutrino Mavors there could be other neutrinos which do not participate in weak interactions. Such neutrinos are called ‘sterile’. In general, the active and sterile neutrinos can mix, and thus the sterile neutrino can interact, albeit with a reduced strength. Big bang nucleosynthesis (BBN) is sensitive to the number of neutrino Mavors which are ultrarelativistic at the ‘freezeout’ when the n ↔ p reactions are no longer in equilibrium and therefore it is perhaps sensitive to (almost) sterile neutrinos. However, analysis of BBN (i.e., of the abundances of 4 He, d, 3 He, and 7 Li) are consistent with 3 neutrinos, and disfavors N = 4 for fully thermalized neutrinos [19]. 2.2. Neutrino mass terms In a 3eld theory of neutrinos the mass is determined by the mass term in the Lagrangian. Since the right-handed neutrinos are absent in the standard electroweak model, one can either generalize the model and de3ne the mass term by using the ideas of the see-saw mechanism, or one can add more possibilities by adding to the three known neutrino 3elds ‘L new 3elds, corresponding to possibly heavy right-handed neutrinos jR . The mass term is then constructed out of the 3elds ‘L , their charge-conjugated and thus right-handed 3elds (‘L )c and from the 3elds jR ; (jR )c . There are, in general, two types of mass terms, − LM = Mi‘D TiR ‘L +
1 M M (TiR )(jR )c + H:c: 2 ij
(2)
(The term with c ‘ L ‘L is left out assuming that the corresponding coeRcients are vanishing or negligibly small.) The 3rst term is a Dirac mass term, analogous to the mass term of charged leptons. It conserves the total lepton number, but might violate the individual lepton Mavor numbers.
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The second term is a Majorana mass term which breaks the total lepton number conservation by two units. It is allowed only if the neutrinos have no additive conserved charges of any kind. If there are 3 left handed neutrinos ‘L , and n additional sterile neutrinos jR it is convenient to de3ne ‘L 0 MD 1 c = ; −LM = M + H:c:; M = : (3) 2 (jR )c (MD )T MM The matrix M is a symmetric complex matrix and MT is the transposed matrix. After diagonalization it has 3 + n mass eigenstates k that represent Majorana neutrinos (ck = k ). From the point of view of phenomenology, there are several cases to be discussed. The seesaw mechanism corresponds to the case when the scale of MM is very large. There are three light active Majorana neutrinos in that case. On the other hand, if the scale of MM is not too high when compared to the electroweak symmetry breaking scale, there could be more than three light Majorana neutrinos, mixtures of active and sterile. Finally, when MM = 0 there are six massive Majorana neutrinos that merge to form three massive Dirac neutrinos. The unitary matrix diagonalizing the mass term is a 3 × 3 matrix in that case. In these latter cases the ‘natural’ explanation of the lightness of neutrinos is missing. We shall not speculate further on the pattern of the mass matrix, even though a vast literature exists on that subject (see e.g. [13] for a partial list of recent references). In the interesting case of N light left-handed Majorana neutrinos, the mass matrix is determined by N neutrino masses, N (N − 1)=2 mixing angles, (N − 1)(N − 2)=2 CP violating phases common to Dirac and Majorana neutrinos, and (N − 1) Majorana phases that aJect only processes which violate the total lepton number (for the discussion of these Majorana phases see e.g. [21]). 2.3. Neutrino oscillation in vacuum As stated earlier, the neutrinos participating in the charged current weak interactions (the usual way neutrinos are observed and the only way their Mavor can be discerned) are characterized by the Mavor (e; ; ). But the neutrinos of de3nite Mavor are not necessarily states of a de3nite mass. Instead, they are generally coherent superpositions of such states, |‘ = U‘i |i : (4) i
When the standard model is extended to include neutrino mass, the mixing matrix U is unitary. In vacuum, the mass eigenstates propagate as plane waves. Leaving out the common phase, a beam of ultrarelativistic neutrinos |i with energy E at the distance L acquires a phase m2i L (5) |i (L) ∼ |i (L = 0) exp −i 2 E Given that, the amplitude of the process ‘ → ‘ is 2 U‘i e−i(mi L=2E) U‘∗ i ; A(‘ → ‘ ) = i
(6)
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and the probability of the Mavor change for ‘ = ‘ is the square of this amplitude. It is obvious that due to the unitarity of U there is no Mavor change if all masses vanish or are exactly degenerate. The existence of the oscillations is a simple consequence of the coherence in Eq. (4). The detailed description of the quantum mechanics of the oscillations in terms of wave packets is subtle and a bit involved (see, e.g. the reviews [22] and references therein). It includes, among other things, the concept of coherence length, a distance after which there is no longer coherence in Eq. (4). For the purpose of this review such issues are irrelevant, and will not be discussed further. The idea of oscillations was discussed early on by Pontecorvo [23,24] and by Maki et al. [25]. Hence, the mixing matrix U is often associated with these names and the notation UMNS or UPMNS is used. The formula for the probability is particularly simple when only two neutrino Mavors, ‘ and ‘ , mix appreciably, since only one mixing angle is then relevant, L(km) ; (7) P(‘ → ‘ =‘ ) = sin2 2 sin2 1:27Dm2 (eV2 ) E (GeV) where the appropriate factors of ˝ and c were included (the same is obtained when the length is in meters and energy in MeV). Here Dm2 ≡ |m22 − m21 | (note that the sign is irrelevant) is the mass squared diJerence. Thus, the oscillations in this simple case are characterized by the oscillation length Losc (km) =
2:48E (GeV) ; Dm2 (eV2 )
(8)
and by the amplitude sin2 2. For obvious reasons the oscillation studies are optimally performed at distances L ∼ Losc from the neutrino source. At shorter distances the oscillation amplitude is reduced and at larger distances the neutrino Mux is reduced making the experiment more diRcult. Note that at distance LLosc the oscillation pattern is smeared out and the oscillation probability (7) approaches (sin2 2)=2 and becomes independent of Dm2 . The mixing matrix of 3 neutrinos is parametrized by three angles, conventionally denoted as 12 ; 13 ; 23 , one CP violating phase % and two Majorana phases &1 ; &2 . Using c for the cosine and s for the sine, we write U as i& =2 c12 c13 s12 c13 s13 e−i% e e 1 1 i% = −s12 c23 − c12 s23 s13 ei% ei&2 =2 2 : (9) c c − s s s e s c 12 23 12 23 13 23 13 i% i% 3 s12 s23 − c12 c23 s13 e −c12 s23 − s12 c23 s13 e c23 c13 By convention the mixing angle 12 is associated with the solar neutrino oscillations, hence the masses m1 and m2 are separated by the smaller interval Dm2sol (we shall assume, again by convention, that m2 ¿ m1 ) while m3 is separated from the 1,2 pair by the larger interval Dm2atm , and can be either lighter or heavier than m1 and m2 . The situation where m3 ¿ m2 is called ‘normal hierarchy’, while the ‘inverse hierarchy’ has m3 ¡ m1 . Not everybody follows these conventions, so caution should be used when comparing the various results appearing in the literature.
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The general formula for the probability that the “transition” ‘ → ‘ happens at L is 2 2 P(‘ → ‘ ) = U‘i U‘∗ i e−i(mi =2E)L i
=
i
|U‘i U‘∗ i |2 + R
i
U‘i U‘∗ i U‘j∗ U‘ j ei
|m2i −m2j |L 2E
:
(10)
j =i
Clearly, the probability (10) is independent of the Majorana phases &. The oscillations described by Eq. (10) violate the individual Mavor lepton numbers, but conserve the total lepton number. The oscillation pattern is identical for Dirac or Majorana neutrinos. The general formula can be simpli3ed in several cases of practical importance. For 3 neutrino Mavors, using the empirical fact that Dm2atm Dm2sol and considering distances comparable to the atmospheric neutrino oscillation length, only three parameters are relevant in the zeroth order, the angles 23 and 13 and )atm ≡ Dm2atm L=4E . However, corrections of the 3rst order in )sol ≡ Dm2sol L=4E should be also considered and are included below (some of the terms with )sol are further reduced by the presence of the emipirically small sin2 213 ): P( → ) cos4 13 sin2 223 sin2 )atm − )sol cos2 13 sin2 223 (cos2 12 − sin2 13 sin2 12 ) sin 2)atm − )sol cos % cos 13 sin 212 sin 213 sin 223 cos 223 sin 2)atm =2 + )sol sin % cos 13 sin 212 sin 213 sin 223 sin2 )atm ;
(11)
P( → e ) sin2 213 sin2 23 sin2 )atm − )sol sin2 23 sin2 12 sin2 213 sin 2)atm + )sol cos % cos 13 sin 213 sin 223 sin 212 sin 2)atm =2 − )sol sin % cos 13 sin 212 sin 213 sin 223 sin2 )atm ; P( → ) = 1 − P( → e ) − P( → ) ;
(12) (13)
where % is the CP phase of Eq. (9) and P(e → x ) sin2 213 [sin2 )atm − )sol sin2 12 sin 2)atm ] + )2sol cos4 13 sin2 212 ;
(14)
where the term linear in )sol is suppressed by the factor sin2 213 and therefore the quadratic term is also included. The e disappearance probability is independent of the CP phase %. Note that some terms of 3rst order in )sol depend on the sign of )atm , i.e., on the hierarchy.
