Fractional Statistics and Anyon Superconductivity

(18)

,--, e 2 ':1ino:j(n (r , r" " ,..) , ~

K(r, r'; 'T) =

Kn(r, r'; 'T) =

f Dr( t) ei/t-l J: dr' L(t') 8( 8' -11- cp -

K"'n ( r,r ,.) ,'T -

M [ - M (2 '2)] 4'1Tlii'T exp 41ii'T r +r

2'1Tn) ,

(19)

(20) where I,,(Z) is the modified Bessel function. In our cas~. v:e have and

K{r, Th~refore.

M -Mr r'.,'T)'_- ,,-~oo (4.".lii'T )exp (2) 21ii'T e

in."

00

Iin-al

ir'l = IPrl = Irl·

r (M2) 21ii'T .

(21)

we arri ve at the result

(22)

This is. as expected. formally divergent. A convergence factor e - ~.~ is inserted in the integrand. with E -+ 0 at the end of the calculation. We use the result [15]

F,,(l + E)

-+

~ (1 + tfi) -".

(23)

V2E

As before. we appeal to eq. (8).

Expa~ding about Fermi statistics. ex = 2 j + 1 + 8.

179 123

D.P. ArOVQS et ale / Statistical mechanics of arryons

and In - al = 1 ± 8,3

± 8, etc. Thus,

B(2j + 1 + 8, T) = ~A2T + 2A\lim

£-0

(24) Expanding about Bose statistics introduces a term 1c5IA2T due to the In-al pi.!~e

=

181

in the sum Making the required periodic extension recovers the earlier result

of eq. (12). In some sense, the path Integral result is more satisfying, because, although one still is presented wit!l the delicacy involved with extracting the (finite) difference of two divergent expressions, there is no necessity to impose a finite volume constraint, which was originally effected in order tv perform the mode counting. One might object to our original calculation on the grounds that the virial coefficient might possibly be sensitive to the manner in which we perform the mode counting, since the dominant terms in the sum of eq. (9) are those at the tail end. As we have seen, this fear is unfounded. A striking result is the nonanalyticity of eq. (12). It would be interesting to know whether cusps also arise in higher-order virial coefficients. Due to the proliferation of the number of relative angles, such higher-order virial coefficients are exceedingly difficult to evaluate. In the high density limit, one might consider averaging the statistical flux over the entire system, and then consider the effect of a net statistical uniform. magnetic field of magnitude B = na~o' where n is the particle density. It is !,ossible t~ rep~oduce the correct form of the free energy to leacing order in n in this manner, however, one ioses perioGicity in a, and only cei tain values of a actuaHy yield the correct result. The most significant feature of the statistical interaction is that it is long rauged, hence perturbation expansions in a yield divergences and resummation is necessary, a situation reminiscent of the electron gas. Nevertheless, this statistics transformation process does yield a viable method for interpolating quantum statistics. The representation of a Fermi gas in terms of a Bose gas may be useful in other contexts, such as lattice field theory. Unless the statistical interaction is treated nonperturbatively, however, divergences may be difficult to handle. Finally, it is interesting to derive the lagrangian of eq. (14) for the solitons of the nonlinear a-model. Let us briefly recall that the model in question has a unit-vector order parameter nQ(x), a = 1,2,3 and a conserved topological current

(25)

180 124

D.P. Arovas et al. / Statistical mechanics of anyons

The conservation of Jp. licenses us to manufacture a U(l) "gauge potential" by the curl equation (26) The crucial point is that we could include a topological term (27) in the action, with 8 a real number (a == B1fT), which is analogous to the 8-parameter in quantunl chromooynamics. H is, in f&ct, the Hopf invariant describing maps of S1 to S2. In a suitable gauge, such as aA = 0, we can sohre for A,~ and so write H a~ a nonlocal interaction :unong th~ nil fields. The solitons are bosons for B = 0 (a = 0) and fermion'S for & = 'IT (a = 1). In a more general context, any conserved current Jp. can be c.oupled to the vector field Ap.. If the only other appearance of Ap. in the laerangian is the Chern-Simolls term [16] £p."pAp.a"Ap, then Ap. represents a nondynamical field [17] which can be eliminated to give a nonlocal interaction, which will impart anomalous statistics to particles carrying charge associated with the current. In ref. [3] the statistics of the solitons in this model were determined by invoking the linking number the:>l·em. Here we will determine the statistics directly by inter~hanging two widely separated solitons, and in the process elucidate the linking numbe!" theorem. For separations large compared to the sizes of the solitons we can approximate the solitons by point particles and the topological current by

JP(x) =

Ef d'T6(3)(X -

q.( 'T») ~q: .

(2S)

a

with a = 1. 2 and qa( 7') describing the trajectories of the two "point solitons. n We evaluate Hby inserting eq. (2S) into eqs. (26) and (27) and keeping only the cross-term~. The divergent self-interaction terms are evidently artifacts of the point approximation. To best understand the situation, we go to euclidean 3-space and think of eq. (26) as one of the time-independent Maxwell's equations V X B = i with the identification of AI' as the magnetic field B. Then H can clearly be interpreted as the work done on a magnetic monopole moving along the trajectory ql( 'T) by the magnetic field generated by an electric current flowing along the curve q2( 'T). With suitable normalization, this is just the number of times curves "1" aJ"d "2" wind around each other. We have thus made contact with the explicit form for the linking number between two curves given in mathematical texts [IS]. This discussion also defines the linking number between two curves which are not closed. TI) evaluate H explicitly, it is easiest to dis!ort one of the curves, say "2,n to a straight line q~( 'T) = 'T8p.o, as we are allowed to do. We find by eq. (26) that

181 D.P. Arovas et al. / Statistical mechanics of anyons

125

Ai = Eijxj /r 2, Ao = 0, a pure (but topologically nontrivial) gauge. Once again, we

could have appealed to (2 + I)-dimensional electrodynamics, this time interpreting J 0 as B. These remarks make clear that the effect here is essentially the BohmAharonov phenomenon. It is sometimes convenient to transform to a singular gauge wherein A = 0 except along string singularities attached to each particle, across which A has a jump discontinuity of constant magnitude. In summary. the action describing N of these point particles is just

(29) Here Xu is a two-dimensional vector locating particle a and (Jab is the angle of pH!"ticle h relative to particle a. :neastlred i'rom the x-axis, say. The precedi~g discussion has boiled BH down to the second term in this equation. As we have seen, although this terrn is a total time derivative and appears as an interaction, it determines the stati$tics of the particles. In the original model, the solitons have topological charge Q = f d 3xJo taking on all integer values. The IQI > 1 solitons are unstable against breakup. In writing down eq. (29) we have included only Q = + 1 particles. It is easy enough, however, to include Q = - 1 particles as well by noting that the (+ -) "interaction" has opposite sign from the (+ + ) and (- - ) "interactions." DPA woula like to thank Stefan Theisen for makir.g the work of Gerry and Singh known to us, and for many useful discussions. This work was supported in part by the National Science Foundation under grants DMR82-16285 and PHY77-27084, supplemented by funds from the National Aeronautics and Space Administration. One of u.; (DPA) is grateful· for the support of an AT&T Bell Laboratories Scholarship.

Note ad~ed This work supersedes the prepri"! "Interpolating quantum statistics", NSF-ITP84-25, by two of the authors (F.W. and A.Z.).

References [1] [2] [3] [4]

F. Wilczek. Phys. Rev. Lett. 49 (1982) 957 Y. Wu. US Dept. of Energy preprint 40048-09P4 (1984) F. Wilczek and A. Zee. Phys. Rev. Lett. 51 (1983) 2250 R.B. Laughlin. Phys. Rev. Lett. 50 (1983) 1395 [5] B.I. Halperin. Phys. Rev. Lett. 52 (1984) 1583 [6] D. Arovas. R. Schrieffer and F. Wilczek. preprint NSF-ITP-84-66. submitted to Phys. Rev. Lett. [1] J.G. Dash, Films on solid surfaces (Academic Press. 1'.:l~8)

182 D.P. ArovQS et al. / Statistical mechanics of anyons

126

M. Abramowitz and I. Stegun, Handbook of mathematical functions (Dover, 1972) R.P. Feynman, Statistical mechar.ics (Benjamin, 1972) Y. Aharonovand D. Bohm, Phys. Rev. 115 (1959) 485 Gerry and V.A. Singh, Phys. Rev. D20 (1979) 2550 A. Inomata and V.A. Singh, J. Math. Phys. 19 (1978) 2318 Gerry and V.A. Singh, Nuovo Cim. 73B (19g3) 161 S.F. Edwards anj Y.V. Gulyaev, Proc. Roy. Soc. London A279 (1964) 229 I.S. Gradshteyn and I.M. Ryzhik, Table of integrals, series. and products (Academic Press. 1980) 1. Schonfeld. Nucl. Phys. B185 (1981) 157; S. Deser, R. Jackiw and S. Templeton. Phys. Rev. Lett. 48 (1982) 975; Y.Wu, Washington preprint (1983) [17j C.R. Hagen, R0Cheste~ preprint (1933) ~181 H. Flanders. Differential forms (Academic Press. 1963)

[8] [9] [10] [11] [12] [13] [141 [15] [16J

c.c.

c.c.

183

PHYSICAL REVIEW LETTERS VOl.UME

II JUNE 19H4

5!

NU~Ulf.M

24

General Theory for Quantum Statistics in Two Dimensions Yong-Shi Wu Deparlmnr; o/Physics, Uitiversiry 0/ Wash,ngtoff,

~Qitle,

WQshillglOn 98195

(Received 9 April 1984) Bec.aus~ or ~omplkatcd h)j')Olog~' of the configuration space for ;ndistinguishable particles in two dimensions. Feynman's path-integral formulation allows exotic statistics. All oossible quantum statistics in two-space are characterized by an anile parameter (J which interpolates between bosons and fermions. The current formalisms in terms of topological action of multivalued wave functions can be derived in 8 model-independent way.

PACS numbers: 05.30.- d, OJ.65.-w

Physics in two spatial dimensions is not always simpler than physics in three or higher dimensions. By now the w~lI-known examples include the appearance of both fractional angular momentum or spin l - 3 and exotic statistics.4-7 Both of them are forbidden in space of dimension ~ 3; in fact. they are related to special topological situations in twospace. s The physical relevance of these exotic st:~'istics is expected to be in condensed matter physics where two-dimensional systems have become experimentally available. 9 Since a wide variety or model Hamiltonians can be constructed for various
~

K(q'(:qr>= a

~E

WI

X(a)f.

q(t}Ea

expUfq'dtLI f-

I

composites I and topological solitons. 2 In particular, are there more exotic statistics other than those found in these models? This Letter is devoted to discussing these questions. I will work in Feynman's path-integral formalism of quantum staiistics. lo The propagator for a system is a sum over all continuous paths in the configuration space connecting the initial state q and the final state q'. Since the configuration space of n ir.distinguishable particles, Mil' is always multiply connected (see below), paths in different homotopy classes cannot be continuously deformed into each other. Thus the propagator is actually a (weighted) sum over "partial amplitudes," each being an integration over paths belonging to a distinct homotopy c1
!/q(r).

The homotopy classes a of paths from q to q' can be made to be identified with the elements of 7T1(MII), the fundamental group of Mil' by choo;)ing a mesh of "standard paths" from a fixed point qo to every point in Mil and adjoining the path qq' to the standard ones qoq and q' qo to form a loop. The point here is that, as in quantum mechanics on any multiply connected space, II the complex weights

(I)

X(a) for different homotopy classes have no

Q

priori reaSO.l to be the same. Invariance under dif-

ferent choices of the standard path mesh and the composition law for the propagator require that X(a) must be a phase factor and form a representation or' 7T1(MII )}O What is Mil? For distinguishable particles, the

© 1984 The American Physical Society

2103

184 VOLUME 52, NUMBER

24

PHYSICAL REVIEW LETTERS

configuration space is just Rtbt. Rtlx ... X Rtl (n factors). where d is the dimension of space. For indistinguishable particles. there is no physical distinction between points in Rtbt which differ from each other only in the ordering of particle indices. So the symmetric points in R· (under the action of the symr.etric group of n objects. S,,) should be identified [e.g .• (1"1. T 2 , · · · , r,,)-(T 2, 1"1,···, r II) J. Because we are nOI always guaranteed that there is a finite probability for two particles to coincide with each other, the so-called diagonal points in Rtbt, D-I('f l , · · · . "f ll ) with r,-T j for some ; ~ jl. have to be excluded too.12 Thus the configuration space of n indistinguishable particles is Mil - (Rdrt- D)/SII' For d?:J, r.1!R 46t _·!)-'l, so that 11'1(1.1.) --5". There are (\nly two one~dimensiona! represen~ations of SrI: )(. (ad - 1 for aU a and )( _ (Q ) - ± 1 ac,=ording as Q is an e\'~n or odd permutation. The physical meaning of )«0) is now cleer: it determines the statistics. In three or h:gher dimensions there are only two kinds of statistics. either Bose-Einstein (with X+) or Fermi-Dirac (with X_ ). with no possibility for exotic ones in a pathintegral formulation. lo In two dimensions ft'l (M.) is muc~ more complicated; it is an infinite non-Abeiian group. Fortunately, its structure has been clarified for some time.13 I give a pictorial illustration as follows. Recall that a c(",,3ed path in M" can be represented by n curves in the three-space (x,y,t) with no intersechons and with the final positions in R2 at !. being just permutations of the initial ones at I. I display the equivalence classes of these curves c: projecting them on a fixed x-I plane. To distinguish, Ihe projections on the plane will be calJed strings. Without loss of generality, we can assume that (0 the initial pos::ions of the strings are all different (i.e .. XI <...: ..• < XII)' (2) at each time slice there is at most one intersection of two neighboring strings. and (3 J the strings are a!ways parallel to the I
11 JUNE 1984

of the strings at the intersection be in front if the corresponding curve in three-space has smaller ordinate at that point. Such a ·configuration of strings is called a braid. 14 (Some examples are shown in Figs. 1 and 2.) The multiplication of two braids follows from that of two closed paths in M", so that the equivalence classes of braids under continuous deformation also form a groul'. called the braid group, B.(R 2 ). From what is said above, it is isomorphic to 1f'1(MIi ) . Some f~atures of the braid group are easily recognized. Denote by U, the operation of interchanging two neighboring strings at X, and x{+ i with the left one in fronl. Then a braid can be algebraically expressed as a produt:t of a sequence c r U (l:s;; i' ~ n - 1). From Figs. 1 and 2 it follow:;

/-1

(2)

There are no further relations among U ,'s. IS From Eq. (2), all one-dimensional unitary representations (\f 1f1(M,,) satisfy )(,(CTI)- ... -)(,(u __ I)-e-".

(3)

and are labeled by 9(0 E:; 9 < 2ft'). As a natural generalization of the interpretation of X(a) from three to two dimensions, the X,(a) in Eq. (I) represent new types of statistics, which we call 9 statistics. It interpolates the Bose-Einstein (9 - 0) and Fermi-Dirac (9 -7r) statistics. (An example for such interpolation is known in one dimen· sion. t6 ) To have more explicit understanding of (I statistics, we need more knowledge of X,(a). et E 7r 1( M,,) is always a product of a sequence of U ll. Physically, Uk represents interchange of only the two particles at r k and T Jc. I along a counter:' clockwise loop with other particles kept outside. Since we label particles by their initial positions and the particles temporarily at r k and T t + I can be any two of them. we can rewrite Eq. (3) as )(,(ut±t)-e;"-exp(-;(9/7r) I~~ul.

(4)

1<1

where ~~u is the change of the azimuthal angle of

(01

(b)

FIG. 1 Two braids for n - 3.

2104

(0)

(b)

FIG. 2. Two braids for n - 4.

185 VOLUME S2, NUMBER

PHYSICAL REVIEW LETTERS

24

.".

f dl ~ I4>v( 1)1·

(5)

ut 1<)

JUNE

1984

Note that the r~ght-hand side is indeed a homotopic invariant. When we insert Eq. (5) into Eq. (I), the righthand side of Eq. (5) includes also the contributions from the standard paths qoq and q'qo. However, they contribute only an overall phase factor to K so that we Cll,", neglect them:

particle; relative to particle j. For each 0'", only [)ne term in the sum is nonvanishing and its value is fr. This formula can be easily generalized to arbilrarya E frl (M.): X,(o) =expl-;!.

II

K(q't'~qt}- f exJiffdliL-!. ~ I4»u(dll9 q(t}. Y\ Tf'ul,<}

f

(6)

,

Here q (t> is a path in R·- D. but paths with inilial particle positions differing only in permutations should be included. Thus, the inclusion of X,(a) is equivalent to addition of a topological action which does not affect the equations of motion but determines slatistics. 6 There is another way to el!minate the phase factors X(a) and the sum over Q'. Let us consider the sel, M". of all equIvalence dasses of paths in M,. with a given fir.al point qo· Paths with diiferent initial points are necessarily in different classes. Mathemai;caJly. M. is identified with the universal cover;ng space of ,'4.. Closed lrops tbrough I point q in Mrc in different homotopy classes can be viewed as ope;, paths in it. fro~ point q to the corresponding points qa on different sheels. (We write the !let ion of 1TI(M,,) _on fl,. to tile rightl Thus the pilth intesral in Eq. (1) over qed E '" in M" car. be viewed as a propagator in M" from qa- I to q' corresponding to the original Lagrangian L:

k(q't'~qa-l,t)- f exp(/f':~ldIL)9Jq(t}.

(7)

Now from the wave function +(q,I), which is single valued in M" and propagates according to K, i.e., t/I(q',t')-

fM dqK(q't',qt)III(q,t}, "

we can define a new wave function ~(q,t) in

~(q,t) -exp( -

;(9/.,,)

(8)

M.:

f:o d( I4>v»).,,(q,t), f

(9)

I<}

wher~ the integral is along a path in M" which is identified with the point and its propagation obeys +(q',I')

-J:. dq K(q'I',q/)+(q,t>

q in M".