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On the other hand, at distances much larger than the atmospheric neutrino oscillation length, the electron neutrino and antineutrino disappearance is governed by P(e → x ) 1 − sin4 13 − cos4 13 [1 − sin2 212 sin2 )sol ] ;
(15)
In both of the latter cases x denotes any neutrino except e . (General expressions for mixing of three neutrinos in vacuum can be found in Eqs. (7)–(12) of Ref. [13].) In the mixing matrix (9) the presence of the phase % signi3es the possibility of CP violation, the expectation that P(‘ → ‘ ) = P(T‘ → T‘ ) ;
(16)
i.e., that for example the probability of oscillating into e is diJerent from the probability of T oscillating into Te . The magnitude of the T or CP violation is characterized by the diJerences P(T → Te ) − P( → e ) = − [P(T → T ) − P( → )] = P(e → ) − P(Te → T ) 2 = − 4c13 s13 c23 s23 c12 s12 sin %[sin 2)12 + sin 2)23 + sin 2)31 ] 2 =16c13 s13 c23 s23 c12 s12 sin )12 sin )23 sin )31 ;
(17)
where, as before, )ij = (m2i − m2j ) × L=4E. Thus, the size of the eJect is the same in all three channels, and CP violation is observable only if all three masses are diJerent (i.e., nondegenerate), and all three angles are nonvanishing. The possibility of CP violation in the lepton sector was 3rst discussed in [26,27]. 2.4. Neutrino oscillations in matter The oscillation phenomenon has its origin in the phase diJerence between the coherent components of the neutrino Mavor eigenstates described by Eq. (5). When neutrinos propagate in matter additional contributions to the phase appear, besides the one caused by the nonvanishing mass of the state i . To see the origin of such phase, consider the eJective Hamiltonian of neutrinos in presence of matter. Obviously, only phase diJerences are of importance. Without invoking any nonstandard interactions two eJects are present. All active neutrinos interact with quarks and electrons by the neutral current weak interactions (Z exchange), but only electron neutrinos and antineutrinos interact with electrons by the exchange of W . The corresponding eJective potential is 3 (VC stands for the charged current) √ √ VC (e ) = 2GF Ne ; and VC (Te ) = − 2GF Ne ; (18)
3
The √ eJective potential was introduced 3rst by Wolfenstein [28], and used also in Refs. [29,30] who corrected the missing 2 in the original paper. Finally, the correct sign was obtained in Ref. [31].
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where Ne is the electron number density. (There are no or leptons in normal matter, hence there is no analogous potential for or .) In practical units VC = 7:6Ye
1 (eV); 1014 (g=cm3 )
Ye =
Ne : Np + N n
(19)
Similarly, any√active neutrino acquires an eJective potential due to the neutral-current interaction VN = −GF Nn = 2. The corresponding eJective potential is absent for sterile neutrinos. Given the eJective potential, Eq. (18), electron neutrinos travelling distance L in matter of constant density Ne acquire an additional phase e (L) = e (0)e−i
√
2GF Ne L
:
(20)
The corresponding matter oscillation length [28] is therefore L0 = √
22 1:7 × 107 (m) : 1(g=cm3 )Ye 2GF Ne
(21)
Unlike the vacuum oscillation length, Eq. (8), the matter oscillation length L0 is independent of the neutrino energy. Note that the matter oscillation length in rock is L0 ≈ 104 km, and in the center of the Sun L0 ≈ 200 km. Considering for simplicity just two mass eigenstates 1 and 2 that are components of the Mavor eigenstates e and & with the mixing angle , we obtain the time (or space) development SchrUodinger equation 2 m1 2 VC sc 1 2E + VC c d 1 = ; (22) i dt 2 m22 2 2 VC sc + VC s 2E where, as before c = cos and s = sin . The 2 × 2 matrix above can be brought to the diagonal form by the transformation 1m = e cos m − & sin m 2m = e sin m + & cos m ;
(23)
where the new mixing angle in matter, m , depends on the vacuum mixing angle and on the vacuum and matter oscillation lengths Losc and L0 , −1 Losc : (24) tan 2m = tan 2 1 − L0 cos 2 The eJective oscillation length in matter is then −1=2 Losc 2 2Losc sin 2m = Losc 1 + − cos 2 ; Lm = Losc sin 2 L0 L0
(25)
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and the probability of detecting e at a distance L from the e source has the usual form, but with → m and Losc → Lm , 2L : (26) P(E ; L; ; Dm2 ) = 1 − sin2 2m sin2 Lm When considering oscillations with two Mavors, the mixing angle can be restricted to the interval (0; 2=2). In vacuum, only sin2 2 is relevant, and hence only half of that interval, (0; 2=4) could be used. However, once matter oscillations are present, also the cos 2 becomes relevant, and thus the whole (0; 2=2) interval might be needed. The part of the parameter space corresponding to the (2=4; 2=2) angles was called the “dark” side in Ref. [32], who suggested using tan2 in the plots instead of sin2 2. (The advantage of tan2 is that when plotted on the log scale, the reMection symmetry around tan2 = 1 for vacuum oscillations is maintained.) The importance of allowing the full range of the mixing angles in the three Mavor analysis was stressed earlier, see e.g. Ref. [33]. With our convention m2 ¿ m1 mixing angles in the interval (2=4; 2=2) would mean that e contains dominantly the heavier component 2 . (This is not the case in practice, see Section 3.1.) It appears now that at least two of the three mixing angle are large (and 6 2=4), and the constraint on the third one, 13 are still such that linear plots are more revealing. Therefore, in the following we use the traditional plots with sin2 2. The same results can be obtained, naturally, by rewriting the equation of motion in the Mavor basis sin 2 cos 2 1 − + e e Losc L0 2Losc d = 22 : (27) i d x & sin 2 & 0 2Losc Here one can see clearly that in matter, unlike in vacuum, the oscillation pattern depends on whether the mixing angle is smaller or larger than 2=4. (For antineutrinos the sign in front of 1=L0 is reversed.) In matter of a constant density we can now consider several special cases: • Low density limit, Losc L0 . In this case matter has a rather small eJect. On Earth, one is able to observe oscillations only provided Losc ¡ Earth diameter, and therefore this limit applies. The matter eJects could be perhaps observed as small day-night variation in the solar neutrino signal. In long baseline oscillation experiments matter eJects could cause diJerence in the oscillation probability of neutrinos and antineutrinos, hence these eJects, together with the hierarchy problem (i.e., whether m3 ¿ m2 or m3 ¡ m1 ) are crucial in the search for CP violation and its interpretation. • High density limit, Losc L0 . The oscillation amplitude is suppressed by the factor L0 =|Losc |. For m22 ¿ m21 the matter mixing angle is m → 90◦ and thus e → 2 . • |Losc | ≈ L0 . In this case the matter eJects can be enhanced. In particular, for Losc =L0 = cos 2 one has sin2 2m = 1, i.e. the maximum mixing, even for small vacuum mixing angle . This is the basis of the Mikheyev-Smirnov-Wolfenstein (MSW) eJect [28,34]. When neutrinos propagate in matter of varying density, the equations of motion (22) or (27) must be solved. There is a vast literature on the subject, in particular on the application to the solar neutrinos. There, the e are often created at densities above the resonance density, and therefore
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are dominantly in the higher mass eigenstate. When the resonance density is reached, the two instantaneous eigenvalues are almost degenerate. In the adiabatic regime, the neutrino will remain in the upper eigenstate, and when it reaches vacuum it will be (for small vacuum mixing angle ) dominantly in the state that we denoted above as & . In the general case, there is a 3nite probability Px for jumping from one eigenstate to the other one, and the conversion might be incomplete. The average survival probability is [35] 1 P(e → e ) = [1 + (1 − 2Px ) cos 2m (1max ) cos 2] ; 2
(28)
Usually cos 2m (1max ) −1 and thus P(e → e ) sin2 + Px cos 2. The transition point between the regime of vacuum and matter oscillations is determined by the ratio √ 1Ye 7 × 10−5 eV2 E Losc 2 2GF Ne E (29) = = 0:22 L0 Dm2 1 MeV 100 g=cm3 Dm2 If this fraction is larger than unity, the matter oscillations dominate, and when this ratio is less than cos 2 the vacuum oscillations dominate. Generally, there is a smooth transition in between these two regimes. The electron neutrino survival probability is illustrated in Fig. 1, where it is plotted against 4E =Dm2 , see Eq. (29). In Fig. 1 it was assumed that the neutrinos originated in the center of the Sun, hence the relatively sharp feature at 4E =Dm2 ∼ 2:6 × 1011 eV−1 where according to the Eq. (29) Losc =L0 = 1 for the central density of the Sun. Note that below and above this dividing line the survival probability is almost independent of the neutrino energy, hence no spectrum distortion is expected. The subsequent increase in the neutrino survival probability for larger values of 4E =Dm2 is caused by the ‘nonadiabatic’ transition, i.e. by the gradual increase of the jump probability towards the limiting value Px → 1. 1 0.9 0.8
0.02
P(νe −> νe )
0.7
0.1
0.6 0.5 0.4
0.3 vacuum osc. 0.8
0.3 0.2
MSW matter osc.
0.1 0 10 10
11
10
12
10
13
10
2
14
10
10
15
16
10
−1
4E ν /∆m (eV ) Fig. 1. Schematic illustration of the survival probability of e created at the solar center. The curves are labelled by the sin2 2 values.