It is single valued in

M"

(10)

M.

since the phase factor in Eq (9) is chosen in accordance to Eq. (5). As J,(qa,1) -X(a-I)~(q,t}

(11)

by identifying all the points qa with q, J, can be also considered as a wave function (though multivalued) in M". Nothing is wrong with this multivalued wav~ function, because all branches have the same modulus, and the muhivalued phase factors a~ejusl rig!:t Lo keep track of the weights X(a). Equation (9) can be rewritten ii. terms of complex coordinat~s z,: ~ (ZI.Z,.~t) "With

1

0, <)(z,- Z)II/"1(:"z,.;1)

single valued and symmetric in iJairs o!" will satisfi' the ordinary SchrOdilller equation without the extra 9-dependent term. When n - 2, an exchange of the particle positions gives rise to a phase factor e""', m being the wind·ing number. As emphasized previously,' for a system having three or more particles, Eq. (12) exhibits even more complicated behavior under interchange of particle positions. Thus 9 statistics can be considered either as a na-

(z"z,.). This

+

(12) I

tural generalization of normal statistics in which 9 appears in the phase that the wave function acquires under exchange of particles, or as due to a peculiar long-range interaction arising from a topological action where 9 appears as a coupling constant. In the firc:t way, the notion of wave functions must be generalized. To conclude, some remarks are in order. First, Eqs. (6) and (12) have already appeared in Refs.

2105

186 VOLUME 52, NUMBER

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PHYSICAL REVIEW LETTERS

5-7. So all the results and conclusions derived there from these equations are generally true. Especially the 9 statistics in those models exhaust all exotic possibilities. Second, mathematically there is a very close analogyS of all this to 9 worlds l7 in gauge theories. In a path-integral formulation in the gauge A 0'" 0, the configuration space of a non-Abelian gauge theory is the quotient space sf I Y ,18 where d is the space of gauge rotentials in three-space and ~ is thl! group 0f giluge transformations ir. lh(~e-spce. Since TT,LcI;'[i )-11'0(:9 )-IT)(G)=Z, all one~imens:onal unitary re.presentations of TTl (." I ~~ ) are characterized by an angle parameter 9 too. The '· ... acuum angl!" (J appears either as a co,",pling constant in a tovoiogical actiun or in th: multi valued phase of a wave function!'1 in sll ~ which is single vallled-ln !he universal ccvering space d. [emphasize this prallel to show that nothing is ill defined or mysterious with 9 statistics. Finally, since the treatment is model independent, we expect the appearance of () statistics in two dimensions on a general ground. It is worthwhile looking fer tl,e signal for it in two-dimensional physical systems. The author has benefited from discussions with F. Wilczek and A. lee. He is indebted to D. C. Ravenel for explaining the braids and introducing 'he relevant mathematical literature. Particularly, the author thanks C. N. Yang for drawing his attention to physics in lower dimenSions This work was supported in part by the U. S. Depal.ment of Energy.

IF. Wilczek, Phy~. Rev. Lett. 48, 1144 (I982); R. Jackiw and A. N. Redlich, Phys. Rev. Lett. SO, 555 (\ 982); see also A. S. Goldhaber, Phys. Rev. Lett. 49, 905 (1982). 2F. Wilczek and A. lee, Phys. Rev. Lett. 51, 2250

2106

11 JUNE 1984

(983). 3See also P. Hasenfralz, Phys. Lett. 858, 338 (979); M. Peshkin, Phys. Rep. 80, 376 (982). 4J. Leinaas and J. Myrlheim, Nuovo Cimento B 37, 1 (977).

SF. Wilczek, Phys. Rev. Lett. 49, 957 (982). 6F. Wilczek and A. Zee, University of California at Santa Barbara Report No. NSF-ITP-84-2S, 1984 (to be published) 7y. S. Wu, University of Washing~on Report No. 40048-07 P4, 1984 ho be published). 'Fractiontllizcl,ion of spin is connecteLi lO thl! fael Ihat the universal coverin& group of SO(2) i~ noncompact. 9A well-known example is MOSFET (metal oxidesemiconductor interface). See, e.g .• K. von Klitzing, G. Dordil, and M. Pellper, ?hys. Rev Let~. 45, 49~ (J ~a(\). The ~s .. ible relevance c,f exotir. statistics to the frac\it)nBI quantized Hall effect has been suggesfed by D. t Halperh, Phy!>. Rev. Leu. 52,1583(984). 10M. G. G. Laidlaw and C. M. De Witt, Phys. Rev. D 3,

DiS

O~ili

ItReference 10 and L. Schulman, Phys. Rev. 176, 1558 (1968), and J. Math. Phys. 12, 304 (1971), and Techniques and Applications 0/ Path InteKlation, (Wiley, New York, 1981); J. S. LJowker, J. Ptlys. A S, 936 (1972), and "Selected Topics in Topology and Quantum Field Theory," Austin lectures, 1979 (unpublished). l2.f the diagonal is included, R·' Sit will be simply connected for any d. We would obtain only the Bose statistics, as expected. llR. Fox and L. Ne'Uwirth, Math. Scand. 10, 119 (961); E. Fadell Jnd J. Van BUskirk. Duke Math. 29, 243 (1962), :4E. Artin. Arln of Math. 48. 101 (941), ISThe proof of this stiitemeOl i:lo highly nontrivial. See Ref. 12 and also F. Bohnenblust, Ann. of Math 48, 127 (947). 16c. N. Y,mg and C. P. Yang, J. Math. Phys. (N.Y'> 10, 1l150969L Callan. R. Dashen, and D. Gross, Phys. Lett. 638, 334 (1976); R. Jackiw and C. Rebbi, Phys. Rev. Lett. 37, 172 (1976), IISee. e.g., J. S. Dowker (979) in Ref. II ~ and R. Rennie, UOIversil}' of CaHornia at ~3nta Barbara Report No. NSF-ITP-84-17. 19P4 (to be published).

17e.

187 Volume 214. number 3

PHYSICS LEliERS B

24 November 1988

ARE CYONS REALLY ANYONS?

Alfred S. GOLDHABER /" ((ifill(' (or Theoretical Ph.r.• i,·s. Stute (. ·"il'crsit.'· of Nell.' York.." SI"n)' Brook, SIn".\' Brook, NY J J794-3840. (lSA

and R. f\IACKENZIE

.\n unrcsol" "d ;ssu~ in f 2 + ( ,-dimensional quanlum m'xhanil"S is whethcr a ~ompo\jtl" ob.iel·1 form~d from a charyo-:i pllnide bound 10 a magn"lic Ilux. tube (:arrics fracllvnai angui .. r momcntum. We argut: Ittal this is indeed so. that the: result conlirms Wilczek's gc:nenllized conne'lion of spin and statistics. and that further confirmation is provided by the recently discovered induced charge and ..pin of nUll tubes in : 2 + I )-dimensional QED. a theory in wh!ch different spin and Slatistics are defined at short· and at long-distance !-Cales.

Several years ago, Wilczek [1] cor.structed and analyzed quantum mechanical and field theoretic systems which exhibit arbitrary values of angular momentum in (2 + I ) dimensions. That such possibilities exist is no surprise when one considers the character of rotational symmetry in (2 + I ~ versus (3+ I) dimensions. In the latter, more familiar case. the nonabelian nature of the rotation group implies that its representations may be labelled by a discrete index, the integer or half-integer angular momentum. In the former case, by contrast, the rotation group is abelian and has a continuum of representatio&s, corresponding to a continuum of allowed angular momenta"l. One can also generalize the notion of statistics beyond the usual cases of Bose and Fermi statistics [ I J. The generalized definition is expressed in terms of the phase change of the wave function upon interchange of two such objects along a counterclockwise path. In familiar cases the phase factor is ± 1, but an arbitrary value may be termed "fractional" statistics. Wilczek argued that the value of the statistics phase is just what one would expect from a generalization

.1

For related. earlier work, see ref. [2].

of the usual spin-statistics connection, namely that spin s corresponds to statistics exp(21ris). For objects with such fractional angular momentum and associated fractional statistics, Wilczek proposed the ilame "anyons". Perhapl:i the cleanest example exhibiting this behavior is the O( 3) nonlinear a-model with a term proportional to the Hopf invariant a~Jed to the lagrangian [3), where solitons in the model acquire an angular momentum as well as a statistics phase proportional to the coefficient of the Hopf term. Since the coefficient is arbitrary, fractional spin and statistics will result in general. Their values are consistent with the generalized spin-statistics connection hypothesis. Another example discussed by Wilczek is a composite object con~isting of a charged pu.:cle orbiling around a magnetic flux tube, called a "cyon" in ref. [4). Wilczek argued that the angular momrntum of such a composite is proportional to the flux, so for arhitrary flux the angular momentum can take on any value. These objects can be studied in (3 + 1 )-dimensional theories which are independent of one spatial direction (a flux tube along the z axis, for example) or in theories which are (2+ 1 )-dimensional at the

© Elsevier Science Publishers B.V. ( North-Holland Physics Publishing Division)

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188 Volume 214, number 3

outset. Both approaches have their subtleties: the return flu,,- in (3 + 1) dimensions, spatial infinity in (2+ 1) dimensions. (For a related discussion in the context of the Aharonov-Bohm effect, see the review by Peshkin [5].) The cyon was re-analyzed by Jackiw and Redlich [ 6 ]. who did a careful study of the intI icacies involving gauge freedom and the distinction between mechanical and canonical angular momentum. They concluded that the angular momentum of the composite or·ject is always integral. in contrast to \\'iicz~k's ass:!rtion. (While this d~bate exists regarriing the spin of cyons, it shouid oe mentioned that th.eir statistics is not In doubt: in gener~! it is fraC!loM.l, e'lu~ to that value one would infer from a generalized spin-statistks cor.nection using WilC7.:ek's propcsal for the spin. ) In this letter we argue that the analysis of Jackiw and Redlich is mathematiUllly correct and consistent with that of Wilczek, but that the discrepancy lies in whether or not the "full" conserved, integer-valued canonical angular momentum should be associated with the composite object. In essence, our conclusion is that the angular momentum of the (2 + 1 )~imen­ sional univene remains integral, but that this angular momentum is divided into a piece localized near the cyon which is fractional in general. and a piece located at spatial infinity (for a static configuration) which is also fractional and which makes up the discrepancy betwee~! the two poinls of vu~w. If we associate the localized angular momentum with the cyon, the diffuse angular momentum being unimportant for description of local phenomena, then the cyon is indeed a Wilczek anyon. In (2 + I ) dimensions, a particle 10 the presence of a flux tube, or solenoid, centered at the origin can be described (following the notat:on of Jackiw and Redlich) by the lagrangian

L= iJW 2 + (e/c),,·A . where the vector potential has the form Ai= (E iirl2xr2) 4)(r) .

This corresponds to magnetic field B = (2xr) - I X d4)(r)/dr. We take 4)(r) to increase from zero at the origin to a constant value 4) for r~ R; and Rand 4) are the solenoid's radius and flux, respectively. For the moment we consider the case where the panicle

472

24 November 1988

PHYSICS LETTERS B

remains outside the solenoid; we shall comment later on what happens if the particle is allowed to penetrate the interior. The two types of angular momentum relevant to the discussion are the mechanical and the canonical; these are given, respectively, by

Lm=rXJll', and L~

=rXp=rx [,w.'+ le/c)A J .

Wilczek asS(!rts that the angular momen!um of the cyon is equal to Lm. In quantum mechanics, replacingp by ·-iVyte11s Lm =rX (_. j'q -e.4/c). A panic!.: outsid~ the solenoid ha~ Ai = (211'r1) - IE iJrjt/J, and hence angular momentum m - 4> for a wave function with an~lar dependence exp(imql~, where t/J is the flux expressed in units of the usual flux quantum: ~=4J l{2ne/e). Jackiw and Redlich, on the other hand. correctly point out that the total angular momentum is equal to the canonical angular momentum, namely Le=rX (-iV), which has the value m for the same wave function. Should one associate Le entirely with the cyon? To clarify this point, one may ask where the angular momentum arises physically. The difference between the canonical and mechanical angular momfmta is that contair.ed in th~ electroma~netic field.

Le":: -

J... 2JtC

J

d 2X' r' ·E(r' )B{r') .

Let us follow the disposition of angular momenlum as a flux tube is created. Imagine turning on the solenoid with, e.g., a CU'T~nt source j=,(4) I nRl )I( t )<S( r- R), wheref(t) is a function which goes f:-om zero to one. This current generales not only the magnetic field of the solenoid, but also an outwardtravelling field whose flux is equal and opposite to that of the solenoid. Initially there is no flux, and the angular momentum is integral. In turning on the solenoid, the total angular momentum is conserV'ed, but a pan of it, in general fractional, is carried away by the radiated field. For phenomena on finite length scales, after a finite time this angular momentum is no longer relevant and the cyon retains fractional angular momentum, even though the total angular momentum of the universe remains integral. . Two remarks are in order. First, the situation in

Volume 214, number 3

189 PHYSICS LEtTERS B

~-1 ) dimensions is very similar to that discussed here. Ttte role of the (~+ 1 )-dimensional radiated electromagnetic field is played by the return flux of the solenoid, assumed long but finite. Peshkin [S ] has examined the (3 + 1 )-dimensional case, and finds, as expected. that it is the return flux which carries the electromagnetic angular momentum. Again, fe-local phenomena (radii much less than the length of the solen.Jid, whence the (2 + I )-dimensional description is sensible) the e!ectromagn,- tic angular momentum is irrelevant. Second. we have assumed that the charJed particle does not penetrate th..: region of magnetic fl~x. This is l'Iot central to the argument. If the charge is allowed into thai P!gior., the nC:l result i~ the same, n&mely that the locaEzed angl1lar momentum is fractional, equal to -
q=-~/2.

Now, if N anyons are composed. together to make an "N-anyontt then the statistics phase aN must be N 2 times the phase a for a single anyon. This follows since in interchanging two such objects, each anyo~ in the first N-anyon must circle around each one in the second, 2nd all the phases add. Ther!fore, the phase for arbitrary flux should be (constant) X tJ 2. The spin-statistics connection then implies a spin of (constant/2Jt) xtJ 2• We may use the considerations in ~~ earlier part of this paper to deduce the value of the constant. Imagine two identical vortices carrying opposite fluxes, initially coincident and then adiabatically separated. Radiation should be negligible, so 1I)at the total angular momentum should remain

24 November 1988

zero. This total angular momentum is the sum of the induced spins of the two vortices (equality follows because the spin is proportional (0 tIJ 2 ), and the angular momentum of their crossed fields, which is equal to - tJ 2/2, independent of the choice of origin for the position coordinate T. There is no other gaugeinvariant local function of the defined variables which could contribute to the angular momentum density. The result for the induced spin of either vortex is

s= l~l, agreeing perfectly in absolute magnitude with the result of Paranjape, the discoverer of spin induction for this system [8 j. I !owever. the sign of Paranjape's spin i:i oppc.3ite to ours, which we b~lie .'e IS ltlribalable to an additive shift due to ttis u!;c of the JackiwRedlich rather than the Wilczek definition of spin_ The aoove argument suggcs,~ the cuthne of a general deriv&llOn for the fractional spin-statistics connection. If instead of a vortex-antivortex we consider two identical vortices, the system angular momentum should be preserved in magnitude but reversed in sign, and therefore should be twice the spin of either particle. A rotation by 1f generated by this system angular momentum is the gauge-invariant way to describe the spatial interchange of a pair [9]. This gives at once the statistics phase factor exp (21fis). A nice consistency check with the generalized spinstatistics hypothesis is found for one flux unit, 4J= I. In this case, adding an electron to a vortex can produce a system in which the charge, the spin, and therefore presumably also the statistics phase, are all conjugated. At the same time, these quantities all shift, bye, - 1/2, -1f, respectively. This confmns the already-deduced values for the vortex alone of - e/ 2, 1/4, x/2, respectively. There is another length S\:8le in this problem which so far we have implicitly assunied to be arbitrarily larg~: the Compton wavel~ngth of the photon, con5e1uent on the "~auge-invariaDt topological mass" term [ 10] in the effective lagrangian which summarizes the induced charge (as well as current) effects already described. On length scales comparable to this, electric or magnetic fields decay exponentially, so that on still larger scales, by Gauss' law the net charge must be zero. This implies that for vortices the total induced charge, and hence the total induced flux, must vanish, along with the spin and statistics. 473

190 Volume 214, number 3

PHYSICS LETTERS B

Since all contributions to the angular momentum are now localized, the Jackiw-Redlich and Wilczek definitions of spin coincide in this context. On the other hand, if an elementary charge Q is introduced the charge must be screened, and therefore.it acquires a net flux. The result is an anyon with spin Q2 / e 2 and corresponding statistics. Thus spin and statistics are defined for length scales either much smaller cr much greater than the photon Compton wavelength (although theIr values at these two scales are not the same). At scaks in between there are apvreciable velocity-dependent forces and torques. and part of the ~Iectromagnetic cloud associated with one particle overlaps y;ith that of the oth~r m~mb·:-r of ~ pair; hcnC'~ spin a:1d s':J.tisti:s are ambiguous. Vv(. conclude that Wilczek's hypothesis of a gcner,hz:.'d spin-statis~i,=s connection applies t~ cycn~. as well as to a variety of object:; which appear in different regimes of (2+ 1 )-dimensional QED. As we were finishing this work, we learned of a study by Hansson, Roeek, Zahed and Zhang [II], with a somewhat different starting point but partly overlapping and entirely compatible conclusions. This work was supported in part by the US National Science Foundation and by ~he US Dep'lrtment of Energy. We appreciate the support of the UK SERC and the hospitality of the Department of Applied Mathematico;, and Theoretical Physics at Cambridge L'ni\·er.. ity dmin/' the early stages of this work. For hospitality during its completion R.M. acknowledges Harvard University. We thank Sudip Chakravarty

474

24 November 1988

and Frank Wilczek for helpful conversations, and H'lns Hansson and Martin Roeek for informing us about their work.

References [IJF. Wilczek. Phys. Rev. Lett. 48 (1982) 1144: 49 (i982) QS". [21 J.M. Leinaas and J. Myrhcim, Nuo\"o Cimento B 37 (1977) I: G.A. Goldin. R. MenikoIT and D.H. Sharp. J. Math. Phys. 22 (I'lSII 1664. (3) F. Wilczek :fcd A.. Zee. Phys. Rev. Lett. S I (1983) 2150. (4) \.5. Goldhaber Phys. Rev. Leu. 49 ll9S21 90S. I S I M. Pesh:':in. Phy:;. R~p. 80 (1981, 375. [ 611\ . .i ad.iw and A.N. Redlic-h, Phys. Re\,. Leu. SO (19a3) S5~.

[ , ) AJ. !..; iemi and G. W. Seme=tofT. Phys. I\.ev. Lett. S I (1983) 2077. . [8) M.S. Paranjape, Phys. Rev. Lett. 55 (1985) 2390; 57 (1986) 500; L.3. Brown. Phys. Rev. Lett. S7 (1986) 499; D. Boyanc .'slcy, R. Blankenbeckler and R. Yahalom, Nucl. Phys. B 270 [FSI6] (1986) 483; R. Blankenbeckler and D. Boyanovsky, Phys. Rev. D 34 (1986) 612; A.P. Polychronakos, Nucl. Phys. B 278 (19tt6) 207; M.B. Paranjape, Phys. Rev. D 36 ( 1987) 3766. [9] R.A. Brandt and l.R. Primack, Intern. l. Theor. "hys. 17 (1978) 267. [ 10) W. Siegel. Nucl. Phys. B I j6 ( 1979· • JS; .f .F. Schon~cld, Nucl. Phys. B 18S ( 1981) 157; S. Deser. R. Jackiw and S. Templeton. Ann. Phys. 140 (:'16:) 372. [ III T.H. Hansson, M. R~k. I. Zahed and S.C. Zhang. Phys. Leu. B 214 (1988) 47S.

19] Modern Phy~ics Letters A, Vol. 4, No.1 (1989) 21-3"1

e World Scientific Publishing Company

FIELD CORRECTIONS TO INDUCED STATISTICS ALFREDS.GOLDHABER Institute Jor Theoretical Physics. State University oj New York at Stony Brook. Ston}' Brook. NY I I 794·3840 R. MACKENZIE Depf.lrtment of Physics. Ohio Stilte Uni:Jersi!y. C(I/umbus. OH 43210

FRANK WJLCZEKLyman Laboraiory oj Physics, :larvard University. Cambridge, MA 02 J38

Received 7 October 1988

The contribution of explicit field terms appearing in the Lagrangian (Chern-Simons terms and 8-terms) to the calculation of particle statistics is analyzed. The Chern-Simons term halves the contribution of current~field coupling terms in 2 + 1 dimensions. The eff~t of the 8-term in 3 + 1 dimensions is contrasted. In both cases, observational differences between fundamental and induced charges arise. The factor 1/2 is necessary to reconcile Paranjapets result on the angular momentum (L == 1/4) of a '{ortex in minimal massive QED with the general spin-statistics connection. Issues ,elated to the (non)screening of the statistical interaction, ultraviolet sensitivity of induced statistics, and duality in the quantized Hall effect are mentioned. It is shown in passing that quite harmless-looking field theories generically contain particles with exotic statistics.

1. Introduction Analysis of the existence, in 2 + 1 dimensions, of statistics interpolating smoothly between bosonic and fermiunic was inspir(d by the discovery of fractional angular momentuffi around gauge-theory vortices, and the desire to maintain a spin-statistics connection. 1 Although it was immediately recognized that electron-flux tube composites in BeS superconductors also undergo statistical transmutation (fermion + boson -. boson), the topic was widely regarded as rather academic. Very recently, however, the situation has changed, and interest in fractional statistics particles-anyons-has increased sharply. The possibility of peculiar Quantum statistics in 2 + I dimensions, and of their implementation in terms of Lagrangian field theory, has been found to be relevant to the description of Quasiparticles in the fractionally quantized Hall effect (FQHE). Indeed,' Girvin and MacDonald, 2 and Read, 3 have formulated and in some measure derived effective Lagrangians for the FQHE that are • On leave from ITP, Santa Barbara. 21

192 22 A. S. Goldhaber, R. Mackenzie & F. Wilczek

essentially identical to ones previously discussed abstractly, as the simplest possible realizations of fractional statistics. Very recently, Laughlin has argued forcefully that the essential physics of the FQHE state, namely, that it is an incompressible quantum liquid, is likely to occur in other contexts. 4 (Anderson has expressed similar views. s) Specifically, he propose~ to describe Mott insulators this way, and has erected a qualitative theory of the high-Tc superconducting ceramics upon this foundation. He proposes "half-fermi" statistics for the ouasi-particles in these materials. In this note, we shall briefly review, correct slightly, but significantly, and apply the abstract fractional statjstics framework. We find that half-fermions appear in the sixrapkst version of 2 + 1 dimensional QED, and that other, more genf!Oll~ types of anyons arise easily in only slightly more ~laborate models. This clarifies Paranjape's earlier, controversial discovery offracticnal angular mornentum 1/4 in th~ simplest version, 6 and strikingly evokes Laughlin's proposal. We shall also make some comr,lents on analogous phenomena in 3 + 1 dimensions, on the !lon-screening of fractional electric charge in & Higgs phase, and on the importance of our factor 1/2 in the" FQHB. 2. Fractional Statistics Let us briefly recall some basics. A point 'vortex-tube' in 2 + I dimensions, like the infinitely thin solenoid it idealizes, involves a vector potential but no field strength outside a singular core. As Aharonov and Bohm taught us, this vector potential, although negligible classically, is of the utmost significance Quantum mechanically.7 Indeed, a charged particle in the presence of such a vortex disp!ays peculiar properties. If one imagines turning on the flux slowly, the particle feels a torque, and the composite object acquires angular momentum L = - q
(1)

where ~ is the vortex flux and q is the charge of the particle. a (Actually, there has been considerable debate, often of a quasi-theological character, concerning what is the uright" definition of angular momentum in this and other allied problems. 6,8 This point is tricky, because in 2 + 1 dimensions, a constant may be added to the "angular momentum without affecting its conservation or its commutators. For a summary and-we trust-final resolution of these contentions, see Ref. 9.) In many instances, flux is quantized: it comes in integer multiples of a basic flux quantum, 21C/ q, where q is the charge carrier. Binding a charge to such a flux

• Note that the U( I ) gauge group need not be, and usually will not be, ordinary electromagnetism. It is convenient, however, to use familiar terms" borrowed from electromagnetism, at least when confusion is unlikely to result.

193 Field Corrections to Induced Statistics 23

tube then results in an object with integral spin. A more interesting situation arises if, as in ordinary superconductivity, the condensate charge is not equal to the fundamental charge. In that case, the condensate consists of Cooper pairs of charge q = 2e. A flux tube combined with a spinless particle of charge e then has spin of magnitude ,(: = e(2tc/2e}/21C = 1/2: fundamental spinless particles combine to produce fennions. We will see below that, in a different context, the same amounts of flux and charge will result in ~omposites with spin 1/4. Next, consider two such objects. If we slowly interchange their positions, by transporting one around the other in a counterclockwise diiectioo, the usual j. A interaction generates a phase

(2) Indeed, the potential is A~ = cI>/2tcr, the distance travelled is nr in the direction of increasing " and there is a factor of two from the symmetry of the problem between the particles. Numerically, it arises as the stationary particle sees boosted potentials from the moving one. Now the phase acquired by identical particles~ as they are slowly interchanged at great distances, is a measure of their quantum statistics. Although it is largely a matter of semantics, the statistics phase is best defined as the negative of the interchange phase (2). The reason for the difference in signs, which has been treat.ed casually in the literature, is that the interchange phase (2) leads, in a gauge where the statistical interaction appears in the boundary condition of wave functions rather than as an explicit term in the T~~I"~"~t~.'1, to wave functions with angular dependence exp i(m":' q(b/tc)" where VI is the relative angle. (There is the potential for a similar confusion in signs with the angular momentum: for a single cyons for example, angular wave functions exp i(m - qc'b/21C)f/J arc associated with a phase of exp + iqct> for a rotation by angle 21C.) In any case, we see from (2) that vonices combined with charged particles can have unusual statistics. An immediate application of the above ideas is the observation that the statistics of 2 + 1 dimensional particles can be modified, by attaching fictitil)uS flux tubes and charges to them. For instance, one might choose to describe a fermion as a boson with additional structures of this kind. It is important to notice that the statistical factors described above are essentially the only dynamical consequence of the additional structures. This follows because the charge and flux added to the particle are entirely distinct from any dynamical fields such as the electromagnetic field. Notice also that the spin (1) and statistics phase (2) are just what one would expect from a generalization of the usual spin-statistics connection to include Ilnusual values of spin and statistics. An alternative, more elegant implementation of the same physics is as follows.

194 24 A. S. Goldhaber, R. Mackenzie & F. Wilczek

Let jp be a conserved current associated with a Lagrangian ¥, and add to the Lagrangian a tenn 10 (3)

Then the field equation for AJI is (4)

which impiies that a stationa~y charg~ of magnitude q (i.e., a unit charge of carries flux -q!/l. Thus it would appear, according to the pi'eViUU5 paragraph, that thi.s construction impl~ments the statistical phase exp(iq2 / J1). The last term in (3) is called a Chern-Simons term. It is readily seen to be gauge invariant. The construction above was implicit in the work ofWiltzek and Zee,11 spelled out by Arovas et al.,12 and rediscovered recently by Polyakov. 13

f d'2xj\)

3. Field Factor T~ere is a slight but significant subtlety involved in the construction, which seems to have escaped notice until recently.'4,'5 That is, in evaluating the action for ~n interchange of two particles, the Chern-Simons term itself generates a phase factor, in addition to the Olle generated by the Aharonov-Bohm factors associated with f d 3xj·A. Indeed, it follows from the field equation (4) tnat the contribution of the Chern-Simons term is minus one-half that of the AharonovBohm term. In other words, when the charge-flux connection arises from a ChernSlmons term, the statistical phase is one-half its nominal value (1). Thus. there is a significant difference between the statistics of ordinary charge/flux tube composit.es, and that of objects where the charge-flux relation is determined dynamically by the equation of motion of the gauge field. The statistics is not detennined exclusively by the value of the charge and flux, but by their origin as well. A notable application of this result is to 2 + I dimensional electrodynamics. A single massive charge-e two-component fennion contributes via vacuum polarization to lowest order in momentum, a Chern-Simons term 16

(5)

that does not vanish as Im I~ 00. This implies that vortices with the basic flux unit (J) = 21t1 e will carry charge e12. Furthennore, whereas ordinary composites with these values of the flux and charge would be fermions, according to our previous discussion, these objects have statistical phase exp(bt/2), i.e., they are

195 Field Corrections to Induced Statistics 2S

"half-fermions". This result is consistent· with a generalized spin-statistics connection, given Paranjape's calculation of induced angular momentum L = 1/4 for the vortex. It is also worth noting that, while these objects then have phase and spin half that of an ordinary fermion, combining two of them together makes a boson, since spin and phase go as the square of the number of units. 17,9 The above angular momentum is one-half that implied by (I); the spin, like the statistics, is half the "naive value" when the Chern-Simons term is responsible for the charge-flux relation. An intuitive explanation of why this factor of one-half arises is as follows. If we turn on the flux slowly, an induced azimuthal electric field will result which Imparts a torque on the charge which has been induced, thereby giving the :;ystem angular momentu:t1. This is exactly analogous to the situation discussed above with a fundamental charge. The distinction between fundamental and induced ("harges is that, since the induced c~arge is proportional to the flux, at in~ermediate stages in the process c,f turning on the flux, the amount of charge present is less than the final amount. As a result, the torque, and resulting angular momentum, is less when the charge is induced. Stated in mathematical terms, the ang~lar momentum is proportional to f qind(~)d~ rather than qfund f d~, giving a factor of two between the results for identical final fluxes and charges. It is most striking that half-fermions, which are central to Laughlin's speculations, appear so naturally in this simplest of 2 + 1 dimensional field theories. It is worth remarking that, just because a massive charged fermion willle~ve behind a term like (5) as 1m 1-+ ex:> but otherwise will decouple, there is a peculiar arbitrariness in the overall coefficient of such a term in the Lagrangian. The coefficient is not determined by other low-energy properties of the theory. We may also consider vortices with different values of the flux. The basic Aux unit = 21l/ e would appear in a Higgs phase if the condensate carried the same charge e as the fermions, but there is no a priori restriction to this. case. If the charge of the condensate is pe, the vortex will carry induced charge e/2p from the fermion field, and the statistical phase wit! be exp(in/2p2). In this \w.'ay, we see that quite simple models contain objects of arbitrarily exotic statistics. It goes without saying that inclusion of several massive fermion fields, perhaps with different charges, leads to still further possibilities. 4. 3

+ 1 Dimensional Analogue

We have seen that the statistics of 2 + 1 dimensional charged vortices depends upon whether the charge is intrinsic, or induced from a Chern-Simons term. In the latter case, the Chern-Simons term itself contributes a phase as the vortices are interchanged, over and above the Aharonov-Bohm phase. There is an analogous situation in 3 + 1 dimensions. It is well-known that a monopole with minimal charge 21l/ e induces, for an

196 26 A. S. Go!dhaber, R. Mackenzie & F. Wilczek

orbiting particle with electric charge e, half-integral orbital angular momentum. Thus, for instance, a spinless boson of charge e orbiting a monopole of minimal flux will form a composite dyon of half-odd-integral angular momentum. Such a composite would be expected to be a fermion, and indeed it is. IS On the other hand. it is also well-known that dyons with more exotic charges can be produced if there is a 8-term in the Lagrangian. 19 Indeed, with 8-term (6)

the basic monopole has (magnetic flux, el~ctric charge) = (21t/e, -efJ/21C) in this case. A naive generalization cf the result for a composite. dyon would lead to statistics phase exp( - j(/2), a disaster in this ~et~ing because, in 3 spatial dimensions, Ferm.i and Bose statistics are the only consistent possibilities. 2o One might be tempted to argue that enforcing consistency gives a q1Jantization condition on 8. There are many difficulties with this vie,,'. For example, in theories with axions, 8 is not just a parameter but a dynamical field which assumes a continuum of values. How do we recover consistency for arbitrary 81 Surely, the answer must lie in field corrections to the induced statistics,21 similar to the above, but now cancelling the naive phase altogether instead of merely halving it. We will now demonstrate in detail how this happens. The following discussion is b:tsically a refonnulation in continuum lo;~iguage of Cardy's treatment of lattice QED with a 8 tenn. 22 Consider the Lagrangian (7)

where we h9ve used the shorthand notation

r

pY

=

apA. - a.Ap-

21C -spY •

(8)

e

Here S is an antisymmetric tensor related to the monopole current mp by (9)

The appearance of spY in this form emphasizes its origin as an excess flux in compact QED. To bring out the physical content of this Lagrangian, it is useful to integrate out A. Since the Lagrangian is quadrati~ in A, this process reduces to Gaussian

197 Field Corrections to Induced Statistics 27

integrals in the familiar way. The effective interaction becomes

(10)

where D is the photon propagator. For our purposes~ since D is sandwiched between conser/ed currents, we m.ay take simply (II)

We organize all this as follows

(12)

where we have defined a new current (13)

The first term is easy to ili.tc~ret; it represents the star:dard interaction between electric currents. The appearance of j' is the precise indication that monopoles have acquired additional electric charge proportional to O. The next two terms may be combined using the identity (in momentum space)

(14)

Thus we find that they represent the "Coulomb" interaction among monopoles of charge 2tc/e. The fourth term gives the Aharonov-Bohm phase between electric and magnetic charges. However,. it.is important to note that it is only the old,

198 28 A. S. Goldhaber. R. Mackenzie & F. Wilczek

theta-independent piece of the electric current that appears here. What about the extra, theta-dependent piece that seemed to generate a paradox? That piece is just the fifth term. Fortuna!ely, as anticipated, there is a field correction: our sixth term. It is not quite obvious that these terms cancel, but the rather amusing identity (15)

shows that they do. This identity is most readily proved by forming a totally antisymm~tric tensor on five indices from the momentum q and two copies of s, and then contracting with q"e":pyj. Since in four dimensions a ten~or anti~ymmetric on Eve indices vanishes) the result mu.st vanish, and this gives us the required tdentity. Thus the potentiai v-dependent statistical interaction neatly cancels. Given this cancellation~ we may illustrate in this context, as in the discussion of Sec. 3, how dyon statistics reveals the coefficieot of the 8-term. As an example, consider SO(3) gauge theory broken to U(l) by a Higgs triplet, and let the theory also contain a scalar isodoublet, with electric charge normalized to ± e for the isodoublet. The theory supports monopoles of flux 21C/ e. If 8 = 0, the sector of the theory with (magnetic flux, electric flux) = (m· 21C/ e, n' 21C/ e) will carry the statistical factor (- )mll, where~s !f e ~ 1t, the factor will be (- )m(II-l). Thus the statistics of the d~/on is dependent not only on its quantum rlumbers, but also on how it acquires them-more precisely, on the value of the 8-term. 5. Comments i) We have been assuming in Secs. 2 and 3 that the Lagrangian con~ains no kinetic energy terms F;. . corresponding to the gauge potential field A. There is some interest in asking whether including a small term of this type changes the situation. Intuitively, we expect that the kinetic energy term should be negligible for lowenergy processes if the vector field acquires a large mass by the Higgs mechanism. A curious point arises here. It is commonly stated that in the Higgs phase electric charge is screened at large distances. If that were true in a strict sense, then the statistical factor exp(iq(l), which does, after all, refer to long-distance quantities, would become trivial. However, the uscreening" of electric charge is not an absolute constraint on the local charge operator, but rather a statement about its average value in the ground state. Now, while a condensate of charge (say) pq particles can adjust its density so as to insure the vanishing of the average value of charge in any volume-and thereby, in particular, screen the Coulomb interaction completely-there are some things it cannot do. In particular, charge pq excitations, however arranged, have no effect on the expectation value (16)