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For the parameters corresponding to the preferred solar solution (sin2 2 0:83 and Dm2 ∼ 7 × 10−5 eV2 ) the pp neutrinos with 4E =Dm2 ¡ 2:4×1010 eV−1 and the 7 Be neutrinos with 4E =Dm2 = 4:9×1010 eV−1 undergo vacuum oscillations, while the 8 B neutrinos with 4E =Dm2 ¿ 2:55×1011 eV−1 undergo MSW matter oscillations. (Clearly, that must be the case since otherwise it would be impossible to understand the ∼ 0:3 suppression of the 8 B neutrino e Mux observed in the charged current reactions, Section 3.1.) 2.5. Tests of CP, T and CPT invariance In vacuum CP conservation implies that P(‘ → ‘ ) = P(T‘ → T‘ ) (see Eq. (18)). Violation of this inequality thus would mean that CP is not conserved in the lepton sector. Substantial eJort is devoted to the CP tests. Matter eJects can induce inequality between P(‘ → ‘ ) and P(T‘ → T‘ ) and so the analysis must carefully account for them. If CP is not conserved, but CPT invariance holds, then T invariance will be also violated, i.e. P(‘ → ‘ ) − P(‘ → ‘ ) need not vanish. Here matter eJects cannot mimic the apparent T invariance violation. Finally, if CPT is not conserved, then P(‘ → ‘ ) − P(T‘ → T‘ ) might be nonvanishing. Many tests of the CPT invariance in the neutrino sector have been suggested, see e.g. [36] or references listed in [13]. CPT invariance is based on Lorentz invariance, hermiticity of the Hamiltonian and locality. Its violation would have, naturally, enormous consequences. Yet there are many proposed scenarios of CPT violation, in particular in the neutrino sector (for a whole series of papers on that topic see e.g. [37]). CPT invariance implies that neutrino and antineutrino masses are equal. If that is not true, then the Dm2 as determined in the solar neutrino experiments (thus involving e ) might not be the same as the Dm2 needed to explain the LSND result which involves T → Te oscillations, see Section 3.1.4. That was the gist of the phenomenological proposal in Ref. [38]. (See also further elaboration in [39].) With the demonstration of the consistency between the observed solar e de3cit and the disappearance of reactor Te by the KamLAND collaboration (Section 3.1.3), this possibility seems unlikely, even though a proposal has been made to accommodate CPT violation in that context [40], see also [42]. As has been shown in [41], the consistency of the solar oscillation solution and the KamLAND reactor result can be interpreted as a test of CPT giving |Dm2 − Dm2T | ¡ 1:3 × 10−3 eV2 90% CL ;
(30)
where Dm2 and Dm2T refer to the mass eigenstates 1 and 2 involved in the observed solar and reactor neutrino oscillations. To test for the CP invariance violation experimentally, one would compare the probabilities P( → e ) and P(T → Te ). This could be done realistically with ∼ 1 GeV beams at a distance L ∼ E =Dm2atm such that the contribution involving Dm2sol are small. The eJect of matter, however, must be included. Using the notation sij = sin ij ;
cij = cos ij ;
)ij = Dm2ij L=4E
(31)
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we obtain the formula 2 2 2 s13 s23 sin2 )31 P( → e ) = 4c13 2 + 8c13 s13 s23 c23 s12 c12 sin )31 [cos )32 cos % − sin )32 sin %] sin )21 2 2 2 2 −8c13 s13 s23 s12 cos )32 sin )31 sin )21 2 2 2 2 2 2 2 + 4c13 s12 [c12 c23 + s12 s23 s13 − 2c12 c23 s12 s23 s13 cos %] sin2 )21 aL sin )31 2 2 2 2 : − 8c13 s13 s23 (1 − 2s13 ) sin )31 cos )32 − 4E )31
(32)
Here the 3rst term gives the largest eJect, while the terms in the third and fourth line represent small CP conserving corrections (proportional to sin )21 and sin2 )21 . The term with sin % in the second line violates CP symmetry, while the term with cos % preserves it. Finally, the term with aL=4E in the last line represents the matter eJects. The matter eJects are characterized by √ a = 2 2GF Ne E = 1:54 × 10−4 Ye 1(g=cm3 )E (GeV)
(a is in (eV2 )) :
(33)
The probability P(T → Te ) is obtained by the substitution % → −% and a → −a. To test for CP symmetry, one would determine ACP =
P( → e ) − P(T → Te ) P( → e ) + P(T → Te )
−)21
sin 212 aL cos )32 sin % − ; sin 13 2E sin )31
(34)
where we used the empirical fact that cos 23 ∼ sin 23 and sin()21 ) ∼ )21 for the distances and energies usually considered. Since 13 is small, the CP asymmetry can be enhanced. However, the individual terms in Eq. (34) depend on 13 , so for smaller 13 it is more diRcult to reach the required statistical precision. There are several parameter degeneracies in Eq. (34) when separate measurements of P( → e ) leading and P(T → Te ) are made at given L and E . (i) There can be two values %; 13 and % ; 13 to the same probabilities. (ii) Sign of )31 and )32 (i.e. the normal or inverted hierarchy), where with the one set %; 13 gives the same oscillation probabilities with one sign, as another set % ; 13 opposite sign of )31 . (iii) Since the mixing angle 23 is determined in experiments sensitive only to sin2 223 there is an ambiguity between 23 and 2=2−23 . However, for the preferred value 23 ∼ 2=4 this ambiguity is essentially irrelevant. Various strategies to overcome the parameter degeneracies have been proposed. The choice of the neutrino energy E (and whether a wide or narrow beam is used) and the distance L play essential role. Clearly, if some of the so far unknown parameters (e.g. 13 or the sign of the hierarchy) could be determined independently, some of these ambiguities would be diminished. (For some of the suggestions how to overcome the parameter degeneracy see e.g. [43,44].)
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2.6. Violation of the total lepton number conservation Neutrino oscillations described so far are insensitive to the transformation properties under charge conjugation, i.e., whether neutrinos are Dirac or Majorana particles. However, if the mass eigenstates i are Majorana particles, then → T oscillations that violate the total lepton number conservation are possible. There are two kinds of such processes. Recall that in the standard model charged current processes the neutrino ‘ produces the negatively charged lepton ‘− while antineutrino T‘ produces the positively charged antilepton ‘+ . The standard model also requires that in order to produce the lepton ‘− the neutrino ‘ must have (almost) purely negative helicity, and similarly for ‘+ the T‘ must have (almost) purely positive helicity. The amplitudes for the ‘wrong’, i.e., suppressed helicity component is only of the order m =E and therefore vanishes for massless neutrinos. But we know now that neutrinos are massive, although light, and the therefore ‘wrong’ helicity states are present. When neutrinos are Majorana particles, the total lepton number may not be conserved. Thus, neutrinos ‘ born together with the leptons ‘+ can create leptons ‘+ (or even a diJerent Mavor ‘ + ) as long as the helicity rules are obeyed. The amplitude of this process is of the order m =E , thus small, independent of the distance the neutrinos travel. Therefore, this 3rst kind of the → T transformation should not be really called oscillations. When such a transformation happens inside a nucleus, it leads to the process of neutrinoless double beta decay discussed in more detail below. For Majorana neutrinos, there are several nonstandard processes involving helicity Mip in which left-handed neutrinos L are converted into right-handed (anti)neutrinos cR . This can happen for Majorana neutrinos with a transition magnetic moment ij . In a transverse magnetic 3eld B⊥ , iL can be connected to cjR . However, the transition probability is proportional to the small quantity | B| that vanishes for massless neutrinos. There are also models that predict neutrino decay involving a helicity-Mip and a Majoron 5 production. Again, the decay rate is expected to be small. If such processes exist one expects, among other things, that solar e could be subdominantly converted into Te . The recent limit on the solar Te Mux, expressed as a fraction of the Standard Solar Model 8 Be Mux is 2:8 × 10−3 [45]. (See also [45] for references to some of the theoretical model expectations for such → T conversion.) In addition, there could be also transitions without helicity Mip, which require that both Dirac and Majorana mass terms are present. These second class oscillations [46] involve transitions L → cL , i.e., the 3nal neutrino is sterile and therefore unobservable. Estimates of the experimental observation possibilities of the neutrino-antineutrino oscillations are not encouraging [47]. Study of the neutrinoless double beta decay (066) appears to be the best way to establish the Majorana nature of the neutrino, and at the same time gain valuable information about the absolute scale of the neutrino masses. Double beta decay is a rare transition between two nuclei with the same mass number A involving change of the nuclear charge Z by two units. The decay can proceed only if the initial nucleus is less bound than the 3nal one, and both must be more bound than the intermediate nucleus. These conditions are ful3lled in nature for many even-even nuclei, and only for them. Typically, the decay can proceed from the ground state (spin and parity always 0+ ) of the initial nucleus to the ground state (also 0+ ) of the 3nal nucleus, although the decay into excited states (0+ or 2+ ) is in some cases also energetically possible. In Fig. 2 we show a typical situation for A = 136. There are 11 candidate nuclei (all for the 6− 6− decay) with Q value above 2 MeV,
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β−
+ β
A=136
7
136 59 Pr
6 5 4 3
136 54 Xe
136 55 Cs
β−
ββ
136 57 La
β+
136 58 Ce
2 1 (MeV)
136 56 Ba
Fig. 2. Masses of nuclei with A=136. The even-even and odd-odd nuclei are connected by dotted lines. 136 Xe is stable against ordinary 6 decay, but unstable against 6− 6− decay. The same is true for 136 Ce, however, the 6+ 6+ decay is expected to be slower than the 6− 6− decay.
thus potentially useful for the study of the 066 decay. (Large Q values are preferable since the rate scales like Q5 and the background suppression is typically easier for larger Q.) There are two modes of the double beta decay. The two-neutrino decay, 266, (Z; A) → (Z + 2; A) + e1− + e2− + Te1 + Te2
(35)
conserves not only electric charge but also lepton number. On the other hand, the neutrinoless decay, (Z; A) → (Z + 2; A) + e1− + e2−
(36)
violates lepton number conservation. One can distinguish the two decay modes by the shape of the electron sum energy spectra, which are determined by the phase space of the outgoing light particles. Since the nuclear masses are so much larger than the decay Q value, the nuclear recoil energy is negligible, and the electron sum energy of the 066 is simply a peak at Te1 + Te2 = Q smeared only by the detector resolution. The 266 decay is an allowed process with a very long lifetime ∼ 1020 years. It has been observed now in a number of cases [12]. Observing the 266 decay is important not only as a proof that the necessary background suppression has been achieved, but also allows one to constrain the nuclear models needed to evaluate the corresponding nuclear matrix elements. The 066 decay involves a vertex changing two neutrons into two protons with the emission of two electrons and nothing else. One can visualize it by assuming that the process involves the exchange of various virtual particles, e.g. light or heavy Majorana neutrinos, right-handed current mediating WR boson, SUSY particles, etc. No matter what the vertex is, the 066 decay can proceed only when neutrinos are massive Majorana particles [48]. In the following we concentrate on the case when the 066 decay is mediated by the exchange of light Majorana neutrinos interacting through the left-handed V − A weak currents. The decay rate is then, 2 0 gV2 0 0 + + −1 0 [T1=2 (0 → 0 )] = G (E0 ; Z) MGT − 2 MF m66 2 ; (37) gA
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where G 0 is the exactly calculable phase space integral, m66 is the eJective neutrino mass and 0 MGT , MF0 are the nuclear matrix elements. The eJective neutrino mass is 2 i&i m66 = (38) |Uei | mi e ; i
where the sum is only over light neutrinos (mi ¡ 10 MeV). 4 The Majorana phases &i were de3ned earlier in Eq. (9). If the neutrinos i are CP eigenstates, &i is either 0 or 2. Due to the presence of these unknown phases, cancellation of terms in the sum in Eq. (38) is possible, and m66 could be smaller than any of the mi . The nuclear matrix elements, Gamow-Teller and Fermi, appear in the combination 2 gV2 0 g V 0 + + MGT − 2 MF ≡ f| (39) H (rlk ; ET m )l k ˜>l · ˜>k − 2 |i : gA g A lk The summation is over all nucleons, |i ; (|f ) are the initial (3nal) nuclear states, and H (rlk ; ET m ) is the ‘neutrino potential’ (Fourier transform of the neutrino propagator) that depends (essentially as 1=r) on the internucleon distance. When evaluating these matrix elements the short-range nucleonnucleon repulsion must be taken into account due to the mild emphasis on small nucleon separations. There is a vast literature devoted to the evaluation of these nuclear matrix elements, going back several decades. It is beyond the scope of the present review to describe this eJort in detail. The interested reader can consult various reviews on the subject, e.g. [12,49,50]. Obviously, any uncertainty in the nuclear matrix elements is reMected as a corresponding uncertainty in the m66 . There is, at present, no model independent way to estimate the uncertainty, and to check which of the calculated values are correct. Good agreement with the known 266 is a necessary but insuRcient condition. The usual guess of the uncertainty is the spread, by a factor of ∼ 3, of the matrix elements calculated by diJerent authors. Clearly, more reliable evaluation of the nuclear matrix element is a matter of considerable importance. (For a recent attempt to reduce and understand the spread of the calculated values, see [51].) The 066 decay is not the only possible observable manifestation of lepton number violation. Muon-positron conversion, − + (A; Z) → e+ + (A; Z − 2) ;
(40)
or rare kaon decays K2 ; Kee2 and Ke2 , K + → + + 2− ;
K + → e + e + 2− ; K + → + e + 2− ;
(41)
are examples of processes that violate total lepton number conservation and where good limits on the corresponding branching ratios exist. (See Ref. [52] for a more complete discussion.) However, it appears that the 066 decay is, at present, the most sensitive tool for the study of the Majorana nature of neutrinos. 4
The same quantity is sometimes denoted as m or mee .