199 Field Corrections to Induced Statistics 29

if = 21C/pq, corresponding to a fundamental vortex. Thus the part of the charge that is not an integer multiple of the condensate charge pq is not fully screened, in the sense that there is an observable that responds to it. This observable, as defined above, is rather global and abstract. What are its physical implications? For one thing, it is precisely what governs the statistics of charge-vortex composites, as we have seen. More concretely, the factor (17)

appears as a prefactor in the Aharonov-Bohm scattering of a charge q off the vortex. 7. 23 Thus this cToss-sec~ion is a dirf!Cl probe of the fractional pa.rt of the charge. relative to the condensate charge. ii) As we lemarkoo above. the vCl.lue of the cceffici~nt of the Chern-Sitnons term can be affected by arbitrarily massive excltations. This has the imponant implication that for a given microscopic theory, it is impossible in principle to determine the correct coefficient by freezing the massive modes, and evaluating the effective action for the light degrees of freedom. The massive modes must be integrated out, not simply frozen. Thus various calculations of the closely-related induced Hopf terms, in which all but the light modes are simply frozen, are suspect (as some of the authors recognized). 24 In a constructive vein, let us note that the theories of interacting mass 1 ~ ~ fermions and gauge bosons mentioned above can be l~tticized readily, and thereby presumably plovide explicit non-perturbative models in the same universality classes as the conjectures of Laughlin4 or Dzyaloshinskii et al. 25 As mentioned before, in the minimal model, half-fermions arise. iii) The factor 1/2 discussed here i3 crucial in deriving the FQHE hierarchy from charge-vortex duality.26 Thus, this elegant duality comes to provide an additional argument for the appropriateness of the Girvin-MacDonald2effective Lagrangian, which in essence embodies a non-dYhamica~ gauge theory with Chern-Simons term. The following simple consideration contains the he:!rt of the matter. Charge q generates, according to the basic field equation (4), magnetic flux 4l = -q/p.. The statistics, taking our factor of 1/2 into account, is then 8 = q2/2p.. Fermions thus correspond to (18)

where m is an odd integer. Now basic vortices will have flux <1>., = 21C/q. According to the same field equation, th.ey carry induced charge Q., = - P.(21C/ q). Hence their statistics is (19)

200 30 A. S. Goldhaber, R. Mackenzie & F. Wilczek

or, using (18), 1[

8v =-·

(20)

m

This is the correct statistics for q'-lasiparticies in the FQHE. If we had missed out on the factor 2, then (1 7)-( 19) would have read (18')

(l9')

4n 8v =-,

(20')

.m

which is wrong. Acknowledgments This work was supported in part by NSF grants No. P:IY-87-14654, PHY-8217853 and PHY-85-07627, and by DOE Contract DE-AC02-76ER01545 .. We thank Alfred Shapere for enlightening conversations. R. M. thanks !..yman Laboratory of Physics, Harvard University for hospitality. F. W. thanks the Smjthsonian Institution for support as a Regents Fellow. References 1. F. Wilczek, Phys. Rev. Lett. 48 (1982) 1144; Phys. Rev. Lett. 49 (1982) 957. 2. S. M. Girvin and A. MacDonald, Phys. Rev. Lett. S8 (1987) 1252. 3. N. Read, unpublished. 4. V. Kalmeyer and R. B. Laughlin, Phys. Rev. Lett. 59 (1987) 2095; R. B. Laughlin, Phys. Rev. Lett. 60 (1988) 2677. 5. P. W. Anderson, Science 235 (1987) 1196. 6. M. B. Paranjape, Phys. Rev. Lett. S5 (1985) 2390; Phys. Rev. Lett. 57 (1986) 500; Phys. Rev. D36 (1987) 3766. 7. Y. Aharonov and D. Bohm, Phys. Rev. liS (1959) 485. 8. R. Jackiw and A. N. Redlich, Phys. Rev. Lett. SO (1'983) 555; L. S. Brown, Phys. Rev. Lett. 57 (1986) 499; G. W. Semenotf and L. C. R. Wijewardhana~ Phys. Lett. 184B (1987) 397. 9. A. S. Goldhaber and R. MacKenzie, Are cyons reallyanyons?, Stony Brook preprint ITP-SB-88-36, to appear in Phys. Lett. 10. C. R. Hagen, Ann. Phys. 157 (1984) 342. II. F. Wilczek and A. Zee, Phys. Rev. Lett. 51 (1983) 2250. 12. D. P. Arovas et al., Nuel. Phys. B2S1 [FS13] (1985) 117. 13. A. M. Polyakov, Mod. Phys. Lett. A3 (1988) 325.

201 Field Co"ections to Induced Statistics 31

14. T. H. Hansson et al., Spin and statistics in massive (2 + l)-dimensional QED, Stony Brook preprint ITP-SB-88-32, to appear in Phys. Lett. 15. X.-G. Wen and A. Zee, Phys. Rev. Lett. 61 (1988) 1025. 16. A. Niemi and G. W. Semenoff, Phys. Rev. Lett. 51 (1983) 277. 17. M. J. Bowick, D. Karabali and L. C. R. Wijewardhana, Nud. Phys. 8271 (1986) 417. 18. A. S. Goldhaber, Phys. Rev. Lett. 36 (1976) 1122. 19. E. Witten, Phys. Lett. 868 (1979) 283. 20. For an elementary treatment, see R. P. Feynman, R. B. Leighton and M. Sands, The Feynman Lectures on Physics, Vol. 3 (Addison-Wesle), 1963-65). For a discussion emphasizing the distinction between two and three dimensions, see Y.-S. Wu, Phys. R£>1J. Lett. 52 (1984) 2103; R. MacKenzie and F. Wilczek, Peculiar spin and statistics in 2 + J dimensions, OSU preprint OOE/ER/O 154.)-406. to appear in Int. Jour. Mod.

pn.vs. 21. F. Wilczek. Phy,;. Rev. Lett. 48 (1982, 1146. 22. J. l. Cardy, Nucl. Phys. 8205 (FSS] (1982) 17. 23. M. Aliord and F. Wilczek, Aharonol'-Bohm inleraction ofcosmic s:rings with matter, Harvard preprint HUTP-88/A04 7. 24. F. D. M. Haldane, Phys. Rev. Lett. 61 (1988) 1029; E. Fradkin and M. Sione, Topolog!cal terms in one- and two-ditnensional quantum Heisenberg antiferroMagnets, Illinois preprint P/88/4/44; T. Dombre and N. Read, On the absence of the HopI invariant in the long wavelength action of2D quantum antiferromagnets, MIT preprint (19g8). 25. I. Dzhaloshinskii, A. M. Polyakov and P. Wiegmann, Phys. Lett. 127A (1988) 112. 26. A. Shapere and F. Wilczek, Self-dual models with 8-ter ms, lAS preprint IASSNS-HEP88/36.

202 Leiters ill Mathematical Physics 16 (1988) 347-358. © 1988 by Kluwer Academic Publishers.

347

Quantum Field Theory of Anyons J FROH L lCH Tht'o"t'(;cal Phs;cs, ETH-Hol/ggerfJerg, CH-8093 Ziirfcl:. SlI'itzerlul1d ':nJ

P , . ,~ 1\ 1,-\ R C H E TTl [Jr·,n(/!'!1/1U'·:r.

F F:'.';t'a ddf'TJllirC'rsira di Pacio"a. IXi":'S Se:;C/1l1 di P(/e/m',;. J-351.H Pacio\'a, lfa~,'

AO';lract. hH a \linko\\'3ki sra(~timt: of dimensi011 three, particles of arbirrary. real spin and intermediate ~tatist\~s, ~alled ·WI,l't)I1/. are studied within the framewo~k of rela~i\'istic quantum field theory. The lo ... alization properties of interpolating fields for anyons and the relation between the spin of anyons and their statistics are discussed on general grounds. A mocel of a quantum field theory exllibiting anyons is described. Our results might be relevant in connection with the fractional quantum Hall effect and twe-dimensional models of high- Te, supercondu~tors. I, ( •• )

I. \\'hat are Anyons? In three spacctilue dimensions, the spin of a relati'\f'istic particle can be an arbitrary real numbt:r ...md the statistics of partidc~ can be intermediate between Bose and Fermi statistics. Particles with real spin and internlediate statistics have been called 'anyons' by 'W·ilczek [1]. He suggested that electrically charged magnetic vortices, occurring in certain Higgs models, may be examples of such particles. He recognized that if a magnetic vortex carries a fractional electric charge, its spin is f:actional and its stat~stics cannot be Bose or Fermi statistics. These findings follow from studying a variant of the ,\nJronov-Bohm effect A more :nathematical ana l yc;is of the statistics of anyons in l~rms of representations of the uraid groups [2] was presented by Y. S. \Vu [3]. Recent interest if} the physical properties of anyons has been triggered by developments in solid state physics: In the Laughlin approach [4] to the fractional quantum Hall effect, excitations carrying fractional charge and fractional angular momentum, whose statistics is intermediate between Bose and Fermi statistics, playa key role. Such excitations are also expected to arise in recent models of high Tc superconductivity [5]. Generally speaking, anyons can appear in two-dimensional physical systems with a (hidden or manifest) local gauge symmetry and broken parity iovariance. In this Letter, we present a general discussion of anyons within the framework oflot;al quantum theory. We discuss the localization properties of fields which have nonvanishing matrix elements between the vacuum and a one-anyon state with fractional spin ¢! 71.. We show that such fields cannot be localized in bounded regions of threedimensional spacetime (see Section 2). We thpn discuss the three-dimensional, noocompact Abelian Higgs model with a Chern-Simons term in the action which breaks parity and show that it describes particles with the properties of anyons (Section 3).

203 J. FROCHLICH AND P.-A. MARCHE1TI

348

[Another field theory describing anyons, mathematically essentially equivalent to the former, is an 0(3) nonlinerar a-model with a Hopftenn in the action, [1].] We construct anyon Green functions in terms of Euclidean region functional integrals, in close analogy to our construction of soliton Green functions [6]. The fractional spin, intermediate statistics and extended-particle structure of the anyons in our medel and a connection between their spin and statistics are investigated. A more detailed account of our results will appear in [7].

2. Spin of Relativistic Particles in Three Spacetime Dimensions Accordillg to \Vigncr. a relativislic part~cIe is describco by a l'nilary, irreducible representation of the quantum mechanical Poincare group, .?jJ T+ which is the un;vcr5al covering gro~lp of tlit.: proper Poi:1c~re group, }I'f+. In three spacetime

dimensions, ,qJT+

= SO(2,

I)

0

1R3 .

A unitary, irreducible represerltation of ~t+ corresponds to a Lorentz-invariant subset VM = {p E Mr-~ : p2 = M 2 }, i~2 real, of momentum space. For those representations which describe a physical particle, M2 is nonnegative, (M ~ 0 is the mass), and one chooses the branch, V ~, of VM for which pO ~ O. Representations corresponding to V ~, M ~ 0, are further classified by a unitary, irreducible representation of the subgroup of SO(2, I) which leaves invariant some momentum P E V ~, isomorphic to the little group. The Lorentz group SO(2, I) is homeomorphic to 1R2 x S I, where S I corresponds to ~ the group of rotations of two-dimensional space. Its universal covering group, SO(2, 1) is thus homeomorphic to 1R3. The little group for massive particles (M> 0) is given by the additive group of Sl = IR, so its irreducible representations are labelled by a real number, s, the spin of the particle. The little g!'oup for massless particles (.M = 0) is also given by IR. Hence, its ~rreducible representations are labelled by a real number, s, the he/icity of the particle. Thus, every relativistic particle in three spacetime dimensions is characterized by its mass M ~ 0 and a real number, s, its 'spin' (sp~ or helicity). These results have been established by Bargmann [8]. N ext, we imagine that a relativistic particle of mass M > 0 and spin s arises as a one-particle state of some local relativistic quantum field theory, more generally of some local quantum theory. As shown by Buchholz and Fredenhagen [9], one can then construct a 'field', v, with non vanishing matrix elements between the physical vacuum, 0, and that one-particle state. This follows from very general principles oflocal quantum theory: Locality, relativistic spectrum condition, existence of one-particle states. However, in general, the field v is neither observable nor local, but - as shown in [9] - v can always be chosen to be localizable ir.. a space-like come, ~ of arbitrarily small opening angle. Physically, rc can be identified with the location of a fluctuating string of (chromo-)electric flux emanating from a charged particle. Such particles can occur in

204 QUANTUM FIELD THEORY OF ANYONS

349

gauge theories; examples of (lattice) gauge theories exhibiting such particles, have been discussed in [10]. Now, it can happen that the field v can actually be chosen to be localizable in a bounded region of spacetime, or even in spacetime points. This is the typical situation in a three-dimensional field theory without local gauge invariance. In this situation, a general analysis, due to Doplicher, Haag and Roberts [11], applies and proves that particles created by applying t' to the vacuum, n~ have necessarily integer or half-integer .;pin, and tht,! U:5ual cunnection between spin and statistics holds. It follows that if the spin of a particie is r...!ither integer nor half-integer, the field v cannot be chosen to be localizable in bounded r~gi\)ns of spacetime~ aJthoueh it is stili localizabie in a space-like c()ne (~ \Ve therefore expc.ct SUC~l particles can only occur in gauge Ltteoli~s, or in thC0rteS with a hidden local gauge invariancc. In this situation, it is usually still f:rue that t . . . . 0 field Jperators, l' ,.,. and v'tt"' , locaiized on space-like separa~ed, space-like cones, c:c, ..~-' , [esp., can be chosen to commute or anticommute~ but this does not imply that the particles crt;ated by applying fields v ~ to th~ vacuum are bosons or fermions. This fact and the possible statistics of particles are discussed in Section 4. See [~, 11, 7] for details.

3. A Quantum Field Theory of Anyons 3.1. DEFINITION OF THE MODEL

starting point is the Euciidean description of quantum field theory (time pureiy imaginary). The objects of main interest are then gauge-invariant Euclidean Green (Schwinger) functions. From this point of view a model is given if there is a recipe for calculating all its Euclide~n Green functions. In this Letter, Euclidean Green functions will be expressed in terms of functional integrals with the help of the Euclidean Gell-Mann-Low formula. We work in the formal continuum limit, but all our fonnulas make sense in a lattice approximation and are then mathematically rigorous; see [7]. As an example of a relativistic quantum field theory in three spacetime dimensions exhibiting anyons, we consider the (noncompact) Abelian Higgs model with a Chern-Simons term in the action. In the Higgs phase, this model has topological solitons, the quantum vortices, which, in the presence of a Chern-Simons term., carry electric charge. We propose to construct a (nonlocal) gauge-invariant interpolating field, v, for the quantum vortices by giving formulas for all its Euclidean Green functions. The fundamental fields of the Higgs model are a complex scalar field, 4>, the matter field, and a real vector field, All' the gauge field. The Euclidean action functional, S, of the model is given by OUf

t

S8(A,4')=

r d 3x[_I_ (O[Il A V)2(X)+! l(oll-iA,J4'1 2 (x)+ 4e 2

JR3

2

+ A 14'14 (x) -

i

M2

.

]

-14>1 2 (x) + ;8, CS + counterterms ,

(3.1)

205

350

J. FROCHLICH AND P.-A. MARCHETTI

where (3.2)

The consta,lt e is the electric charge', A. and M~ are positive constants, and () is a real constant. The counterterms consist of vacuum counterterms and mass countertenns proportional to , ¢! 2, but e and ,t do not require infinite renorma!izations in three dimensions. and () does not renormalize. The action (3.1) is gauge-invariant if AlA tends to a pure gauge, asymptotically. A lattice approxilnation to this theory is discussed in [7 J. using a notion of l~ttice wedge product defined in [12]. For th~ lattice theory, the fOrlr'al discllssion that follows can be rendered mathelna~ica1ly rigorou~. If e'"!. and i.IM() are chosen small enough, the model describes a massh·e Higgs phase in which charges which are integer multiples of e are screened. If lJ:I= 0, AIJ is always massive. In the Higgs phase. the physical Hilbert space of the model contains massive, stable one-particle states carrying magnetic flux + 1 or - 1 which correspond to the vortex solutions of the classical equations of motion. [While this is a somewhat fonnal statement for the continuum Higgs model, it can be .proven rigorously for the lattice theory. See [6, 7].] These particles are called quantum vortices. For () i= 0, the qUaLtum vortices also carry an electric charge of ±28e and have spin ± (). They are then called 'anyons'. Classical 'anyon' solutions have been constructed in [13]. In the next section, we construct an interpolating field, v, for the anyons by giving formulas for it~ Euclidean Green functions.

3.2. 'ANYON' GREEN FUNCTIONS

Let F~oJ be tile classical electronJ.agnetic field strength of n magnetic (Dirac) :;nonopoles locate<': at the points x = {Xl' ... , Xn} in 1R3 with magnetic charges 10·= {m,},_ I, ... ,"' WitL OJr conventions, each m, mus~ be an integer multiple of 21t, and we require that 1:7_ 1 m, = O. Let be real-valued vector fields with supports in cones with the apex at x and contained in the half spaces {y: yO ~ XO}, {y: yO ~ XO}, respectively. Moreover, the fields are required to satisfy

E:-

fI:-

E:

(J .

E: )(y) = a(x - y) .

(3.3)

Suppose that, for i = 1, ... , k, x? < 0, while, for i = k + 1, ... , n, an electromagnetic field strength F [x, Ot, E] by setting

F#,v[x, m, E] = B;..tvp {

i

m,(E;')P +

1=1

f

m,(E;; )p} .

x? ~ O. We introduce (3.4)

l-k+1

By (3.3) and the definition of F(O), e#'VP

a#,(F~~) - F"p[x,

m, ED (x)

= O.

Therefore, by the Poincare lemma, there exists a globally defined vector field,

206 QUANTUM FIELD THEORY OF ANYONS (X1l[X,

351

m, E], such that

Fllv[x, m, E] = F~~ + O[Il(XV] [x, m, E).

(3.5)

We now define a modified Higgs action functional SO(A.

lP, [x, m, E))

+ counterterms] .

(3.6)

On the r.h.s. of (3.6), '1(0) denotes the covariant derivative on sections of the Dirac mOl&op' ,~",: bundle with curvature given by F(O). The action (3.6) has a di\'ergent contribution from the term

(the self-energies of the point monopoles). The divergence is removed by replacing that term by

1" m;mj - - L2e 2 i<j 47t Ix; - xjl

(3.7)

Note also that the cross-term

Finally, we add a gauge-fixing tenn, e.g.

to the actions (3.1) and (3.6). The resulting gauge-fixed, finite actions are denoted by S8(A, cp) and S8(A, lP, [x, m, E)), respectively. The Euclidean Green functions of 'anyon' fields, v[ x, m, carrying a magnetic

E:- ],

207 J. FROCHLICH AND P.-A. MARCHETTI

352

flux m and an electric charge 28me, are given by the following functional integrals tI(

G" x,m,

E) = rJL fl)AIl§Jq,exp[ -SO(A,-q" [x, m, En].] . J §JAil ftq, exp[ -SO(A, q,)]

(3.8)

reno

In (3.8), we require that r~',", 1111; = O. The subscript 'ren.' indicates that a mUltiplicative renormalilation of the quotient in [ ... Jren. is required: Diamagnetic inequalities [16] indicate that this quotient tends to O. as an ultraviolet cutoff. K, is removed. To make the expression well defined, one must multiply that quotient by n7_ I C(K)"", where C(K) div~rgcs. as the ultraviolet cutoff K is removed. See also [6. 7]. This explains the meaning o~ til~ r.b.s. of (3.8). It is important to note that (3.8) does not depend on our choice of F~:'j, subject tc the condition that the lilagnctic charges. III I' ...• In,;. and their locations. x I' ...• -'"". are kept fixed, i.e. (3.8) only depends on the cohomology dass of F~~.). In order to obtain a complete set of Euclidean Green functions. we can insert, in the numerator on the r.h.s. of (3.8), gauge-invariant functionals of All and q" such as products of :exp

[i J./"dX" + iLF\?~ dq"J. :, II'

,2; (x) • . .. .

Here, .!f?is a loop in 1R3 , l: is a compact surface boun~t:d by :R, and dO'llv is the surface element on r. The doublecolon~ indicate normal ordering. Formally, these Euclidean Green functions satisfy a varia~t [14] of the Osterwalder-Schrader axioms; Proof~ of these [Osterwalder-Schrader positivity is tied to our choice of the fields claims for the lattice models are outlined in [7].] Using a variant of Osterwaldt:r-Schrader reconstruction [14], we obtain a Hilbert space, .1f9, of physical states containing a vacuum, n 9, and a unitary representation of PI '!.. on .1f 9 satisfying the relativistic spectrum condition and leaving n 9 invariant. .Tf 9 contains 'multi-anyon states',

E:- .

I x, m, E +

),

(X? > 0, i = 1, ... , n) .

(3.9)

The scalar product between these states is given by


=

G:

+

,,(ri, x, - lit, m, r t

+,

E+ )

,

(3.10)

. where r denotes reflection at the time-zero plane, and n

L

i-I

r1

mj

=

L mi·

(3.11)

1 .. "1

If(3.11) is violated, the scalar product (3.10) is zero. The fields v[x,m,E;], XO ~ 0, are obtained from the Green functions (3.8) by Osterwalder-Schrader reconstruction, and n

I x, m, E +

)

=

n v[x 1-1

for 0 ::s:;;

x? ::s:;; xg ::s:;; ••• ::s:;; x~.

l , lill'

E:' ]0 9 ,

(3.12)

208 QUANTUM FIELD THEORY OF ANYONS

353

The Hilbert space .tf 9 decomposes into superselection sectors total magnetic flux (vorticity) m. Each vector m E JIf! satisfies

JIf!, labelled by the (3.13)

where QE is the total electric charge operator (in units, where e = 1). Hence, anyon states of total vorticity m carry an electric charge qE = 2mB. In particular, the states ! x, m, E + describe electrically charged, magnetic vortices, and 2m/BE:; describes the electric flux emanating from the vortex at Xi' i = 1, ... ,n. Because of Gauss' law, see (3.3), the electric charge of a vortex localized at X can be measured at an arbitrary distarl~e from x, if 9¢ ~l. Hence, for B$ ~l, the fieid operators v[x, ± C E; 1creating anyons cannot be repla~ed by operators localizable in hcundol spacetime regions, as expected, i.e. a1lyons are 'extended parLcles'. It is in.;tructive to explicitly construct the llnitary representation of space rotations on the states I x, m, E + ). Let 9l(cp) denote a space rotation through an angle
>

fJt,(cp)x

9l(cp) x ,

if Ixl <. r,

fJt«1 - t)cp)x,

if jxj = r + I, 0 ~ I ~ 1 ,

x,

iflxl>r+ 1.

I

=

Then the unitary represen tation lJ (cp) of d( ¢) on

U(cp) I x, m, E +

)

=

lim

r .... ex:>

1'~r(CP)x,

(3.14)

Jt".8 is defined b:,'

m, E + o9tr(cp»

(3.15)

which is compatible with equations, (3.8), (3.10); (for finite r, the diffeomorphisms rJt,(cp) do not affect the boundary conditions used in the definition of the r.h.s. of(3.8». From (3.15), (3.10) and (3.8), one easily derives, using the fact that the Chern-Simons term measures linking properties of electromagnetic field lines and cluster properties of the functional integrals defining G:(x, m, E), that (3.16)

.1f,:.

for all states m E Hence, states m E JIf': have total angular momentum Bm mod.l. In particular, a one-anyon state (m = ± 1) has spin 8 mod.l. Detailed calculations substantiating these claims can be found in [7]. 4~

2

Collision Tneory, Asymptotic Particle States

For the lattice theory, one can prove [6, 7] that for e2 and 1 small, M~ large enough, the fields v[x, ± 1, E: ] couple the vacuum 0 9 to a stable o;it-parlicle slate of mass M> 0, separated (at small momenta) fr~m the multi-particle thresholds by a positive upper gap.

209 J. FROCHLICH AND P.-A. MARCHETTI

354

It is likely, although not proven mathematically, that the same result is true in the continuum limit. We shall assume this henceforth. We then show how to construct collision states of asymptotic (incoming or outgoing) anyons with the help of a variant of Haag-Ruelle theory [9, 11]. For this purpose, we return to our basic formula (3.8) for anyon Green functions. We let the supports of the distributions E ~ shrink to wedges contained in two-dimensional olanes. These planes are supposed to arise from rotations of the time-zero plane by angJes 4>i around the x 2-axis, with 4>1 < ... < tPk < 0 < 4>k + I < 4>k + 2 < ... < 4>". The resulting Green functions can be analytically continued in the boost angles back to the Minkowski space region. From their boundary values, one can reconstruct ~field operators', v(m, ce), carrying magnetic flux m and localizaed in s?ace-like cones t:t in three-dimensional Minkowski space. Moreover, l'( ± 1. ~;) couples n tJ to a stable one-anyon state. The fields v( ± 1, CC) can be used tG con~truct a Ha:ig-Ruelll! collision theory in the form developed.by Buchholz and Fredenhagen in [9]. The collision theory yields Hilbert spaces Jf.Jt!,/out of incoming, or outgoing, multi-anyon stat~s. These spaces carry unitary representations, ~in/out' of ?J t+ • Let

I m,f<"»as, m = {m i }7= I' ml = ± 1, for all i, be

an asymptotic n-anyon state with momentum-space wave fune-tion /(")(m l , PI' ... ' m", p,,), PIE V~ == {p: p2 = M2, pO> O}. Then from fonnula (3.16) and the construction of asymptotic anyon states, it follows that

U..(27r) Im,j
=

exp [27ri8

(J. m,)'] Im,/(n» ...

(4.1)

This equation shows that the spin of an asymptotic anyon is

s = 8mod.Z and that there is a peculiar corr..position la\v for the spins of asymptotic anyons consistent with viewing an anyon as a particle carrying magnetic flux:n = ± 1 and an electric charge

28. Next, we tum to a discussion of the statistics of asymptotic anyon wavt"functions. From the discussion of Haag-Ruelle collisic,n theory in [9, 11] (and refs. given there) it follows that n-anyon states 1m, f<"» as can be obtained as limits of vectors, CI>" in the physical Hilbert space .1f 8, as t -+ ± 00, only for wave functions, /(")(m l , PI' ... ' m", p,,) which vanish for overlapping momenta, (Pi = Pi' for some i :F j), i.e. for wavefunctions defined on the space

,

(4.2) with

D"

= {Pi'· ... ,p,,: Pi = Pi' for some

i:F j} .

In fact, a more careful analysis [7] reveals that /(,,) must vanish except on simply connected components, M", of {PI' ... ,p,,: PI =# Pi' for i:F j}, for Im,/("» as to be a

210 355

QUANTUM FIELD THEORY OF ANYONS

limit of states cf), E Jlf9, as t -+ ± 00. Here, p is the spatial direction of a 3-momentum P E VA'i; the vectors cf) t are constructed by applying a monomial of degree n in the fields v(± 1, ~), smeared out with test functions that vanish for (Pl, ... ,Pn)tlMn, to the vacuum n 8. By taking limits of linear combinations of such asymptotic states, one can extend the space of wavefuuctions, f(n), to a space I)f functions defined over Mn, but vanishing on Dn' The space Mn is not simply connected. Its fundamental group, 1t t (Mn), is the pur~ braid group, Pn' which is a subg:-oup of the braid group, B on n strings [2, 31: Bn has generators'ti,i+t,; = 1, ... ,n - 1,describingacounter-clockwiseexchangeinM,,0i'the m;)ment3, Pi and PI..;. i ' of the ith and i +- 1st particle; (the inverse, !i~l+., describe~ a dockwise e,(chcia,ge f)f Pi and Pi+ 1 in M,,). The generators !/.i+ 1 satisfy the relations II'

-r... i +

t 'ti + 1,/+2 ti,i+ 1

'tit ; +

1

~ • .i +

1

= ~,j +

=

!;+ 1,/+2 ri,i+ 1

1 'ti ,; + I?

~i+I,;+2'

for I i - j I ~ 2 .

The pure braid group, Pn, is the subgrcup of Bn generated by the elements "iu = ti, 1+ 1 •••

.-2

-I

~ - 2, j - 1 'rj- I, j'tj - 2, j - 1 ••• t

-I i+ 1 •

(4.3)

"

Wavefunctions on nonsimply connected spaces, such as M n , need not be singlevalued, but correspond to single-valued functions defined on the universal covering space, here denoted by M,,; see [15]. A space of such multi-valued wavefunctiuns defined on M,. carries a representation, .1i, of 1t J (Mn) = Pn. This representation describes the ~statistics' of asymptotic particles. General quantum mechanical principles imply that yt is unitary in the scalar product of .J'f ~s. We now consider the transformation properties of n-particle wavefunction, f(n), under a rotation, U (21t), through an angle 21[ around the pO -axis: This rotation corresponds to the element

n n Yij "

rn =

j-.

(4.4)

j - 2 i- 1

of Pn , where Yij has been defined in (4.3). Hence

U (21t)f(n)

=

qJ(rn)f(n) .

(4.5)

This equation establishes a general connection between the angular momentum mod. I, or spin, of f(n) and the statistics of f(n), as described by the representation 9t. In order to explain more clearly in which way 91 describes the statistics of asymptotic particles, we consider the wave function f(n) (p), p = {p l ' ... ,Pn}' of an asymptotic state of n identical particles. Since all particles are identical, f(n) should really be viewed as a multi-valued function over

M!

= [(VA'i )xn\Dnl/1:n'

where 1:n is the pennutation group of n elements. The fundamental group of M~ is (4.6)

211 J. FROCHLICH AND P.·A. MARCHE1TI

356

A point in the universal cover M! = Mn of M! can be written as a pair [b, p], beBn' p eM!. A multi-valued wavefunction f(n) describing n identical particles is given by a single-valued function, 1 (n) [b, p], on Mn that transforms according to a representation, 9l, of Bn under covering transfonnations,

1 (n)[b, p] =

~(b 8 -I)

1 (n)[b, p] .

(4.7)

The values j (II'[b, p], bE Bn , correspond to the different branches of the wavefunction at the point p e M~. We have arguments suggesting that, independently of specific models, Yt is always given by the product of a one-dimensional representation, Xn' of Bit and a representation. Pn' of!',. de!)cribing para-Bose or para-Fermi statisti~s offinite order deterrrlii.1e(~ by the statistics ofthejields Vlif. The spin of fen) then determines X,~. thanks to Ec;uation (4.5): Every one-dimens!onal, unitary representation Xn has the fonn

/(f1)

(4.8)

; = 1, ... , n -

1; (4.8) clearly determines Xn(b), for all be Bn. Since I(n) describes n identical particles, 9~n) == 9(n) is independent of ; and, assuming standard cluster properties, 9(n) == 9 must be independent of n. By (4.4) and (4.8) Equation (4.5) now takes the fonn U(27t)/(n)

= exp[21ti9n(n -

1)]/(n) ,

(4.9)

i.e. the angular momentum of f(n) (mod. Z) determines the statistics parameter 9 (mod·4Z). If the angular momentum of f(n) is integer-valued, i.e. U(21t) = 1, Equation (4.9) implies that

9 = 0 or 1/2, mod.l.

(4.10)

By (4.8) this means that Xn describes Bose (9 = 0) or Fermi (9 = 1/2) statistics. By (4.3), Xn 1Pn is then trivial, and thus /(n) is a sir,gle-valued function on Mn. But if e #: 0, mod. Z, Xn 1 Pn is nontrivial and, hence, f(n) is multi-valued. For e#:o (mod. Z), (4.8) implies that

4,

f(n)(p) = 0, ifPI

= Pi' for

some i #= j .

(4.11)

This is a generalized exclusion principle. If the particles described bv f(II) have spin s, then (4.9) implies that U as (21t)I/(n»as = exp[21ti(9n(n - 1) + ns)] I/(n»as'

We do not know any models for which

9 #= s or

#= s + 1/2, mod.l.

So far, our analysis of statistics has been model-independent. We now specialize to the Abelian Higgs model with Chern-Simons term. We consider an asymptotic state

212 357

QUANTUM FIELD THEORY OF A"NYONS

Im,/(n) >as describing n asymptotic anyons with magnetic flux m.

= m2 = ...

=

mn

=

m

=

±1 .

Then Equation (4.1) implies that

U(21t)f(n) = exp[21ti(8n 2 =

-

sn)]f(n)

exp[21ti8n(n - i)]/(n),

(4.12)

() is the coefficient in front of the Chern-Simons term, and s = 0 mod. Z. Comparison with (4.9) shows that the statistics parameter 9 of anyons with magnetic flux m = ± 1 satisfies \"her~

e = f) = s mod. I

.

(4.13)

This is a 'spin-statistIcs theorem' f')r aIlyons. For allyons, the reprt:sentation Pn of Ln is trivial, and we conclude that if s is integer the anyon is a OOSOfl, if s is half-integer the anyons satisfy 'intermediate (or 8- ) statistics'. Some anyon is a [ennion, while if s ~ of these results have been sketched in [17], some details will appear in [7].

4z,

5. 'Statistical

Me~hanics'

of Anyons

If.the Abelian Higgs model with Chern-S~mons term is put on the lattic.; and e and liMo are cho.ien small enough, one can rewrite the model as a dilute gas of magnetic vortex loops of integer vorticity with topological interacti0ns. The statistics of this loop gas depends on the coefficient, 8, in front of the Chern-Simons term: The statistical weight of a configuration of vortex loops has a complex phase given by exp21tiOw, where w is the 'writhe' of the configuration, (in the sense of knot theory). Hence, for 8 = 0 the statistics corresponds to that of bosonic loops, for 8 = the statistics corresponds to that of fermionic loops, while for 8:F O. ~ the statistics is intermediate, and statistical sums become highly nontlivial: see [18 J. (7]. Similar phenomena can appear in the statistical mechanics of random surfaces in five dimensions. The important concept in this connection is that two objects of dimen~ion k and of dimension I, respectively, can be nontrivially linked in a spacetjTl1e of dimension k + 1 + 1. Recently, related observations have also appeared in [19], but the ideas underlying [19] are closely related to those expressed earlier in [1, 3,4].

4

References 1. Wilczek, F., Phys. Rev. Lett. 48, 1144 (1982); 49,957 (192); see also: Wilczek, F. and Zee, A., Phys. Rev. . Lett. 51, 2250 ( 1983). 2. see, e.g., J. FrOhlich 'Statistics of fields, the Yang-Baxter equation and the theory of knots and links', to appear in the proceedings of Cargese S..mmer School 1987. 3. Wu, Y. S., Phys: Rev. Lett. 52,2106 (1984); 53, 111 (1984). 4. LaUghlin, R. B.,Phys. Rev. Lett. SO, 1395(1983); Halperin, B. I.,Phys. Rev. Lett. 52, 1583 (1984); Arovas, D. A., Schrieffer, R., and Wilczek, F., Phys. Rey. Lett. 53, 722 (1984); Tao, R.; and Wu, Y. S., Phys. ReY.

213

358

5.

6.

7. 8. 9. 10. II.

11. 13. 14.

15. 16. 17. 18. 19.

1. FROCHLICH AND P.-A. MARCHEITI

831,6859 (1985); Thouless, D. 1., Wu, Y. S., Phys. Re,. 831, 1191 (1985); Arovas, D. A., Schrieft'er, R., Wilczek, F., and Zee, A., Nucl. Phys. ml, 117 (1985). Wiegmann, P. B., Superconductivity in strongly correlated electronic systems and confinement vs. deconfinement phenomena, preprint 1987; Laughlin, R. B., Superconducting ground state of noninteracting particles obeying fractional statistics, preprint 1988. FrOhlich, 1. and Marchetti, P. A., Commun. Math. Phys. 111,343 (1987); Marchetti, P. A., Europhys. Ll!u. 4,663 (1987); FrOhlich,l. 8t"ld Marchetti, P. A., Bosonization, topological solitons and fractional charges in two-dimensional quantum field theory, Commun. Math. Phys. 116, 127 (1988). FrOhlich, 1. and Marchetti, P. A .. Quantum field theories of vortices and anyoDS, preprint 1988. Wigner, E., Ann. of Math. 40,149 (1939); Bargmann, V.,AM. of Math. 41,568 (1947). Buchholz, D. and Fredenhagen. K., Commun. Math. Phys. 84, 1 (1982). Doplicher. S., Haag, R., and Roberts, J. E., Commun. Math. Phys. 13, 1 (1969); IS, 173 (1969); 13, 199 (1971); 35, 49 (1974) FrOhlich. 1. and Marchetti, P. A.. COntmUIJ . •"Iatl:. Phys. t12. 343 (1987); Barata, 1. C. A., and Fredenhagen, K., Commun. Math. Phys. lI3, 403 (1987); Marchetti, P. A., Particlt: structure analysis or soliton sectors in mas.iive lattice field theories. t~ appear in Commun. Math. P.~}s. Becher. P. and Jovs. H.. Z. Phy!;. C1S, 343 (1982). Paul, K. and Khore, A., Phys. Lett. B17., 420 (1986); see also: Deser, S., Jackiw, R., and Tcatpleton, S., Ann. Physics 140, 372 (1982). FrOhlich, J., Osterwalder, K., and Seiler, E., Ann: of Math. 118,461 (1981); Seiler, E., Gauge Theories as a Problem in ConstiUcti,e Quantum Field Theory and Statistical Mechanics, Lecture Notes in Physics Vol. 159, Springer, Berlin, Heidelberg, New lork, 1982; see also [6]. see e.g., Isham, C. J., Phys. Len. BI06, 188 (1981). Brydges, D., FrOhlich, J., and Seiler, E., AM. Phys. (NY) Jll, 227 (1979). FrOhlich, J., Statistics and monodromy in two- and three dimensional quantum field theory,:o K. Bleuler et. al. (eds.). Proc. 1987 Como Conftrence, Arovas, D. A., Schriefl'er, R., Wilczek, F., and Zee, A., NucJ. Phys. ml, [FS13], 117 (1985). (Some additional details have been worked out in: S. Kind, diploma thesis, ETH 1988.) Po1yakov, A. M., Preprint, Landau Institute, 1988.