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2.7. Direct measurement of neutrino mass Conceptually the simplest way to explore the neutrino mass is to determine its eJect on the momenta and energies of charged particles emitted in weak decays. In two-body decays, e.g. 2+ → + + the analysis is particularly straightforward, at least in principle. In the system where the decaying pion is at rest, the energy and momentum conservation requirements mean that m2 = m22 + m2 − 2m2 m2 + p2 : (42) However, the neutrino mass squared appears as a diJerence of two very large numbers. Hence the uncertainties in m2 ; p and even m mean that the corresponding mass limit is only m ¡ 170 keV. This problem can be avoided by studying the three body decays, in particular the nuclear beta decay. Near the endpoint of the beta spectrum a massive neutrino has so little kinetic energy that the eJects of its 3nite mass become more visible. The electron spectrum of an allowed beta decay is given by the corresponding phase-space factor dN ∼ F(Z; Ee )pe Ee (E0 − Ee )[(E0 − Ee )2 − m2 ]1=2 ; (43) dE where Ee ; pe is the electron energy and momentum and F(Z; Ee ) describes the Coulomb eJect on the outgoing electron. The quantity E0 is the endpoint energy, the diJerence of total energies of the initial and 3nal systems. Clearly, the eJect of 3nite neutrino mass becomes visible if (E0 −Ee ) ∼ m , i.e. very near the threshold. For the case of several massive neutrinos with mixing, the beta decay spectrum is an incoherent superposition of spectra like (43) with corresponding weights |Uei |2 for each mass eigenstate m2i . If the experiment has insuRcient energy resolution the quantity ) = |Uei |2 m2i (44) m2(eJ e i
(using the RPP notation [20]) could be determined from the electron spectrum near its endpoint, where the sum is over all the experimentally unresolved neutrino masses mi . ) Based on an upper limit for the m2(eJ one can deduce several limits that do not depend on the e 2 mixing parameters |Uei | . First, at least one of the neutrinos (i.e. the one with the smallest mass) ) . Moreover, if all (with an emphasis on all) has a mass less or equal to that limit, m2min 6 m2(eJ e ) 2 |Dmij | values are known, an upper limit of all neutrino masses is m2max 6 m2(eJ + i¡j |Dm2ij |. e Thus, if we assume that the |Dm2ij | values deduced from the experiments on solar (and reactor) and atmospheric oscillation studies (and include also the LSND result) cover all possibilities, and ) combine that knowledge with the m2(eJ limit from tritium beta decay, we may conclude that no e active neutrinos with mass more than ‘a few’ eV exists. 3. Experimental results and interpretation Positive evidence for neutrino mass has so far been obtained only in measurements of neutrino oscillations. As noted above in Section 2.3, the neutrino masses enter only through the diJerences
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in squared masses (i.e., Dm2ij = m2i − m2j ), so although these measurements provide lower limits to the mass values (e.g., m2i ¿ Dm2ij ) they do not actually determine the masses. As discussed in Sections 2.6 and 2.7, direct neutrino mass measurements and neutrinoless double beta decay allow determination of quantities related to the values of the masses themselves (as opposed to diJerences). However, these experiments have so far yielded only limits. Finally, additional constraints have been derived from measurements of the cosmic microwave background radiation and galaxy distribution surveys through the eJect of neutrinos on the distribution of matter in the early universe. In this section, we summarize the positive observations from oscillation measurements followed by brief discussion of the direct measurements, double beta decay searches, and recent constraints from cosmology. 3.1. Neutrino oscillation results The 3rst hints that neutrino oscillations actually occur were serendipitously obtained through early studies of solar neutrinos and neutrinos produced in the atmosphere by cosmic rays (“atmospheric neutrinos”). In fact, the atmospheric neutrino measurements were a byproduct of the search for X proton decay using large water Cerenkov detectors. So it is somewhat ironic that although there was substantial interest in searching for neutrino oscillations, the 3rst evidence for this phenomena came from experiments designed for very diJerent purposes. As shown below, recent studies de3nitively establish that the solar neutrino Mux is reduced due to Mavor oscillations, and so it is now clear that the 3rst real signal of neutrino oscillations was the long-standing de3cit of solar neutrinos observed by Ray Davis and collaborators using the Chlorine radiochemical experiment in the Homestake mine. While it took almost three decades to demonstrate the real origin of this de3cit, the persistent observations by Davis et al. and many other subsequent solar- experiments were actually indications of neutrino oscillations. We will discuss this subject in more detail below. 3.1.1. Atmospheric neutrinos The Kamiokande experiment in Japan [53] and the IMB experiment in the US [54] were pioneering X experimental projects to develop large volume water Cerenkov detectors with the primary goal of detecting nucleon decay, as predicted by Grand Uni3ed Theories developed in the 1970s [55]. Although these detectors were located deep underground to avoid cosmic ray-induced background, they both encountered the potential background events produced by atmospheric neutrinos (both e and ) that easily penetrated to these subterranean labs and (rarely) produced energetic events in the huge detectors. And indeed, both experiments [56–58] (along with the Soudan experiment [59]) observed that the ratio of -induced events to e -induced events was substantially reduced from the expected value of ∼ 2. The decay chain of 2± produced in the upper atmosphere would produce (through the subsequent -decay) a , T , and a e (or Te ). Thus, based on rather simple basic arguments one expects the ratio of =e events to be about ∼ 2—and this is supported by more detailed Monte Carlo simulations. The observed values were closer to ∼ 1, which was viewed as an anomaly for many years. Here again, although -oscillations could clearly cause this anomaly there was not enough corroborative evidence to substantiate this explanation.
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Fig. 3. Distribution of observed atmospheric neutrino events vs. zenith angle from the SuperKamiokande experiment, compared with Monte Carlo simulations [60]. The blue hatched region represents the prediction without neutrino oscillations and the red line includes the eJect neutrino oscillations. (PC means ‘partially contained’ events.)
However, the situation dramatically changed in 1998 when the larger experiment, SuperKamiokande, reported a clearly anomalous zenith angle dependence of the events [61]. The measurements indicated a de3cit of upward-going -induced events (produced ∼ 104 km away at the opposite side of the earth) relative to the downward-going events (produced ∼ 20 km above). The e events displayed a normal zenith angle behavior consistent with Monte Carlo simulations. Since at that time (1998) the solar- problem was still unresolved, these data represent the 3rst really solid evidence for -oscillations. More recent measurements [60] are displayed in Fig. 3, and the conclusion that Mavor oscillations are responsible is essentially inescapable. Moreover, the deduced values of sin2 2 ¿ 0:90 (90% C.L.) indicate a surprisingly strong mixing scenario where the muon-type neutrino seems to be a fully-mixed superposition of all three mass eigenstates. (This situation is completely contrary to the quark sector, where the mixing between generations is generally small.) The most recently reported value of Dm2 derived from the Super-Kamiokande results is 0:0020 eV2 [62] (for the error bars, see Table 1). The observation of the angular distribution of upward-going muons produced by atmospheric neutrinos in the rock below the MACRO detector [63] supports the conclusion that the observed eJect is due to the → oscillations, and disfavors → sterile assignment. Although the parallel eJort to detect neutrino disappearance at nuclear reactors had made steady progress (setting upper limits and establishing exclusion plots) for many years, these experiments made a strong contribution at this point: the failure to observe Te disappearance at CHOOZ [64] and Palo Verde [65] in the region near Dm2 0:0020 eV2 implies that the disappearance observed by Super-Kamiokande does not involve substantial e appearance. (This inference assumes that m = mT
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Fig. 4. Energy distribution of the Mux of solar neutrinos predicted by the standard solar model [66], as computed by J. Bahcall [67]. The ranges of energies associated with the various experiments are indicated at the top of the 3gure.