214 Modern Physics Letters A, VoL 3, No.3 (1988) 325 - 328 © World Scientific Publishing Company

FERMI-BOSE TRANSMUTATIONS INDUCED BY GAUGE FIELDS A. M. POLYAKOV LAndau Institute of Theoretical Physics,

M08COW,

USS,:1.

Re;:e!.ed De~m:'er 1987 We show that in (2 + l) -dimensional abelian gauge theory with the Chern -Simons term in the action, charged particles reverse their statistics.

In this letter I will demonstrate that in some theories with gauge fields, statistics of elementary excitations is reversed. The interest of this result is at least twofold. First, as was shown by Anderson et al., I it is possible that high T c superconductivity is connected with some unusual properties of Heisenberg antiferromagnets or perhaps, of Hubbard model. The way for quantitative L~plementation of these ideas was found by Wiegrllan)2 '.vho showed that poss!ble confinement r regime in this model may be responsible for the Cooper pairing of electrons. It was noticed in Ref. 3 that, under the same premises, it is highly probable that antiferromagnetic magnons are fermions - the fact envisioned by Pomeranchuck.4 Here I shall try to give more technical and quantitative arguments in favor of the physical picture, adv')cated in Refs. I_A. Essentially, it will be demonstrateJ by f:he explicit comput~tion of the propagators in the gauge fields, that the Chern-Simons termS in the gaugp. art;on turns bosons to fermions and vice versa. In particular, electrons in this field behave as spin one and zelO bosons, which seems to confinn the statements made in Rtf. 1. Let us start with the (2 + I)-dimensional antiferromagnet. We shall accept a conjecture, put forward in Refs. 2 and 3, that antiferromagnetic magnons can be described by the field of unit vector, with "-term, which is Hopfinvariant for the map S3 - S 2. It is convement to use Cpl representation for this n-field, and discuss the Lagrangian I

Y=-. Yo

'1

L

la"Zk+ iA "Zk I2 +

k=l

.,

--2

16"

e"..,).,A"a.,A).,.

(I)

Here the field z = ( Z 1 ' Z 2) defines a point of s 3 such that Zt

z ==

I;

I

12 + 1Z 2 12 = I .

The first term is a standard representation for the n-field, namely if we define n = (z t 0 z), it can be rewritten as (a", n).2 . The last term is Chern-Simons invariant, which was first discussed in gauge theories in Ref. 5. Modulo higher derivatives, it is equal to Hopf invariant for the n- field. 325

215 326

A. M. Polytlkov

+

The conjecture in Refs. 2 and 3 was that spin - antiferromagnet corresponds to (J = 1r in (l). We shall now prove that the main long-range effect of the gauge field is that z-quanta become fermions. For the proof, let us look at the expression for the transition amplitude of z -quanta in the form of the path integral. We have

L

G(x,x')=

e-mLCP"x') (ei!pxx,A"dx")

(2)

(Pxx')

(Here L is the length of P"x')' It will be convenient to start with the partition function

z::

L e -mL(P) (exp { ; J..

Jp

(p)

A"dx" } )

where P - are closed paths, and m - the mass. The averaging over A" is easily performed

+t §p

(eifpA"dx''> = exp ( -

dx"dy. (A" (x)A.{)I)

I

(3) We performed averaging here using only the last tefCl in (I) because vacuum polarization by z-quanta produce terms ex with higher number of derivatives. The integral in (3) wou!d be Gauss integral for Unking number, if we had integrated over two different loops P and P. In our case, we have to regularize this integral carefully. We do it by

F:.,

/ = -] " "dxlJdYIl 411' '1p lp _

=

~

P

dx" ~

~p

a

eIJilA A _,

_1_,

x-Y

(4)

d ly" .5 (x - y) .

(Here we used Stokes' theorem. The ~P is an arbitrary surface bounded by P. Let us replace

6 (x - Y)

=>

1

2

6 (x - Y) = - - 2 e-(x - Y) IE . E (211'e)3/

(5)

The integral (5) is now well defmed and is dominated at e ... 0 by the region close to the boundary. Parametrizing this region, we get I = _I "dx. [0 x it]

21r

x

l;.

= x(s);

it

do

= ds

(6)

Here dx is a tangent vector to the path, while 0 is a normal along the ~p. It is easy to rewrite this in terms of normal connection defmed by the relations

216 Fermi·Bose TrtlnsmutatiolU Induced by Gtluge FIeld,

327

1

e= L

B; D;; X ex e

;= 1

(7)

D; = Ce;kRk -B;e. The integral (6) is expressed in terms of normal connection C which is also called a torsion of the curve, 1 1=27r

IL C(s)ds.

(8)

0

1~ is easily seen from (7) tha~ C is essentially a Dirac patef'Atial on a s~here e l = 1 with a magnetic monopole in the center. Namely; one can represent C in the form

C(s) = where we introduced

r oe ae] dueolJo as au I

int~!"pOlating

e(s,u) =

(9)

X -

field

I

e(s)

u=1

const

u=O

The integral (9) is defined modulo some integer, which is irrelevant, due to Dirac quan· tization. So, our result is

~(P) = (ei~A~dx~)= exp~

C ~>u ds

e· [:: X : : ] )

(10)

We shall now show that the right-hand side of (10) is precisely equal to the spin factor for the spin- -t particle. Namely, the sum over closed paths for the Dirac electron in thrte dimensional space time is given by

z=

I

e -mL(P) t/J(P) .

(II)

(P)

The main reason why (11) is true lies in the fact that if we consider e(s) as a dynamical variable, and the Dirac potential in (10) as an action, defining Poisson brackets, then it is obvious that

(12) (Structure constants in Poisson brackets are in general defined by the closed 2-form in the action.) More technically, it means that

J

ill

~·ee

;f~ cIs C(s)

1

(

6 (~ -1) eQ1 (sl)." eQN sN

= Tr (0011 , • • OON)

)

(13)

217 328

A. M. PoIYfllcov

where 00l are Pauli matrices (for th~ monopole of the charge 1$, we would have (1$ + 1)dimensional representation in (13) ). Therefore, we can replace

e(s)

~ 0

(14)

.

The propagator in the momentum space for our particle is given by

G(p) =

C

dLe- mL

{Deeif~ Cds 6(e' -

1) •

eipf~ .d. .

(IS)

Using (14), we get

G(p) = ,- dLe- mL ei(u.

Jo

p)L

= __ .1 _ _

m-,(g·p)

s~mewhat more than (13) which refers to traces. But careful treatment of boundary terms in (13) permits us to use it for matrix elef!1ents as well. So, we have proved t:1at dressmg of z-quanta by the gauge fields turns them into Dirac fermions X. In this proof, we neglected higher derivative tenns in the action. They will lead to short range· interaction among these fermions. Let us notice also, that the mass tenn for z -quanta, z t z turns into X..,Il because the XX term violates p -parity. So, we arrived at the rather peculiar picture: z-quanta being bosons eat large momenta behave as fermions at small momenta. Vice-versa, if we had a charged fermion in our theory it would transmute into two bosons with spins one and zero. It is very tempting to find similar phenomena in different dimensionalities and also for strings rather than for particles. At the moment, this problem is not solved.

whkh is correct fermion propagator. Strictly speaking, we used

all x

Acknowledgments I am grateful to P. B. Wiegman for many stimulating discussions.

References 1. P. W. Anderson et al., Princeton preprints (1987). 2. P. B. Wiegman, Phys. Rev. Lett., to be published. 3. I. E. Dzialoshinsky, A. M. Polyakov and P. B. Wiegman, Phys. Rev. Lett., to be published. 4. I. Va Pomeranchuck,J. Phys. USSR, 4 (1941) 356. 5. R. Jaclow and S. Templeton, Phys. Rev. B185 (1981) 157.

4. Anyons in Model Field Theories F. Wilczek and A Zoo, "Linking Numbers, Spin, and Statistics or Solitons", Phys. Reu. utt. 51 (1983) 2250-2252 ........................................................................... 222 A. N. Redlich, "Parity Violation and Gauge Noninvarianc
Y.-H. Chen and F. Wilczek, "Ind~'ccd Quantum Numbers in some 2 + 1 Dimensional Models", Int. J. Mod. Phys. B3 (1989) 117-128 ..................................... 234

221

4. Anyons in Model Field Theories In this chapter three papers demonstrating the appearance of anyons in simple model field theories are collected. The pape~ [1] by Wilczek and Zee introduces the possibility of the Hopf term into the classic non-linear (1' model, and shows that it gives rise to fractional spin and statistics for collective excitations in that model. The germ of the Chern-Simons construction also appears there. Redlich [2] demonstrates in great detail how integrating out heavy fermions can give rise to Chern-Simons terms (and thereby, implicitly, ~o anyons). This paper not only containd ~n important r~sult but also exemplifies formal techniques of great power and generality. Finally, the paper [3] by Chen and others contains some further examples of the calculation of induced charges and currents in 2+ 1 dimensional field theories. It is show ~hat in a non-abelian framework the fractional part of the spin and statistics is quantized. Also, the interesting phenomenon that the form of induced currents can be a discontinuous function of the parameters, leading to peculiar effects on the surfaces of discontinuity, shows up here.

REFERENCES 1. F. Wilczek and A. Zee Phy,. Rev. Lett 51 (1983) 2250. *

2. A. N. Redlich Phy,. Rev. D29 (1984) 2366. *

3. Y.-H. Chen and F. Wilczek Int: J. Mod. Ph,!!" B3 (1989) 1. *

222 VOLUME

51,

NUMBER

25

PHYSICAL REVIEW LETTERS

19

DE(;EMBER

1983

Llnklnl Numbers, Spin, and Statistics of Solitons Frank Wilczek l1IstUute loy TMorettcal Physics, UnlversUy 01 Call/omla, Santa Barba'Ytl, Call/omiIJ 93106

and

A. Zee Ir.stUute lor TMontical Physics, UniversUy 01 Calffomla, Slmta Bayba'Ytl, Call/omiIJ 93106, arul Depewt,*rIl 01 Pltysics,(a) UnweysUy 01 Washington, Seattle, WashingtOft 98195 (Received 24 October 1983)

The sp1n. and statistics of solitons in. the (2 + 1)- and (3 ... 1)~tmec9iODa1 nocl1near (T !!Iodele Is considered. For the (2 + l)-
The existence of solilona in the (2 ~ 1)-dimen:.ionai 0(3) nonlin('ar (f model, as fiest discussed ~y Belavin and Polyakov, l is ilnplieu by the homotopy 1'2(S,) = z. The model is described by the functional E=j".%(1/2f)(8."·)2,

i=1,2; a=1,2,3,

(1)

giving the energy of a static configuration specified by ,,-(i). The "order parameter" ,.- is a three-dimensional unit vector: "-,.- =1. If we describe the ground state by n(i) =(0, 0, 1), then the baSic soliton Is described by n(i) = (is~D/, cost).

(2)

Here i =i/lil denotes the two-dimensional unit radial vector and 1("= IiI> is a function varying smoothly and monotonically from 1(0) = _ to I( 110) = 0 as " increases. We refer to such a topological configuration as a skyrmion. a The topological current in this model is (3)

The space-time indices p., II, ••• run over 0,1,2One easily verifies the conservation of this. The topological charge of this current, Q=jd'-xJO = (1/81')jtP% ('/(--,.- 8, .. ' 8 / ,.e,

(4)

clearly describes the homotopy of the mapping s, - sa for satisfying the boundary condition ri(i =110) = const. The skyrmion has Q= 1. By using Bogomolny's inequality, one can solve exactly the problem of minimizing the energy functional for a given Q. Finally, we mention that this model' provides a phenomenological description of Heisenberg ferromagnets in a two-dimensional system and thus the phenomena exhibited in this mod-

n

2250

el Ir.y conceivably be ftccessibie gxperimentaLiy. In trod paper we point out that the skyrmion may possess fractional angular r.lomentum and obey peculiar quantum statistics. One of us had previoutlly proposed4 -e the possibUlty of fractional angular momentum and of statistics w!u~h are neither Bose-Einstein nor Fermi-nirae. As we will see, the (2 + 1)-dimensional 0(3) nonlinear t1 model provides an amusing a:ld explicit ftelCltheoretic realization of these ideas. Our discussion is also related to severai othe-r field-theoretic phenomena discovered in recent years. The relevant mathematics which allows 3yrmionCJ to have these peculiar properties is the bomotopy Jf,( ~I) = Z [whicil is perhaps somewhat less obvious than tbe homotopy -a( SI) =-=.c responsible for the skyrmion's eXistence). It is easy to exhiblt the baSic Hopf map of s, - Sao In Cact, physicists should be familiar with this fact from elementary discussions of the Pauli matrices"'. Define ,,- = Z t".% where

is a comli1ex two~omponent spinor with th" constraint IZll1 + Iz,11 = 1. Notice that the U(1) transformation z-e,ez leaves" in.variant and so th-, inverse image of any point on ~ Is a circle OIl Sse What is not so obvious is the const!"lction of a Hopt invariant to describe -,(Sa) (just as Q de· scribes _,(s,)]. This will be explained below. Let us first address the phYSical question of the spin of the skyrmion. To dete rmine the spin, we rotate the skyrmion adiabatically through 2w over a long time period T. According to Feynman, at the end of this rotatiOI'l the wave function acquires a phase factor p" s where S is the action corre-

© 1983 The American Physical Society

223 VOLUME

SI.

NUMttER

PHYSICAL REVIEW LETTERS

2S

sponding to the adiabatic rotation. The angular momentum J of the skyrmion is given by efS

g(r) =

Now, if 5 has simply the standard form [cf. Eq.

»

5 0 =Jd'x(I/2/)(8"nG)2,

(5)

then it is easy to see that 50 is of order l/T-O as T - 00. The skyrmion has J = O. However, we have not taken into account the possibility of including in 5 a topological terna. This possibility is by now familiar from the discussion of the 8 vacua 7 in quantum chromodynamics, from studies of three-dimensional Yang-Mills theory and gravity, I and from recent work of Witten' on strongly interacting skyrmions· o (based on e3.rlier work of Wess and ZumiDc)lJ). In general, we can ~ve S = 5f, + 8 H wher~ 0 is a real parameter ar.d His the Hopf innriant which we now define. The conserlation of J" licenses us to manufacture a "gauge potential" A" by the curl equation: (6) A ,. is defined up to the gauge freedom A ,. - .4 ,. - 8 A. Note that A,. depends nonlocally on n·(x). ,.

2

In the gauge 8A~0, we haveA,.=-8- £"1I).8,,J).. [An alternative construction is to write A,. = iz t 8"z. The U(I) phase rotation on I. induces the gauge transformation on A". ] The Hopt invariant is defined by H= - (1/411>j d'x

£,."~,,F,,).

=- (1/211)1 d'.A"J".

(7)

H is obviously invariant under gauge transforma-

tion on A ". [We note that this is Just the Abelian version of the topological term studied by Deser, Jackiw, and Templeton,' but since II is gauge invariant, 8 is not quantized. Fu rther II' ore, here H is to be regarded as a functional of n(i). In the language of Zumino, Wu, and Zee,1I II is propor .. tional to f "'so. If jJ denotes a four-dimensional index then 8,.£~D~pFH=t£~HF~Ft~, conne'!ting the Hopf invariant to the chiral anomaly. ] Spatial rotation of a single skrymion is equivalent to an isospin rotation and tlus we evaluate H for the time-varyi.. configuration fit :i:ins=e 6 'G(t) x (;il :i: ml)(i), lis = I\(i). .j;trtctly speaking, this defines It map of 8 1 x [0, 1] - 82"} It is not necessary to know the explicit form of From Eq. (3) we .find

n..

J, = - (1/811)(da /dt)£u 8Jns-

DECEMBER

1983

f;

dr' r' Jo(r'). This and Eq. (8) allow us to determine Ao =- (do /dt)ns- Inserling into Eq. (7) we find

=e'2.". (1

19

(8)

It suffices to know that Jo(r) = £u 8,.4 J is a function of r to determine AJ =- £u,x"g(r)/r where

H=g(oo)lIs(OO)[ a(T) - 0(0)]/2r= 1.

(9)

The skyrmion has angular momentum 8/211. For a ferromagnet, 8 should be determined by the microscopic theory underlying the phenomenological a model. It is easy to show that H is a homotopic invariant for Ba- 8a• Consider a map with R
=£,.,,).8 I1 0A).

(10)

and we find OH= (-1/2It)2j d'x 6A ,.-J"= O.

There is a deep theorellI lS which equates the Hopf invariant to the linking number between two curves in R'. To have a heuristic understanding of this, consider the maps Ss - Szo ,.he reverse image of a point in 8 a is a C'.lrve in S, which by a stereographic projection we can think of as a curve in gs (with 00 identified as one !)Oint). Thus, for the basic map given expliCitly above, D= (0, 0, 1) corresponds to the great circle l.ell =1, .ez = 0 on 8 s , while =(0, 0, - 1) corresponds to ZI =0, ~zl =1. Write the real componants of (.ell za) as (cos"" SiDiI/cos8, sin",sin8 cosq>, sin,sin8 Siilcp) -nd stereographically project this point to r(1P) x (cos8, sin8 cost', siD8 sinq» in gs where r(,) ranges monotonically from 00 to 0 as , raaces from 0 to.. We see that the curves corresponding to n~ (0, 0, 1) and to n= (0, 0, -1) liat once. The reader may find it amusing to wom out the curves corresponding to other points. Using this linking theorem, we can easily determine the spin and statistics of a akyrmion. Consider the following process in 2 +! dimensions. At some time create a pair vf *yrmion and antiskyrmion and pull them apart. Rotate the skyrr.lion through 211. Allow the pair ~ come together. Since at GO we have the physical ftcuum this defines a map 8 a - 8 1 • Were the skyrmion not rotated, the map would be homotopicallJ trivial. Here, the corresponding map bas Hopt iDftriant 1. The two curves traced out by two specific values of nwill be linked once as indicated in Fig. 1. To determine the statistics obeyed bJ a akyrmion we consider a process in which we create two skyrmion-anti9kyrmion pairs and subsequently

n

22S1

224 VOLUME

51.

NUM8ER

PHYSICAL REVIEW LETTERS

25

19

DECEMKER

1983

(0)

-"'-

(b)

F {G. 1. The creation and allDihUatlon of a skyrmionIlntlskyrmion pair. with !l 2r rotation of the ::::kyrmlon. Tbe two curves correspond to If = (0.0, 1) and (1,0,0), 9l\y.

bring them to annihilation but after interchanging the two skyrmions. We see, '>y the maneuvering i.ndlcatl)d in Fig. 2, that the linking number is 1 fo r this process. The map of 8, - S2 corresponding to this process therefore has Hopf invariant 1. Thus, the skyrmion obeys exotic statistics which interpolates continuously betwp.en Bose and Fermi statistics as described in Ret 6. (By the way, the alternative of directly computing the Hopf integral corresponding to rotating a pair of skyrmions through 11 appears to be quite difficult.) Note that the diSCUSSion there is for a gauge theory. Here, we do not have a gauge theory but, curiously, one can manufacture a gauge potential A II" Given a map/:s,,-sn one can always cnnstruct13 a mapl:s,.H -sn+l (called the Freudenthal suspension of I) by [(I, (1- 12)112%) =(I, (1- t 2)I/",I(X» where sE and t E [0, 1]. This induces a homomorphismu F:~"(s,, )-11'''+I(Sn+l) of the bomotoliY classes of I and ,. Our discussion can thus be "suspended" in~o (3 + 1 )-dimensional space-time: 1'2(S2) -lI,(S,) =% and 1I',(sa) - ~..(s,) =%2. The first of these is an isomorphism, the second is onto: The suspension of a map Sl - r to a map s" _ s' is nontrivial if and only if the map has odd 'fopf invariant. Since s, =SU(2) manifold, the homotopy .,(s,) impUes the existence of skyrmions in the SU(2) 0 SU(2) nonlinear a model. The fact that I'..(s,) = %2 allows one to quantize the skyrmion as a spin-t fermion as discussed by Witten. 9 It is consistent with the standard three-space angular momentum analysis and with the well-known facts 1'1 f30(2»= % and 111 (SO(3»= %2 that 1I'.. (s,) is %2 rather than %. This material is based upon research supported in part by the National Science Foundation under Grant No. PHY77 -27084, supplemented by funds from the National Aeronautics and Space Admini-

_II

2252

FIG. 2. (a) Th~ creation and Ilnnlhllatlon of two skyrmlon-antiskyrmion pairs. (b) The prooess in (a) but with an Interchange of the two skyrmlons. (c) The two curves In (b) after a homotopic deformation.

stration.

leave of absence 1983-1984. IA. A. ~ela~"" and A. M. Polyakov. Pis'ma Zh. !lesp. Teor. Fiz.!!, 503 (1975) [JETP Lett.!!. 245 (1975)). 2T. H. R. Sc:yrme, Proc. Roy. Soc. London, Ser. A 247 260 (1958). ~or a review of the material In this Introductory paragraph, see R. RaJaraman. SolUons and InsttJ1ltOfts (North-Holland, Amsterdam, 1982). 'Some aspects of this subject appear to have been anticipated In the remarkable paper of D. Finkelstein and J. Rubinstein, J. Math. Phys.!. 1762 (1968). sp. Hasenfrat:t, Phys. Lett. 85B, 338 (1979); J. SChODfeld, Nucl. Phys. BI85. 157 (1981). IF. WRczek, Pbys. Rev. Lett.1!, 1144 (1982), and 49, 957 (1982). TG. 't Hooft. Phys. Rev. Lett. ~ 8 (1976); C. Callan, R. Dashen, and D. Gross, Phys. Lett. 63B, 334 (1976): R. Jacldw and C. Rebbl, Phys. Rev. Lett.!1. 172 (1976). 8J. SchODfeld, Ref. 5: S. Deser, R. Jacklw, and S. Templeton, Phys. Rev. LeU.!!, 975 (1982). and Ann. Phys. (N.Y.) lli. 372 (1982): see also Y.-S. Wu and A. Zee, to be publlshed. 'E. Witten, Nucl. Phys. B223. 422, 433 (1983). lOA. P. Balashandran, V. P. Nair, and C. G. Trahern. Phys. Rev. Lett.!!, 1124 (1982). lIJ. Wess and B. Zumino, Phys. Lett.!!!!.. 95 (1971). 12R. Zumlno, Y.-S. Wu, and A. Zee, to be published. I'For example, P. J. Hilton, An I"troducUon to Homotopy Theory (Cambridge Univ. Press, Cambridge, England, 1953), Chap. VI. (.)011

225 VOLUME 29, NUMBER. 10

PHYSICAL REVIEW D

15 MAY 1984

Parity violation and gauge noninvariance of the effective gauge field action in three dimensions A. N. Redlich Center for Theoretical Physics, Laboratory for Nuclear Science, Massachusetts Institute of Technology, OJmbridge, Massachwetts 02139 (Received 17 January 1984) The effective gauge field action due to fermions coupled to SU(M gauge fields in three dimensions is found to change by ±.".I n 1 under a homotopicaUy nontrivial gauge transformation with winding number n. This gauge noninvarianc:e can be eliminated by adding a parity-violating topological term to the action, or by regulating the theory in a way which produces this term automatically in the effective action. The Euler-Heisenberg effective action is calculated in the SU(2) theory and in QED.

I. INTRODUCTION

Gauge invariance of the Euler-Lagrange field equations does not necessarily imply gauge invariance "f the ac~ion I[A].1.2 In a functional integral formulation of a field theory. we require expU/[A]) to be gauge invariant. If under a gauge transformation the action changes by a number, the parameters of the theory must be quantized so thr.t number is an integral mUltiple of 217. 1-4 Another important feature of quantum field theory is .. jmmetry under space-time transfomlations. If the action i~ invariant under such a symmetry, but physical quantities which are invariant in the classical field theory are not iI'variant when the theory is quantized, then we say that the symmetlY is spontaneously broken. In this paper, we di$Cuss a theory in which the spontaneous breakdown of a space-time symmetry, coordinate reflection invariance, and the gauge noninvariance of the action are intimately connected. We consider threedimens~Jnal (2+ I space-time or 3 Euclidean space) SU( N) gauge theories coupled to fermions in the fundamental representation, as well as the U()) theory, QEDour results, however, may be genn-alized to higher odd dimensions. The action I[A,1/1] for an odd number of massless fermions coupled to SU( N) gauge fields in three dimensions is invariant under both gauge transformations anu space-time reflections (which we shall call parity). We establish here, however, that the effective gauge fiel~ action 1eff[ A] must violate either one or the other of these symmetries-Ieff[ A] is obt2ined by integrating out the fermionic degrees of freedom. Which symmetry IS violated depends on the procedure used to regulate tho: ultraviolet divergence'"l in I.[A]: If we regulate in a way ~'hich maintains parity u a good symmetry, then Icft-[A] is found to change by an odd multiple of 17 under a topologically nontrivial gauge transformation. On the other hand, if we introduce a heavy Pauli-Villars regulator field and subtract limM __ I.[A,M) from leff[A], then we ~I the gauge noninvariance in I.[A], but introduce parity violation through leff[A]. A three-dimensional mass term M# violates parity and for M,pO the fermions have The calparity-violating spin equal t02 M / IM I = ± zulations performed here were discussed in a Letter in

t

t.

~

which only th~ results "..ere presented. s Pauli-Villars regularization restores gauge invaria:tce because limM_,/efl'[A,M] contains the parity-violating topological term :tl7W[A]--W[A] is the Chern-Simons secondary characteristic class2.6-w hich changes by ±l7n under a homotopically nontrivial gauge transformation UrI with windins number n. The gauge noninvariance of ±l7W[A] cancels the gauge noninvariance of 1eff[A]. As an 31ternative to introducing a parity-violating PauliVillars regulator to restore gauge invariance at the expense of parity conservation, one may simply add the term ±l7W[A] to the gauge field action. Thus the tutal action consists of the parity-conserving, gauge-noninvariant piece coming from the fermions and the parity-violating, gauge-noninvariant gauge field action. The gauge noninvariance disappears in the sum, but parity violation remains. Another way to restore gauge invariance is to change the fermion content of the theory: one works with an even number of fermion species, so that the effective action changes by 217n under a large gauge transformation. In this case, parity need not be violated, which is not surprising, since an even number of fermions in three dimensions can be paired to form Dirac fermions with parityconserving mass terms. We shall not concern ourselves here with this well-known :nethod of removing gaugeinvariance anomalies, but focus on the novel mechanism which retains the fermion content while modifying the gauge field action. We shall begin with a demonstration that the effective gauge field action I.[A] changes by ±1T I n I under a homotopically nontrivial gauge transformation UrI with winding number n. While this was proved in the Letter,' the derivation presented here is an alternative one. This calculation does not require any approximations. The bulk of the paper contains approximate calculations of the effective gauge field action I.[A]. In Sec. UI, we use perturbation theory to demonstrate that __ . 1~[A]=/eff[A]- Mlim I.[A,M)

contains the Chern-Simons term ±1TW{A]. We then calculate I.[A], exactly, for the special cue of gauge fields 2366

@1984 The American Physical Society

226 PAIUTY VIOLATION AND GAUGE NONINVAlUANCE OF 11fE ...

2367

which produce " constant field strength tensor FPY; 1df[ A] in this case is the three-dimensional analog of the Euler-Heisenberg effective action. In Sec. IV this calculation is performed for the Abelian theory QED, and in Sec. V it is presented for SU(2). For gauge fields with constant field strength tensor FPY we find

To show that det;l'l[A p ] is gauge noninvariant, we vary the gauge field along a continuous path, parametrized by l' from A p (x P ,1')=O at 1'= - 00 to the pure gauge

1eff[A] = ±a1TW[A]+1~A[A] , with

at 1'= +

a= I for SU(2) ,

a=2 for QED. where W[A) is the Chern-SilLons term and 1HA [A) is a nonanalytic contribution to the effective action. The lack of ;ma1yti~ity is related to th~ inira.r~ divergences which a!': charact(;;ristic of three-dimensiolaa! gauge theorie!.7 Il. GAUGE. NONINV ARIANCE OF THE. ACTION (SU(N)

r

0 show that fermions induce a gauge-noninvariant teml in the action, we integrate over the fermion fields in the functional integral:

Z= f

d~d1/ldA exp [i f

= fdA exp [;

+

dlx[ trFp..,FPY +i~a+..4 )tIr] )

[J dlx +trFp..,FPY+1cft"[A])] ,

(2.1)

where IefF[A) = -i lndet«'+~) .

(2.2)

We USf' the usual matrix notation Ap=gTIIA:, with Til anti-Hermitian generators of the group. For defmiteness, we work with SU(2) and a doublet of fermions-where Til =oA IU, and oA.are the Pauli matrices--but our results hold in any group for which 1Tl is the additive group of integers. Z, and the fermioos are in the fundamental representation. The Dirac matrices in three dimensions are Pauli m&trlces (O'l';0'2,iO'I)' In our Letter,5 we demonstrated tht det( a + ~ ) and, by (2.2), the effective action Idf[A] are not gauge invariant. More precisely, we showed, following closely the analogous calculation performed by Witten in four dimensions,l that det(i+.4 )_( -I) l"ldet(a+~)

where U.. belongs to the 11th homotopy class II). The value at 1'= - 00 of the determinant detil'l[A], which is real because ;l'l[A] is Hermitian, can differ from its value at l' = + 00 only by a sign. This is because two three-dimensional fennioDS can be combined to form one Dirac fermion with an effective action 21cfl'[A]. Therefore, (detiBl[A])2 must be invariant under both large and small gauge transformations. As T is varied frore. - 00 to + 00, tbe gauge field must PMS through configurations in tield space which are not pure gauge, because U.. is not continuously deformable to the identity. Therefore, one or more eige!lvalues which are positive (nega.tive) at 1'= - or;, may cross zero and Uecorn~ negative (positive) at 1'= + 00 (see Fig. 1). Since the deta-minant is defined as the product of all the eigenvalues, it will change sign if the number of eigenvalues which cross from positive to negative minus the number which cross from negative to positive is odd. We now recognize that the family of vector potentials A p (x P ,1') is equivalent to an instantonlike fourdimensional gauge f.eld A' in the gauge A·=O [the space x'=(x P ,1'), ; = 1,2,3,4 is the cylinder Sl XR]. The remaining coLlponents of .4'(x P ,1') vary adiabatically as functions of 1'=x· along the path considered above. The number of zero crossings of the eigenvalues of il' l[ A P( 1')] is related to the number of zero modes of the four-dimeru:ional Dirac operator

(U..

00,

has winding number

=

(2.4) with

(2.3)

under a homotopically nontrivial gauge transformation U.. , with winding number II. (We use here a Euclidean fonnulation of the three-dimensional theory, and we consider gauge transformations which approach the identity at large d!st.ances; hence o~r base manifold is S) rather than R 3') The proof makes use of the observation that detiO'p(ap+A p ), 1'=1,2,3 may be written det1ni.B., where ».=rp(a +Ap) and rp are 4X4 matrices. The square root is dd1ned as the product of the positive eigenvalues of ;I'.[A p ], and we showed that tbe determinant changes sign under a large gauge transformation. We shall now rederive (2.3), without introducing 4 X 4 matrices, remaining instead with deli 0'p( a p + A p ) =detiJJl[A p ] in terms of2x2 Pauli matrices.

FIG. I. The eigenvalues of ;.1')[...4"(1')] are plotted along the vertical axis. One eigenvalue is shown to pass from a positive to a negative value as l' varies from - 00 to + 00 •

227 A.N.RBDUar

Y.=I!-:. ~ I· Y'=!~I -;,11·

(2.S)

Y'=!~ ~II· To see this, we write the Dirac equatiOll""'=O 88

~=-r4r,.(a,,+.4,.¥ ,

(2.6)

where #'= 1,2,3. Using the repmentation (2.S) of the r matrices,

values of iJ))[..4"] which pass from negative (positive) to positive (negative) values 88 'T varies from - 00 to + 00. If n _ -n + is odd, then the determinant detiJ')[..4"] will change sign. Moreover, it is well known from instanton studies that (2.11) where n is the instanton number (the winding number of U.). This completes the proof of (2.3): the determinant cbanps sign under a large gause tnDsformation with odd winding number. It 15 possible to regulate detiu,,(a,,+.4,,) in a gaugeinvariant way by introducing a massive, parity-violating Pauli-Villars regulator field. The regulated effective action (2.12)

(2.7)

,,"'±

We now choose the eigenfunctions of =0 to be eigenfunctioru of r,: rs"/lt = ± With the repr:sentation (2.S) of r" Eq. (2.7) becomes

"'t.

du+. . -.u,,(a,,+.4,,)u+ = -.,,)[.4"]u+ ,

~=

du_

~=;u,,(a,,+.4,,)u_

=1,,)[.4"]u_ .

(2.8a)

;")[..4 "]4J1'(x") =A,( 'T¥1'(x") • The eigenvalues A,(-r) vary as a function of'T along the curves of Fig. I. In the adiabatic approximation, Eqs. (2.8) are

dl+

(2.9a)

dl_

(2.9b)

d;""=A.('T)1 _('T)

with solutions

f o1'd-r' A,(-r') ) ,

[fo1'd-r'A,(-r') ) •

(2.10.) (2. JOb)

if _('T» is normaIizable only if A, is positive (negative) for 'T= + 00 and neptive (positive) for 'T= - 00. Therefore, there eDIts a on~to-one correspondence between the number of nonnalizabIe zero modes " .+ =0 ("",_=0), denoted n+(n_), and the number of eigen-

1 +('T)

m-oo)=±1TW[.4],

(2.13)

where

wr.4]=-I-f ~"-l..4".4YAaE" d )xtr(..F" • 8~ ) "YG ) , (2.14) ·P'=tE"IU'IlFa/I

is the Chern-Simons 5eC'mdary characteristic class, known to be odd under coordinate reflection and changing by ft under a homotopica1ly nontrivial gauge transformation with winding number ft. O\lterDatively, one may regulate in a parity-preserving but ga~e-nol,jnvariant way by defining detiu,,(a,,+.4,,) as dell iJ)40 as was done in our earlier paper.' The regulation is performed by standard rauli-Villars methods on deti.4- These now preserve parity because we are dealing is with 4 X 4 Dirac matrices where the mass term m equal to the sum of a positive plus a negative mass term in the 2X2 r-matrix formulation.'