for each eigenstate as required by CPT invariance.) Thus, it would seem that the ’s must be oscillating into or other more exotic particles, such as “sterile” neutrinos. 3.1.2. Solar neutrinos The interpretation of solar neutrino measurements involves substantial input from solar physics and the nuclear physics involved in the complex chain of reactions that together are termed the “Standard Solar Model” (SSM) [66]. The predicted Mux of solar neutrinos from the SSM is shown in Fig. 4 as a function of neutrino energy. The low energy p–p neutrinos are the most abundant and, since they arise from reactions that are responsible for most of the energy output of the sun, the predicted Mux of these neutrinos is constrained very precisely (±2%) by the solar luminosity. The higher energy neutrinos are more accessible experimentally, but the Muxes are less certain due to uncertainties in the nuclear and solar physics required to compute them. The early measurements included radiochemical experiments sensitive to integrated e Mux such as the Chlorine [68] (threshold energy 0:814 Mev) and Gallium [69] (0:233 MeV) experiments. Live counting was developed by the Kamiokande [70] and then the SuperKamiokande [71] experiments, based on neutrino-electron scattering, enabling measurements of both the Mux and energy spectrum. Together these experiments sampled the solar neutrino Mux over a wide range of energies. As can be seen in Fig. 8, all these experiments reported a substantial de3cit in neutrino Mux relative to the SSM. While it was realized that neutrino oscillations could be an attractive solution to this problem, it was problematic to establish this explanation with certainty due
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to the dependence on the SSM, its assumptions, and sensitivity to input from nuclear and solar physics. However, it is now clear that the precise measurements of solar neutrino Muxes over a range of energies coupled with the amazing Mavor transformation properties of neutrinos validates the SSM in a beautiful and satisfying manner. The solar- experiments and the SSM have been reviewed in detail elsewhere [20], and so it is not appropriate to repeat the details here. We present these solar neutrino Mux measurements together in Fig. 8, where we also display the excellent agreement with the best 3t solution to the combined solar- and KamLAND reactor data [72] to demonstrate the remarkably coherent picture that is strongly supported by these impressive experimental measurements and the associated SSM input. 3.1.3. SNO and KamLAND With the advent of the new millenium, the stage was set for a synthesis of the study of solar X neutrinos using a powerful new water Cerenkov detector (Sudbury Neutrino Observatory, SNO) with the study of the disappearance of reactor antineutrinos using a large scintillation detector located deep underground (KamLAND). The results of these experiments provide de3nitive evidence that the solar de3cit is indeed due to Mavor oscillations, and that this eJect is demonstrable in a “laboratory” experiment on earth. X The SNO experiment combines the now high-developed capability of water Cerenkov detectors with the unique opportunities aJorded by using deuterium to detect the solar neutrinos [73,74]. Low energy neutrinos can dissociate deuterium via the charged current (CC) reaction e + d → e− + 2p
(45)
or the neutral current (NC) reaction ‘ + d → ‘ + p + n :
(46)
Only e can produce the CC reaction, but all Mavors ‘ = e; ; can contribute to the NC rate. The CC reaction is detected via the energetic spectrum of e− which closely follows the 8 B solar e spectrum. The NC reaction involves three methods for detection of the produced neutron: (a) capture on deuterium and detection of the 6:25 MeV -ray, (b) capture on Cl (due to salt added to the D2 O) and detection of the 8:6 MeV -ray, or (c) capture in 3 He proportional counters immersed in the detector. There are also some events associated with the elastic scattering of the solar- on e− in the detector which is dominated by the charged current reaction (again only e ) but has some ∼ 20% contribution from neutral currents (all Mavors equally contribute). The SNO collaboration has published data on the CC and NC rate (from processes (a) and (b)). Additional data from the NC process (c) will be forthcoming in the future. Nevertheless, the reported results (see Fig. 5) demonstrate very clearly that the total neutrino Mux (e + + as determined from NC) is in good agreement with the SSM, but that the e Mux is suppressed (as determined from CC). This represents rather de3nitive evidence that the e suppression is due to Mavor-changing processes that convert the e to the other Mavors, as expected from -oscillations. Furthermore, the observed value of e Mux and the observed energy spectrum, when combined with the other solar- measurements strongly favor another large mixing angle scenario at a lower value of Dm2 ∼ 7 × 10−5 eV2 .
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Fig. 5. Measured solar neutrino Muxes from SNO for the NC and CC processes [74], along with elastic scattering events (ES) and the SSM prediction [66].
The KamLAND experiment [75] represents a major advance in the development of reactor Te measurements. In order to reach the low values of Dm2 ∼ 10−5 eV2 indicated by the solar- data, distances of order ∼ 200 km are necessary (given the 3xed range of Te energies from reactors). The loss of rate due to 1=r 2 scaling is severe, which requires substantial increases in source strength (i.e., reactor power) and detector size. Such a huge detector, sensitive to low-energy inverse beta-decay events from reactor Te , would be very susceptible to background from cosmic radiation and so must be located deep underground. By fortunate coincidence, the old Kamiokande site in Japan is very deep (∼ 1000 mwe) and located an average distance of ∼ 200 km from a substantial number of large nuclear power reactors. Thus the KamLAND experiment, a large liquid scintillator detector, was built at this site to study the disappearance of Te from nuclear reactors. For the 3rst time in the long history of reactor Te experiments (dating back to the original discovery of the Te by Reines and Cowan [2]) a substantial de3cit in event rate was observed (Fig. 6). In fact this de3cit is just as predicted by the solar solution to the solar- oscillation solution, and the more precisely constrained values of Dm2 and sin2 2 from a global analysis of the solar- and KamLAND data are shown in Fig. 7. As mentioned previously, Fig. 8 shows a summary of the solar- data compared with the SSM with and without neutrino oscillations. The plotted experiments are: Ga, combined gallium measurements from GALLEX and SAGE [69]; Cl, chlorine measurement from Homestake mine [68]; SNO, Sudbury Neutrino Observatory (CC and NC) [72–74]; and SK, Super-Kamiokande [71]. In this plot one can clearly see the decreasing survival fraction with increasing energy in the progression Ga → Cl → SNOCC (all sensitive only to the e component). The SNONC measurement shows no suppression, whereas the SK data exhibit the intermediate suppression due to the partial contribution of NC events to their elastic scattering signal. In summary, the experimental studies of solar neutrinos, atmospheric neutrinos, and reactor antineutrinos have established neutrino oscillations with two diJerent mass scales, Dm2 ∼ 2:0 × 10−3 eV2 and Dm2 ∼ 7:1 × 10−5 eV2 , both with large associated mixing angles. The allowed regions are shown in Fig. 9, and together these experiments constrain many of the matrix elements in the 3 × 3
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Nobs /Nexp
1.0 0.8 ILL Savannah River Bugey Rovno Goesgen Krasnoyarsk Palo Verde Chooz
0.6 0.4 0.2
KamLAND 0.0 1
10
2
3
4
10 10 10 Distance to Reactor (m)
5
10
Fig. 6. Ratio of observed to expected rates (without neutrino oscillations) for reactor neutrino experiments as a function of distance, including the recent result from the KamLAND experiment [75]. The shaded region is that expected due to neutrino oscillations with large mixing angle parameters as determined from solar neutrino data.
Fig. 7. Region of parameter space constrained by simultaneous 3t to solar- and KamLAND data, from [72]. The best 3t values are Dm2 = 7:1 × 10−5 eV2 and tan2 = 0:41.
mixing matrix for the neutrinos along with the mass diJerences Dm212 and Dm232 . The results for these parameters are listed in Table 1. 3.1.4. LSND There is one other experiment that claims to observe neutrino oscillations: the Liquid Scintillator Neutrino Detector (LSND) [76] at Los Alamos. In this experiment, the neutrino source was the beam dump of an intense 800 MeV proton beam where a large number of charged pions were created and stopped. Since the 2− capture on nuclei with very high probability, essentially only the 2+ decay,
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Fig. 8. Ratio of solar neutrino Mux to SSM (without neutrino oscillations) for various experiments (see text). Filled circles are experimental data (with experimental uncertainties only) and open circles are theoretical expectations based on SSM with best 3t parameters to KamLAND and solar- data (uncertainties from SSM and oscillation 3t combined). All charged current experiments show a substantial de3cit and all are in excellent agreement with the expected values.
Fig. 9. Allowed regions of parameter space (90% C.L.) determined by atmospheric neutrino measurements [62] (2 ↔ 3 mixing) and by solar- and reactor-T measurements [72] (1 ↔ 2 mixing).
producing and then T and e (from + decay). These neutrinos all have very well de3ned energy spectra (from decays of particles at rest) and note that there are no Te produced in this process. The 160 ton detector is then used to search for Te events via inverse beta decay on protons at a distance of 30 m from the neutrino source. The experiment detected an excess of 87:9 ± 22:4 ± 6:0 events corresponding to an oscillation probability of 0:264 ± 0:067 ± 0:45%. (Note that such an appearance experiment aJords access to very small mixing parameters.) The observed spectrum of events is shown in Fig. 10. Other experiments, especially the KARMEN accelerator experiment [77] and the Bugey reactor experiment [78], rule out much of the allowed region of parameter space but there is a small region remaining at 90% con3dence in the mass range 0:2 ¡ Dm2 ¡ 10 eV2 , indicating a minimum mass of m ¿ 0:4 eV (see Fig. 12). It is also signi3cant that the KARMEN experiment studied a shorter baseline, which seems to rule out the possibility that the Te are produced at the source.
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Table 1 Neutrino oscillation parameters determined from various experiments (2003) Parameter
Value ±1>
Reference
Dm212 12 Dm232 sin2 223 sin2 213
−5 7:1+1:2 eV2 −0:6 × 10 ◦ +2:4 32:50−2:3 −3 2:0+0:6 eV2 −0:4 × 10 ¿ 0:94 ¡ 0:11
[72] [72] [62] [62] [64]
Comment For 13 = 0 For 13 = 0 For Dm2atm = 2 × 10−3 eV2
Fig. 10. Observed spectrum of positron energy from beam-induced events in the LSND experiment [76]. The black dots are the experimental data, the lower histogram is the estimate of various backgrounds, the middle histogram includes the estimated Te contamination from the source, and the top histogram includes the neutrino oscillation hypothesis.
Given the well established values of Dm2 in Table 1, it is not possible to form a third value of Dm2 consistent with the LSND data (note that Dm212 + Dm223 + Dm231 = 0). So one would need to either break CPT invariance (allowing m = mT), or invoke additional “sterile” neutrinos that do not have the normal weak interactions. In addition, the LSND range of Dm2 is marginally at variance with recent studies of the cosmic microwave background (see below). Therefore, it is of great importance to attempt to obtain independent veri3cation of this result (see Section 4.1). 3.2. Direct mass measurements Beta decays with low endpoints, in particular tritium (Q = 18:6 keV), have been used for a long time in attempts to measure or constrain neutrino mass. (187 Re with Q = 2:5 keV has been also
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) Fig. 11. Results of measurements of m2(eJ from tritium 6-decay experiments since 1990 [20], showing the steady imTe provement in the achieved precision.
explored recently [79], but the sensitivity is still only 21:7 eV, a factor of about ten worse than for tritium.) There are several diRculties one has to overcome to reach sensitivities to small neutrino masses in the study of the electron spectrum of 6 decay. Obviously, the mass sensitivity can be only as good as the energy resolution of the electron spectrometer. Also, since by de3nition the energy spectrum vanishes at the endpoint, the fraction of events in the interval DE near the Q = E0 − me kinetic energy endpoint decreases rapidly, as (DE=Q)3 . Finally, since the decaying system is a molecule or at best an atom, one has to take into account the possibility that the sudden change of the nuclear charge causes excitations or ionization of the electron cloud, i.e., the presence of multiple endpoints. Although there are as yet no positive results from direct neutrino mass measurements, these eJorts have made (and continue to make) substantial progress (see Fig. 11). Curiously, many past experiments have been apparently plagued by systematic eJects that tended to mimic a negative ) 2 m(eJ . As the experiments were improved, it seems that these problems have been largely overcome. Te Two recent tritium 6 decay experiments have reached impressive results [80,81] quoting 95% upper (eJ ) 2 2 limits for the eJective mass parameter mTe = i |Uei | mi of 2.5 and 2:8 eV. However, these values are now at the level where atomic and chemical eJects on the phase space distributions are signi3cant, and it seems that some of these experiments [81] still occasionally observe anomalous structures near the endpoint. This will likely be a substantial challenge for future measurements of this type. Constraints from direct mass measurements also exist on m and m , but these are much larger (see Section 2.7 for explanation). Given reasonable assumptions, it seems that the oscillation results discussed above and the constraints from tritium decay would imply that the mass values are far below the direct measurements for and so we do not discuss them in detail here. 3.3. Double beta decay Over the last decade, the methodology for double beta decay experiments has markedly improved. Larger volumes of high-purity enriched materials are being utilized, and careful selection of materials
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along with deep-underground siting have lowered backgrounds and increased sensitivity. The most sensitive experiments use the isotopes 76 Ge, 100 Mo, 116 Cd, 130 Te, and 136 Xe. For 76 Ge, the lifetime limit has reached an impressive value exceeding 1025 years. (The experimental results are listed in [20], for the latest review of the 3eld, see [12].) The conversion of the observed lifetime limits to eJective neutrino mass values requires use of calculated nuclear matrix elements, and there is some uncertainty associated with them. Nevertheless, the experimental lifetime limits have been interpreted to yield eJective mass limits of typically a few eV and in 76 Ge of 0.3–1:0 eV. One recent report [82], analyzing the 76 Ge data from the Moscow-Heidelberg experiment, claims to observe a positive signal corresponding to the eJective mass m66 = 0:39+0:17 −0:28 eV. That report has been followed by a lively discussion [83–86]. If this 3nding were to be con3rmed then it would be a major advance in our knowledge of neutrino properties, and in particular it would not only prove that neutrinos are Majorana particles, but it would also strongly indicate that neutrinos follow a degenerate mass pattern, where Dm2 =m2 1. 3.4. Cosmological constraints In the early universe, when the temperature was T ¿ 1 MeV, the high density of particles allowed weak interactions to occur proli3cally leading to a substantial density of neutrinos. As the universe cooled to T ¡ 1 MeV, these reactions became much slower than the expansion rate and the neutrinos decoupled from the remaining ionized plasma and radiation (photons). Much later (∼ 100; 000 years), the universe cooled enough that atoms formed and the radiation decoupled from the matter. The cosmic microwave background (CMB) is the further cooled (through expansion) relic of this period, and contains information on the distribution of matter at that time in the history of the universe. The presently observed distribution of matter (through high resolution galaxy surveys) and distribution of radiation (CMB) would both be aJected by the presence of massive neutrinos in the early universe. Although the power spectra of the CMB and the density Muctuations are both sensitive to massive neutrinos, a combined analysis of both observables is especially eJective in addressing the existence of massive neutrinos [17,87]. Thus comparison of the power spectrum of CMB with the observed distribution of galaxies can provide information on the sum (over all Mavors) of light neutrino masses. An analysis of the recent WMAP data [88] yields the result f mf ¡ 0:7 eV (95% con3dence). It is interesting to note that this signi3cantly constrains the remaining region of parameter space allowed by LSND. (In addition, interpretation of the LSND result in light of cosmological constraints requires careful consideration of issues related to the behavior (e.g. thermalization) of sterile neutrinos in the early universe [89].) However, it has been argued that the obtained limit also relies upon input from Lyman-& forest measurements, and that a somewhat less restrictive limit should be quoted [90,91]. Other recent work [92] also obtains a less restrictive limit without the use of strong prior on galaxy bias. 4. Near-term future The recent discoveries and revolutionary breakthroughs in the study of neutrino properties have motivated a new generation of experimental eJorts aimed at resolving the remaining issues and establishing new launching points for future explorations. These near-term plans and proposals are,
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Fig. 12. Projected sensitivity [93] of the MiniBooNE experiment compared with LSND and other previous experiments.
for the most part, initiatives that advance the themes we have emphasized above. Neutrino oscillation studies are planned to address the LSND result, higher precision measurements of 23 and Dm223 , and determination of 13 . Direct mass measurements are aimed at reducing (or discovering) the absolute value of neutrino masses and establishing the hierarchical nature of the neutrino mass spectrum. And future experiments in double beta decay hope to demonstrate the Majorana nature of neutrinos and constrain (or discover) the mass scale. Finally, the ever increasing precision of cosmological constraints from measurements of the cosmic microwave background promise to provide tighter limits on neutrino masses. 4.1. MiniBooNE A major near-term priority for the 3eld is to resolve the issue of the validity of the results of the LSND experiment. To this end, the Mini-Boone experiment [93] has been built at FermiLab with the main goal to explore the same region of parameter space with higher sensitivity. This experiment X uses a new 0.5–1:5 GeV high-intensity neutrino source and a 800 ton mineral oil-based Cerenkov detector located at 500 m from the source to search for the oscillation modes → e and at a later stage T → Te . The projected sensitivity for 2 years running at full beam intensity in each mode is shown in Fig. 12. This experiment has the potential to test the validity of the LSND results with high sensitivity and to explore the possible role of CPT violation by studying both e and Te appearance. If an eJect is seen the collaboration plans to mount another detector at further distance for additional studies. The initial run with the beam began in the Fall of 2002 and 3rst results on oscillations are expected in 2005.
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4.2. Determination of Dm232 and 23 The phenomenon of neutrino oscillation observed by Super-Kamiokande in the atmospheric neutrino experiment requires further study to precisely determine the mixing parameters Dm232 and 23 . There are two long baseline accelerator experiments with prospects for results in the near term: K2K in Japan and MINOS in the US. The K2K experiment utilizes the 12 GeV proton beam at KEK to produce a beam with average energy 1:3 GeV which is directed at the Super-Kamiokande detector a distance of 250 km [94]. Use of GPS receivers enables clean selection of events with the proper time relative to the beam spill. Careful monitoring of the muons in the 2+ -decay region ensures proper aiming of the beam over the long baseline to the detector. In addition, a combination of detectors located 300 m downstream of the 2+ production target is used to measure the Mux and energy spectrum of the beam. During the 3rst data run in 1999–2001, the experiment detected 56 fully-contained events, compared to the expected 80:1+6:2 −5:4 events without neutrino oscillations. Thus, so far the K2K experiment appears to con3rm the oscillation interpretation of the observed anomaly in the atmospheric neutrino distribution with high probability. By combining the observed energy spectrum and the reduced event rate, the allowed regions shown in Fig. 13 are obtained. The experimental plan is to continue running and roughly double this data sample. The MINOS experiment uses the Main Injector proton beam at Fermilab to produce a beam in the energy range 3–18 GeV directed at a large detector located in the Soudan mine at a distance of 735 km [95]. The lowest energy neutrino beam will be optimal for studying the region of Dm232 2:0 × 10−3 eV2 . This experiment also uses a near detector (1 kton) in addition to the far detector (5:4 kton). The large far detector consists of 486 layers of alternating steel plates with 3nely (4 cm) segmented plastic scintillator planes. The steel is magnetized by the return Mux of a coil located on
Fig. 13. Allowed regions from analysis of the 3rst results from the K2K [94] experiment, representing about 12 of the expected total future luminosity. Dashed, solid and dot-dashed lines are 68.4, 90 and 99% C.L. contours, respectively. The best 3t point is indicated by the star.
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the detector axis, which enables measurement of muon momentum. With 10 kton-years of exposure at full luminosity, the MINOS experiment should easily con3rm the disappearance observed with atmospheric neutrinos and determine the mass parameter Dm232 to about 10%. By searching for e appearance the MINOS experiment will be sensitive to 13 in the region sin2 13 ¿ 0:015 over the presently allowed range of Dm232 , a factor of ∼ 2 lower than the CHOOZ limit. The far detector is complete and operating. It is expected that the MINOS experiment will receive the neutrino beam and start operation in 2005. There is also a long baseline facility from CERN to the Gran Sasso Laboratory (L 730 km). This is a higher energy beam (ET 17 GeV), suitable for production of ’s associated with → oscillations. There will be two detectors at Gran Sasso for this study: OPERA and ICARUS [96]. The OPERA experiment will employ photographic emulsion to identify the ‘kinks’ associated with the short (few m) decays. The ICARUS experiment utilizes a liquid argon time projection chamber to kinematically reconstruct the leptons. Both experiments expect 1–5 events per year to 3rmly establish this oscillation mode, and will begin operation in 2006. Future experiments providing precision measurements of the corresponding parameters are highly desirable. 4.3. Studies of 13 , neutrino mass hierarchy, and CP violation The role of mixing between the 3rst and third generation neutrino mass eigenstates is pivotal in terms of the phenomenological consequences. The possibility of CP violation and implications for leptogenesis scenarios in generating the matter-antimatter asymmetry in the universe require mixing between these two states. Therefore, establishment of non-vanishing mixing (i.e., 13 = 0) is of paramount importance and substantial experimental eJorts are planned to address this issue. 4.3.1. High-precision reactor neutrino experiments When completed, the KamLAND experiment will signi3cantly reduce the allowed region for Dm212 and tan2 12 relative to the present results shown in Fig. 7. The next major goal for the reactor neutrino program will be to attempt a measurement of sin2 13 . As can be seen from Eq. (14) and as discussed in Section 2 above, such experiments have the potential to determine 13 without ambiguity from CP violation or matter eJects. The strategy is to capitalize on the success of CHOOZ and KamLAND to build a new experiment at the 1.5–2 km distance now indicated by the Super-Kamiokande atmospheric neutrino results. Obtaining the necessary statistical precision on this challenging disappearance experiment will require large reactor power (¿ 5 GWth) and large detector size (probably ∼ 100 ton). Systematic errors must be carefully controlled through reduced background and utilization of a two detector scheme (probably in a con3guration with a near detector at a close ∼ 100–200 m distance from the reactor(s) in addition to the far detector at ∼ 2 km). Backgrounds must be controlled through suRcient depth underground (¿ 300 m to reduce cosmic ray induced spallation products), careful choice of detector materials, and passive shielding (e.g., buJer of water or mineral oil) to reduce events due to radioactive decays in the surrounding rock and other material. Comparison of the rates and observed spectral shapes in the two detectors will reduce the sensitivity to reactor source uncertainties and provide substantial improvements in the sensitivity to reactor antineutrino disappearance in this region. With the anticipated statistical and systematic uncertainties held to a total of order 1%, the sensitivity will reach about sin2 213 0:01–0.02 at the optimal value of Dm2 ∼ 2:0 × 10−3 eV2 derived from the Super-Kamiokande
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Fig. 14. Projected sensitivity (90% C.L.) of future reactor neutrino experiments to 13 as a function of integrated luminosity [97]. The authors assumed Dm231 =3:0×10−3 eV2 and studied scenarios with various systematic errors for Mux normalization (>norm ) and energy calibration (>cal ). The JHF-SK line is the projected sensitivity of the future JPARC-SK experiment (see Section 4.3.2), Reactor-I scenario is 400 GW-ton-years, and Reactor-II scenario is 8000 GW-ton-years.
results. Signi3cant eJorts are presently underway to optimize the experimental design and select appropriate sites for this future experiment [97–99]. Fig. 14 shows the results of a general study of the potential of such experiments [97]. With reasonable systematic errors (¡ 1%) for normalization and energy calibration it is apparent that about 400 GW-ton-years of luminosity would provide a determination of sin2 213 with sensitivity better than 0.02. In addition, the 3gure shows interesting behavior at higher luminosity (∼ 8000 √ GW-ton-years) where the sensitivity improves as 1= L due to the high statistical precision in determination of the relative spectral shapes at the near and far detectors. (The curve with >norm and >cal = ∞ describes the situation where the absolute eRciency and energy calibration of the near and far detectors becomes irrelevant, but the relative eRciencies and energy calibrations must be still tightly controlled.) Although the realization of siting KamLAND-scale detectors near large power reactor plants would be very challenging, the increased sensitivity to sin2 213 at such high luminosities would be of great interest. 4.3.2. Long baseline accelerator experiments While the reactor neutrino studies at ∼ 2 km have the potential to establish a non-vanishing mixing and a numerical value for 13 , further studies related to the neutrino mass hierarchy and the role of CP violation will require mounting new long baseline accelerator experiments (see Sections 2.4 and 2.5). The potential for studying these phenomena is evident in Eq. (32) and illustrated in Fig. 15 where one can see the eJects of varying the CP-violation parameters and changing the mass hierarchy. There is a great deal of activity in this subject at the moment, and a variety of proposed scenarios are under discussion.
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Fig. 15. Oscillation probabilities for neutrino and antineutrino oscillations [103] for the NUMI experiment parameters [102]. The diJerent loops correspond to various values of sin2 13 and normal and inverted hierarchy. The variation of the CP-violating phase % traces the loops.
The main features of the next generation long baseline accelerator experiments include: (1) (2) (3) (4) (5)
ability to detect the → e process as the major goal, L=E ∼ 500 km=GeV to optimize the oscillation probability for Dm223 ∼ 2 × 10−3 eV2 , propagation of the beam through the earth, resulting in sensitivity to matter eJects, possibility to vary L=E by adjusting the focusing horn, target position, and/or detector location, possibility to switch to T .
A signi3cant new concept, the “oJ-axis” neutrino beam [100] (combined with a substantial increase in the neutrino Mux and often referred to as a “super-beam”), appears to be a very attractive option for these experiments. By positioning the detector oJ the symmetry axis (at a so-called “magic” angle), the neutrino energy becomes stationary with respect to the pion energy and an essentially monoenergetic neutrino beam is obtained. Although generally some loss of Mux results from the oJ-axis geometry, the advantages include a tuneable narrow-band beam with signi3cant suppression of the higher energy tail and very low e contamination. Thus one can selectively scan a range of Dm2 to measure the dependence of the oscillation probability with reduced background rates (both due to the low e Mux and the suppressed 20 production from higher energy neutral current events). This method has been adopted for both the JPARC-SK experiment and the Fermilab NUMI proposal. The JPARC facility [101] will be a high luminosity (0:77 MW beam power) 50 GeV proton synchrotron with neutrino production facility aimed about 2◦ oJ-axis from the Super-Kamiokande detector (L 295 km). The resulting low-energy (∼ 0:7 GeV) neutrino beam at the Super-K site will enable sensitivity to sin2 213 ∼ 6 × 10−3 after about 5 years running. It is expected that the experiment would start in 2008, and upgrades involving siting a 1Mt detector and increased beam intensity are envisioned for the future, with potential sensitivity to sin2 213 ¡ 1:5 × 10−3 and CP-violating phase % ∼ ±20◦ . The “OJ-axis NUMI” proposal [102] is to site a new detector either near Soudan at L ∼ 715 km or at a further site in Canada at L ∼ 950 km, at an angle of about 14 mrad with respect to the
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Fig. 16. Conceptual design for the KATRIN experiment for higher-precision measurements of the neutrino mass using the endpoint of the tritium 6-decay spectrum.
beam axis. The neutrino beam energy will be about 2 GeV, and with a 20 kt detector (still under development) the sensitivity would be comparable to the JPARC-SK experiment, but the higher energy of this neutrino beam improves the sensitivity to the sign of Dm223 (i.e., normal vs. inverted hierarchy). There is also discussion of a plan [104] to upgrade the AGS at Brookhaven to higher beam power and construct a neutrino beam aimed at the proposed new underground laboratory (NUSL) in Lead, South Dakota or some other underground location at a similar distance. This would aJord the opportunity to perform measurements with a large (0:5 Mt) detector at a distance of L ∼ 2500 km, as well as with a closer oJ-axis detector (L ∼ 400 km). Such measurements could convincingly demonstrate the oscillatory behavior of the Mavor transformations with high statistics. 4.4. Future direct mass measurements A new experiment to study the tritium 6-endpoint spectrum, KATRIN, is under development [105]. In order to improve the sensitivity to neutrino masses by an order of magnitude it is necessary to both reduce the energy resolution of the spectrometer (to ∼ 1 eV) and increase the tritium source strength (to improve statistical precision). The basic strategy for achieving these goals is to scale up the previous spectrometer design used in the successful Mainz and Troitsk experiments to a larger physical size, as shown in Fig. 16. The larger size enables a higher ratio of magnetic 3elds Bmax =Bmin to improve the energy resolution and a larger source acceptance to increase the luminosity. The previous spectrometers were 1–1:5 m in diameter and the new KATRIN design is 7 m diameter. Both a windowless gaseous tritium source and a condensed source will be utilized to enable studies of potential systematic eJects associated with the diJerent source methods. The goal for the tritium source is a column density of 5 × 1017 molecules=cm2 and a maximum accepted take-oJ angle of max =51◦ . These parameters will enable sensitivity to neutrino masses in the sub-eV range, hopefully down to ∼ 0:35 eV. 4.5. Plans for double beta decay In contrast to the future direct neutrino mass measurements described in Section 4.4, the next generation of double beta decay searches is poised to address the mass scale indicated by the atmospheric measurements of Super-Kamiokande (Dm232 2:0 × 10−3 eV2 ) and perhaps even
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Fig. 17. Dependence of the eJective Majorana mass m66 derived from the rate of neutrinoless double beta decay 0 (1=T1=2 ∼ m66 2 ) on the absolute mass of the lightest neutrino. The stripes region indicates the range related to the unknown Majorana phases, while the cross hatched region is covered if one > errors on the oscillation parameters are also included. The arrows indicate the three possible neutrino mass patterns or “hierarchies”.
approach the scale of the solar solution (Dm212 1 × 10−4 eV2 ). For the case of normal hierarchy (m1 ∼ m2 m3 ), there is a rather 3rm prediction for the eJective mass relevant to 66 decay: 2 2 m3 |2 ; |sin 12 m2 |2 ] ∼ 10−5 eV2 : m266 Max[|Ue3
(47)
This mass scale of less than 3 meV is diRcult to address in the near future, but if nature actually chooses the inverted hierarchy (m1 ∼ m2 m3 ), then the 66 prediction becomes 2 2 m1 |2 + |Ue2 m2 |2 ∼ m21 ∼ 10−3 eV2 ; m266 |Ue1
(48)
corresponding to an eJective mass scale of about 30 meV. For the degenerate neutrino mass pattern (m1 ∼ m2 ∼ m3 Dm232 ) the eJective mass is larger than ∼ 50 meV, constrained from above by the mass limit from the tritium beta decay. The relation between the eJective mass in 66 decay and the mass of the lightest neutrino, evaluated for the solar solution (see Table 1), is shown in Fig. 17. Present estimates of the nuclear matrix elements [12] involved in the 66 process indicate that, with of order several tons of enriched material, experiments could reach this interesting mass range. There are many proposed experiments to address this issue in the near future. Most are still in the development stage, and of course the issue of backgrounds is critical in every case. Since the source mass is ∼ 100 times larger than in the present experiments, the background per unit
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Table 2 Proposed or suggested future 066 experiments separated into two groups based on the magnitude of the proposed isotope massa Experiment
Source
Detector Description
Sensitivity to 0 T1=2 (y)
Range of m66 (meV)
COBRA [106] DCBA [107]
130 150
Te Nd
1 × 1024 2 × 1025
700–2400 35–50
NEMO 3 [108]
100
Mo
4 × 1024
270–1000
CAMEO [109]
116
Cd
¿ 1026
¿ (70–220)
CANDLES [110]
48
1 × 1026
160–300
CUORE [111] EXO [112] GEM [113]
130
2 × 1026 8 × 1026 7 × 1027
50–170 50–120 15–50
GENIUS [114]
76
1 × 1028
13–42
GSO [115,116]
160
2 × 1026
65
Majorana [117]
76
3 × 1027
24–77
MOON [118]
100
Mo
1 × 1027
17–60
Xe [119]
136
Xe
5 × 1026
60–150
XMASS [120]
136
Xe
10 kg CdTe semicond. 20 kg enr Nd layers between tracking chambers 10 kg of 066 isotopes (7 kg Mo) with tracking 1 t CdWO4 crystals in liq. scint. several tons of CaF2 crystals in liq. scint. 750 kgTeO2 bolom. 1 t enr Xe TPC 1 t enr Ge diodes in liq. nitrogen 1 t 86% enr Ge diodes in liq. nitrogen 2 t Gd 2 SiO5 :Ce crystal scint. in liq. scint. 0.5 t 86% segmented enr Ge diodes 34 t nat Mo sheets between plastic scint. 1.56 t of enr Xe in liq. scint. 10 t of liq. Xe
3 × 1026
80–200
Ca
Te Xe 76 Ge 136
Ge Gd
Ge
a 0 Adopted from [12]. The T1=2 sensitivities are those estimated by the collaborators but scaled for 5 years of data taking. These anticipated limits should be used with caution since they are based on assumptions about backgrounds for experiments that do not yet exist. Since some proposals are more conservative than others in their background estimates, one should refrain from using this table to contrast the experiments. The range of the eJective masses m66 reMects the range of the calculated nuclear matrix elements (see Table 2 of Ref. [12]). Again, caution should be used since for some nuclei, in particular the deformed 150 Nd and 160 Gd, only few calculations exist and the approximations are even more severe than in the other cases.
mass must be correspondingly smaller. DiJerent proposals approach this issue diJerently, and in all cases substantial R&D is required. Nevertheless, it appears that several experiments will be mounted during the next decade with the goal of studying 066 with sensitivity below 30 meV. The Table 2 lists the proposed experiments we are aware of, together with the claimed sensitivity to the 066 halMife, and its crude translation into the sensitivity to the Majorana mass. Among the listed experiments, a few relatively small scale searches are actually running (COBRA, NEMO3). Others, particularly the ‘ton size’, will be mounted in stages with prototypes of 50–200 kg approved and funded at the present time for several of them (one, CUORICINO, the prototype of CUORE, with 40 kg of natural Te, is already running in Gran Sasso). Clearly, the 3nal decision as to
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351
which of these many ideas will be realized depends on the outcome of these prototype installations. Nevertheless, since they are still ∼ 10 times larger than the present experiments, we can expect substantial improvement in sensitivity relatively soon. 4.6. Cosmological input In the near-term future, substantial additional information from cosmologicalobservations will become available that will further constrain the sum of light neutrino masses f mf compared to the limits quoted in Section 3.4. Already in progress is the Sloan Digital Sky Survey (SDSS) [121] that will map over 106 galaxy redshifts and provide relative sensitivity in the relevant region of the power spectrum of ∼ 1%. When combined with the WMAP CMB data, this should provide sensitivity to m ∼ 0:3 eV [87]. Further in the future, higher precision measurements of the f f CMB anisotropies are expected from the PLANCK mission (now expected to be launched in 2007) [122]. An estimate of the combined sensitivity (1>) of the expected PLANCK data and the SDSS observations to neutrino masses gives the result f mf ∼ 0:06 eV [123]. 5. Longer-term outlook It is diRcult to envision what direction the study of neutrino mass and oscillations will take in the longer-term. Thus, we can only oJer educated guesses. Some of the future research will, obviously, depend on the results of the near-term future described in the preceding section. Among the results that might force a revision of the experimental program are the MiniBooNE tests of the LSND evidence for the T → Te oscillations corresponding to %m2 ¿ 0:2 eV2 , and the attempts to determine the mixing angle 13 . If MiniBooNE con3rms the LSND observation, many more experiments will be needed because the nature of the neutrino mixing matrix would be much richer (more than just three active neutrinos or CPT invariance violation) than envisioned so far and summarized in Table 1. Also, if it turns out that the mixing angle 13 is relatively large, perhaps close to the present upper limit, the study of the possible CP invariance violation in the neutrino sector will become considerably easier and will proceed faster than if 13 is very small. In addition, neutrino astrophysics will undoubtedly advance. Further studies of the solar neutrino spectrum will be conducted, and in particular the low energy part, the Mux of the 7 Be and pp neutrinos, will be likely determined in live counting experiments. These measurements will be valuable not only for the further re3nement of the determination of the oscillation parameters Dm212 and 12 , but also for better understanding of solar energy production. Moreover, with several experiments on-line, we can hope that in the near future the next galactic core collapse supernova will be observed by a variety of neutrino detectors. If that happens, the observation of SN1987A [124] which launched this 3eld of neutrino astrophysics, will be exceeded many times, and all components of the supernova neutrino spectrum will be observed. Again, such observations would not only advance our knowledge of neutrino physics, but will oJer a unique insight in the physics of the stellar core collapse. We also anticipate that during the next decade other applications of neutrino physics will become reality. For example, the observation of ‘geoneutrinos’, i.e. the Te emitted by the decay of radioactive
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U and Th series in the Earth crust [75,125,126] would enable determination of the corresponding radiogenic heat generation and provide an important contribution to geophysics. Another application is the measurement of the relic supernova Mux: the diJuse Mux resulting from past supernovae in the surrounding space, roughly up to redshift z = 1. By measuring this Mux one can determine the average supernova rate over a substantial part of the visible universe (for an estimate of the rate see e.g. [127]). At present, neutrino telescopes have not yet seen the TeV and higher energy neutrinos which likely accompany the production of cosmic rays with such energies. This might change in near future when the relatively small underground detectors are supplemented by much larger ones in open water or ice. Observation of the direction of high energy neutrinos would considerably advance the study of the origin of high energy cosmic rays [128]. It will also make it possible to search for neutrinos from annihilation of the so far hypothetical dark matter particles (WIMPs). It is beyond the scope of this review to describe the projects being built or proposed (see [129] for a recent review). Naturally, the whole 3eld of particle physics will advance at the same time. With the launch of LHC one can imagine that the existence of the Higgs boson will be experimentally con3rmed, and its properties will be, at least crudely, determined. One can also hope that the quest for the understanding of the two main paradigm of present day physics, the nature of the ‘dark matter’ and ‘dark energy’, responsible for most of the energy density of the Universe, will be advanced. These more general areas of particle physics could be, in fact, intimately related to the quest for neutrino mass and oscillations. Taking the see-saw formula ML = MD2 =MH , and using for ML the neutrino mass scale ∼ 10 meV and for MD the electroweak symmetry breaking mass scale ∼ 100 GeV, we arrive at ML ∼ 1015 GeV, i.e. close to the GUT uni3cation mass scale. If that relation could be 3rmed up, we will arrive at another determination of that important mass scale. The relation of the neutrino mass scale to the ‘dark energy’ is based on the, perhaps accidental, numerical coincidence. Remembering that the energy density of the ‘dark energy’ is about 0:7@c ∼ 3:5 × 103 eV=cm3 and rewriting it in the ‘natural units’ as A4 =(˝c)3 we arrive at the dark energy mass scale A ∼ 2 meV, close to the neutrino mass scale, and unlike any other mass scale in particle physics. (For recent attempts to relate these two mass scales see e.g., [130].) Future large scale projects to generate powerful new beams of neutrinos are already envisioned. To explore the possibility of CP violation in the lepton sector one would like to have a well-understood and collimated neutrino beam of a well de3ned Mavor and energy spectrum. Such a beam could be aimed at a distant large detector. Recently it has been pointed out that technology exists to construct pure e and Te beams of the required properties. Accelerating radioactive ions, 6 He (E0 = 3:5 MeV, T1=2 = 0:8 s, produces Te ) or 18 Ne (E0 = 4:45 MeV, T1=2 = 0:8 s, produces e ) to 100, would produce beams of precisely known energy pro3le extending to 2E0 , and collimated to 1=. The oscillation signature in the far away detector would be the appearance of + or − [131]. It is expected that O(1018 ) decays per year can be achieved. The beams of and T as well as e and Te neutrinos could be obtained in a neutrino factory where accelerated muons are stored in a ring with long straight sections. Such a beam will produce neutrino beams with equal mixtures of T and e if + are stored, and with Te if − are stored. It is expected that neutrino factories could provide O(1020 ) useful muon decays per year. Neutrino factories would thus provide beams with small systematic uncertainties in the beam Mux and spectrum. CP violation, and determination of sin2 213 even if it is as small as 10−4 − 10−5 can be achieved with such beams and baselines of several thousand km [132].
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Erratum
Erratum to “Computational modelling of thermo-mechanical and transport properties of carbon nanotubes” [Phys. Rep. 390 (2004) 235–452] H. Ra,i-Tabar Computational Physical Sciences Research Laboratory, Department of Nano-Science, Institute for Studies in Theoretical Physics and Mathematics (IPM), P.O. Box 19395-5531, Tehran, Iran
(1) In Sections 7.1.2, 7.1.5, 7.3.1, principle strains, principle axis, principle stresses, principle planes and principle plane, should be corrected to principal strains, principal axis, principal stresses, principal planes and principal plane wherever they occur. (2) In Sections 7.1.7, 7.1.9, 7.3, 7.4.1, 7.4.2, 7.8.1, Hook’s law should be corrected to Hooke’s law wherever it appears.
doi of original article 10.1016/j.physrep.2003.10.012. E-mail address: ra,
[email protected] (H. Ra,i-Tabar).
c 2004 Elsevier B.V. All rights reserved. 0370-1573/$ - see front matter doi:10.1016/j.physrep.2004.01.002
359
CONTENTS VOLUME 394 S.J. Lee, S. Souma, G. Ihm, K.J. Chang. Magnetic quantum dots and magnetic edge states
1
M. Caselle, U. Magnea. Random matrix theory and symmetric spaces
41
T. Andersen. Atomic negative ions: structure, dynamics and collisions
157
R.D. McKeown, P. Vogel. Neutrino masses and oscillations: triumphs and challenges
315
H. Rafii-Tabar. Erratum to ‘‘Computational modelling of thermo-mechanical and transport properties of carbon nanotubes’’ [Phys. Rep. 390 (2004) 235–452]
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Contents of volume
359
doi:10.1016/S0370-1573(04)00144-9