H

IIi. PERTURBATION THEORY

1 +('T)=I + (O)er.p [1 _('T)=I _(O)exp

I~finite,

(2.8b)

Equations (2.8) are soluble'! in the adiabatic approximation: let 14 t (;%".,.) =1±~'lx"), where 411'(;,,") satisfies the eigenvalue equation

d;""=-).('T)/+('T) ,

is gacgr. invariUlt, but as we show in 5«:. I!I, a parity,·iolaling term survives in 1en<m) at m·= 00; its finite part is [for SU{2)]

We shall begin by establishing that the Chern-Simons tmn (2.14) must appear in IcfF[.4] when ultraviolet divergences are regulated in a gaule-invariant way. To de so, we use Pauli-Villars regularization and defme the regulated actiOll as in Eq. (2.12). Oaly the vacuum polarization graph and the t.riansle graph are ultraviolet diveqent, therd'ore only these two require regularization. For fmite m, I., in the SU(2) theory is, to order ,),

(3.1)

228 PAlUTY VIOLATION AND GAUGE NONINVAlUANCE OF 1HE .•. where D''''(p)= _;g2 D"W(q,k) = •

f

d3k - - 3 tr(r"S(p +k)r"S(k)] , (217')

for the special case of gauge fields which produce a CODstant field strength tensor, F""=coostant: (3.2)

-i3f ~tr[r"S(p)r"S(p +q) (217')3 XraS(p +q +k)]

(3.3)

with S(p)= -;/(p -m). The vacuum polarization graph is linearly divergent; for m - .. oc it is equal t02.9 2

11'''('71

-+

oc

1= -

2

L,t\g;:"..,.. ~ .1_( -ie /""-';p ...:1 , Im I

317'-

417'

0.4) ·..·h'!1 ~ -\ is a momentum cutoff and symmetric integration !'"l. r:r!1'Jved the loga~thmic divergen-:e in both Eqs. (3.2) and 11.31; for rr. - 00 we lind

<3.5)

All higher-order graphs in lelf [Eq. (3.1)] must be propordonai to powers of 1/ m and therefore they vanish when m - 00. Using Eqs. (3.4) and l3.S) in Eq. (3.0, we obtain the effective action for m - 00 :

I.(m-oo)=-~ f d l xtrA2-

I: I 17'W[A] ,

EI'GII • A"(x)=--2-xa FfJ·

To obtain the complete effective action in this case, it is necessary to calculate both I eer[ A] and (J"), the groundstate current in the presence of the background field (4.2). This is because the Chem-SimO'lS term W[A] formally ',anishes for A" given by (4.2), but its variation with respect to A IJ produces a nOflvanishing current. On the other fu~nd, I ef! contains terms which are functions of F,u'F"". These produce terms in (JI') which contain derivatives of F"'v. For pIlv = constant, these tenns vanish: a calculation of (JI') alone is not sufficient. We shall the:efcre calculate If"IT[A] and (J") separately, and use

~df

!:r. = Iecr(finite, m =O)±

W[A],

(3.7)

where the ± comes from m / i m I which depends on the sign of m. The effective action in QED differs from leer [Eq. (3.1)] only by group factors; it follows that the regularized effective action in QED is 2

R e Icff=/:rr(finite, m =I)± 817'

to deduce that the Cherrt-Simons teno is present in the effective action. The rnethod us.!d here w~ developed by Schwinger to perform similar calculations in fourdimensional QED. to It is a gauge-invariant procedure and therefore gives results which agree wi.th the solutions 0btained :ISing Pauli-Villars regularization. We begin by writing (J"(x» in terms of a Green's function G(x,x') for the Dirac operator, [j)(x)+m]G(x,x')=8(x -x') , (J"(x» =;e tr(y"G(x,x')]

1.11-.11' •

(4.3) (4.4)

The limit x -x' is performed by taking the aVer&pe of the terms obained by letting x-x' from the future and from the past.tO Introducing the operator notation G(x,x')=(x I G ix')=(x I 1 Ix'), D +m

we write G =; fo"'tlslexp[ -;(-j)j)+m 2)S]!(-j)+m).

f dlx A" ·F"

=/cfflfinite, m =0)::217'W[A) .

=(J")

('A"

(3.6) where W[A] is the Chern-Simons term (2.14). The divergent term in leff at m =0 (which appears in 17''''') cancels the divergent term in I err (m - 00), (3.6), and we obtain lhe finite effective action

(4.2)

(4.5) 0.8)

Again, the second term is the Chem·Simons secondary chara~teristic [for a U(!I theory]. Also, the only parity violation in Eqs. (3.7) and (3.8) k due to the ChernSimons terms, because for m =0 both QED and the SU(2) theory are parity conserving and therefore lelf(m =0) [in both Eqs. (3.7) and (3.8)] conserves parity (coordinate reflection).

In addition, J1)( -x)=jJ(x) for A"(x) given by Eq. (4.2), so that

f dlx trln[J1)(xH-m] = t f d 3xtrln(-J1)J1)+m2),

lefF[/.]=;

(4.6)

or

2 en<x)=t tr(x Iln(-j)j)+m 2) Ix),

(4.7)

which can be written

IV. le«[..4] IN THR.EE-DIMENSIONAL QED We shall now calculate the total one-loop contribution to the effp.ctive action in three-dimensional QED,

;

2 ea<x)=-tr(x I 2

l"'tIs -(exp[ -i(-j)j)+m 2)s] s 0

-exp[ -i(O+m 2 )s]! Ix) ,

leer[A]=; trln(J"+m) , (4.0 J1)(x) = -;i-eA(x) ,

(4.8)

229 A. N. REDLICH

1370

where the second term has been added so .!Zelf will vanish when A"=O. We now define (x' I U(s) Ix")=(x',s Ix",O), where

U(s)=e- lJr., K=

-",,=

-D"D"- fu""F"v ,

ia.(x',s Ix",O)=(x',s 1'lx",O) , [-;a~-eA,,(x')](x',s

(4.11)

Ix",O)=(x',s I D,,(s) Ix",O) , (4.12)

[-ia",,-eA,,(."<")](x',s Ix",O)=(x',s ID,,(O) Ix",a> , and the boundary condition

and

(x',s Ix",a> IJ_o=6(x' -x") .

i

u"v="2[r,.,rv] . U(s) may be understood as an operator which describes the developm.!l1t of a quantum-mechanical system in the "proper time" s, governed by the "Hamiltonian" K. To determine (x',s Ix",O) and (x',s I D"I x",O), we first solve the ;woper-time dynamical equations dx" ' d.;""='[K,~,.]

(4.13'

.!t'df and (J") can then be written in terms of (x',s I x",O) and (x',s I D" I x",O):

~.r:IX)=f f;~ e-.... [trb.'lx.. O) _

e )w/4 '

_1-]1

s)/2 4~/2

1.11-.11"

(4.9)

t

(4.14)

dD"

aF,.v ~ =i[K,D,,] = -2eF""Dv -ie axv

(J"(x» = -e try#' X

(4.10)

f o" ds e-u.. J(x,s I-"+m Ix',O) I .11-.11' • 2

(4.IS)

and then use

Solving the set of equations (4.9)-(4.13), one finds

Ix'., Ix".O)=Clx'.x"lexp [- fix' -x ").leF CO!hleF,)!,,'lx' -x"

)0] e:,~" exp [ ;~ "• .r

o

] ,

(4.16)

(x',s ID"(O)lx",O)=+[eFcoth(eFs)-I],,v(x'-x")V(x',s Ix",O), C(x',:;")=Cexp

L (s)=

lie f:"A"(x)dx" ],

(4.17)

c= ;;~ ,

+trln[(eFs)-lsinh(eFs)] ,

(4.18)

where the trace (4.18) extends over space-time indices only. These solution a differ only slightly from their four-dimeosional counterparts: the factor 1/s)/2 is 1/s 2 in four dimeasioDs; also, the coostant C is different. " The quantities .!Zelf and (J"), as they were defined above, are not gauge invariant: they contain linearly divergent, gauge-DODinvariant terms - A"A" /(x -x') 1.11_.11' and A" Ibc -x') 1.11'-.11' respectively. However, the corresponding plI8~invariant quantities

(J"(X»OI=ie tr[y"G(x,x')] IJr_J1' (.!Z.)01=1 trln[G -I(x,x')] ,

(J"Ix))=-emC

,

with

are free of divergences. Taking the limit x-x' in Eqs. (4.14) and (4.1S), and dropping the terms which are not gauge invariant, or equ!valently def'miDg (J") and .!Z• to be the gauge-invariant quantities (4.19), we obtain

(4.19)

f; :!'"e-...... Ir"e-Ll"exp [ITa.oF""

II.

~..,b)=IC f; ,t.-.... ,-I- II" Ie -Ll"up II ~ a..p"°ll-l \. Using y"=(u),iu .. iu2), we fmd

(4.2Oa)

(4.2Ob)

230 PAlUTY VIOLATION AND GAUGE NONINVAlUANCE OF nIB ...

2371

(4.21)

where I·FI =(B2_E2)ln. Since L (s) is detennined by a detenninant, we can evaluate it from the eigenvalues of F, which can be fou.id with the assistance of the relation (in three dimensions)

F""Ftl(JF"~= ....... !.

.~..l=

F2 F"~= _(eF2)F"~ . 2

(4.22)

,t.:rate the eigenvalue equalion £"'·=1.", and find - ~ e F: 2A: A=O, ±i!·F I. Th'!ll, by Eq. (4.18),

where the sign of the first tenn depends on the sign of m. V.

lef1'[~]

IN NON·ABELIAN GAUGE THEOR.IES

We now extend our calculation of leff to non-Abelian gauge fields. For definiteness we consider an SU(2) gauge theory coupled to fermions in the fundramental representation, btll th~e result:> are easily generalized tl) other gauge groups. (n non-Abelian theories, there are two types of gauge fieljs which Ptroduce a constant field-strmgth tensor F:" = constant. I Tlte first is an "Abelian" gauge field, .4/:= -fl.

X

l

sinh(es\2~

es ,eFI sin(es

(4.23)

I-F I ) ~4.23)

Inserting Eqs. (4.21) and tain the solutions (J"(x)=-2ie 2mC

.?en<x)=iC

J:o

tis

into Eq. (4.20), we 0bm

2

e2

l o -e-IIrIUF"=---·F", sin I m I 41r ID

(4.24)

ID

~n e -IIIIZ'[es I· F I cot(es I· F 1)-1]

s

1 rID tis = 8~n Jo s5n

~ x" eF~,

(5.1)

where F!:"='r;QF"", 77" is a constant unit vector in iscspin 5t>ace, and F"'" is a constant. The second is a constant gauge field ..c!:=constant, which is truly non-Abelian: [.4",.4"]*0, where A"=gA:,. and '.=Tt:I'U (.,- are the Pauli matrices). We shall not consider here the spt.:ial case £:"=0, which corresponds to a pure puge A"(X)=U-1(x)r"U(x) .

In the non-Abelian theory, I . and

U:(x»

are

Idf[A]=-itrln\jJ+m),

(5.2)

(Jf(z» = -g tr 11"'" .!m ]. (5.3)

Xe-lIIz'[d I·Flcoth(es I·F i)-I],

where in the last expression we IulVe defonned the path of integration: s - - is. If we now let m -0, U"(x) remains unchanged, and

1

.!L' elf becomes

IIn

·F ~ .".

Ole

Ldf(X)=8

=_1_I'(.!)

1..",2 ~

wh~e ;(

Z

ds r ID --(scoths-I) J o ssn

1

2

2

(J:(x) =

ll2

~

The calculation of (J:(x» and'?cn<x) for an Abelian gauge field (5.1) follows direc~ly from the calculation of these quantities in QED (Sec. IV). Keeping track of factors of 2 which comes from group-theoretical termstr[t.,.]-we find

(4.2S) J

+is the I' iemann , function

I: I -:;. F: ,

Idf[.4J=±W[A]+~;(t)f dlx

(S.4a)

[+tr-F

2

r'2.

(S.4b)

I

Using Eq. (4.25) and functionally integrating Eq. (4.24), we deduce that I.[A], for m =0 and F""=constant, is 2

1.[.4] = 1: ;.".

f dlx A"· F"

+2~M)fd'zl'I;1

r

=±2..W[.41+ 2~~t)f d'z

I·I:FI

We shall DOW calculate lemA] [Eq. (5.2)], for a constant non-Abelian gauge field A"=CODStant. In this case, it is easy to demonstrate that (J:(x» is indeed given by 8I.[ A ]l8A: 80 that a separate calculation of (J:(x» is unnecessary. For A"=constant Eq. (5.2) becomes, with dlx =vol,

f

f ..!!\trID(P-IA +m) (217') =i(vol) f ..!!\lndet(P-IA +m) , (2.".) .?cOIA]=i f ..!!\Jndet(P'-IA +m) . (2.".)

1.[A]=i(vol)

r, (4.26)

(S.S)

It is more convenient to calculate the effective action in

231 A. N. REDUCH where ·F"=gl. ·Ff. AU Lorentz- and gauge-invariant quantities can be written in terms of .4 1', II

Euclidean space. We therefore Wick rotate and introduce (_i.4 0,.4 1,.4 2)_(.4 3,.4 1,.4 2):

1~lndet[o"(pI'-i.41')+im] . (S.6) (2.".) The 3 X 3 matrix .4: can be diagonalized by performing

+tr·F2=4.,,6 ~.4.2.462,

1'dr=

rotations in space and in isospin space

.4:_RrR!'Y.4:=U- I (R/)U- I (R,MU(R/(U(R,) , where U(R/)=exp(i6/\ ..,.) and U(R,)=exP(i6,·0'). If we

define the diqonal matrix 0 I L.4:=.4 = 0 .42 0 , 2 1 0 0 .43 .41

0

where D h p=ۥ .4:.Ji'f. The Chern-Simons term (2.14) can also be written in tenns of .4:

(S.7)

1I'W[.4] = .L(det.4 )(vol) .

then the determinant irA Eq. (S.6) can be calculated simply:

de<

[0" ~·-f":T.1 +im

I

I dr. U given by Eq. (S.S), contains a linear ullrz'liolet divergc:nce. We shall renormaiize this expression using Pauli- Villars regularization. We define

=~p2+m2)2+~~rA2 _4(PI

(S.10)

1r

2

.412+pl.422+P32.432)+~ , (S.8)

(S.I1)

(S.12)

(S.13) with a -.412+.4l+.433, b=.4. 2-.4 22+.4l. C =.4. 2 +.4 22-.4l. The ultraviolet divergence in 1'", has become a divapac:e in the 1 integral at 1 =0. We can ~ulate this divergence by introducina a cutoff'1 and intearatina by parts. The resultiq infinite and finite parts of l 'dr are

=

1'...< 00)=

:n I- + 1.(1)

1

" l 'eff(fmite)= -

•

~n 1 (1) 2

1

I-,

(S.14)

•

ds [d d ' 10- ,,3n d: 1.(1)-/2(1)+2s dsI2(S)

I

(S.1S)

•

where

1~(exp{ -1[(q2+m2,)Z+2s(q12a+qlb+qlC>+IA2]) -exp[ _;(q2+ m 2, )Z]), (2.".) d' / 2(1)=-1+ f ~[2(q.2a +qlb+q32c)]exp{ -1[(q2+m 2.r)Z+2s(q.2a +q22b+qlc)+12~]) . (211') /.(1)=+

(S.16) (S.17)

F'maUy. if we introduce ~ • -11,1+_2,)2

= (t.".)-ln

f:_

the .1Ir2.21C,1+_J,br •

(S.18)

thea the resultiq momaatum intepatioas can be performed easily. and we find 1 I.(.r)=~f" tbc e llr 2e 2llll2a 12V2

--

I

e

-.~

[%-as]II2[% ...... b.f].I2[% _cs].n

1

%3n

I

•

(S.19)

232 PAIUTY VlOLAnON AND GAUGE NONINVAlUANCE OF THE .•.

+

[x -cs)ln[x

_~)I/2[X

_",) 112

2373

(5.20)

1'

The infinite part of .!L' ecr [Eq. (5.14)] is independent of the fermion mass m. In addition, the finite part of .!L' eff IEq. (5.15)] for m*O is

y. o:n
1r

(5.21)

and

(5.22) wIth

11($I=-lle-is2A_1f-f"'c!xeij(zl 1 12V2 -aD (x -as]I12{x -bS]1/2[X -CS]I12

14(s)=~e-ISZ4 faD 1211'2

-aD

uebe21

a

[x _as]ln[x -bs]I12[x -cs]l12

+

T:,~:efore,

C

[x _cs]l12[x -as]I12[x -bs]l12

the renormalized effective action (5.11) is

I:rr=/ecr(m =O>±1TW[A] ,

(5.25)

where lerr(m =0) = (vol).!L'~m =0) is given tJy (5.22), and W[A] is the Chern-Simons term (2.14). The first term is nonllllalytic in A, as can be verified by calculating Eq. (5.6) for special cases such as .A I = A 2 =.A l' More imponantly, leff(m =0) is manifestly invariant under coordinate inversion and therefore the only parity violation ir. I~rf is conteined in the Chern-Simons term W[A]. CONCLUSIONS

The action I [A, ",,] for an odd number vf ma...sless f.:rmions coupled LO SUI N) gauge Gelds in three dimensions is invariant under parity. We have found, however, that the ~round-state current =6/./6A"., a physical quantity, does not possess a well-defined parity transformation, i.e., parity is spontaneously broken. This is explicidy seen for the very special case of a constant, Abelian field-st.rength tensor: (I:) =,2 /81r· F:, the vector current, is given by the pseudovector· FP. It is easy to generalize these results to higher odd dimensions. The calculation of I . has been performed in five dimensions, where it again contains the parityviolating tenn ±1TW5 [A], with Ws[A] the fivedimensional Chern-Simons secondary characteristic class. 12 In three dimensions, there exists a "physical" reason for expecting the ground-state current (J~ > to be proportion-

u:>

+

I

'

t5.

23

)

b

[x _bs]l12[x -as]I12[x _CS]I12

I

(5.24)

.

al to • F"-we consider first QED. If we introduce a heavy Pauli-Villars regulator field with Dl8S3 M, thl".l1 all regulator fermions heve spin M / IMI. This alignment of the fermion spins "polarizes" the vacuum, and the ground-state expectation value of the spin-density operator (0 I ~,t %." I 0> does not vanish. We observe that ."to'%. is obtained from the current associated with the spin part of the Lorentz transformation

t

to'

8a11 =[rCl ,r"]1/! , that is,

",to'%.=

Epall

2

Jr''' ,

where

Jr'=~[rCl,r"]1/!. The heavy fermions induce an electromagnetic current

J'::! =F""5a11A" so that,

=FfMZA"-F"'lA CI

,

(01 tE,..."J~ 10)~+EpaIIJ::'=·F"A~ .

On dimCllSional grounds we therefore obtain, to lowest order in e aDd 1/M -without calcu1atioas-

t e2 • (OI.O'%.lo)=alMT F"A,., where a is a dimensionless constant, while a lowest-order calculation gives a= 1/41T. In three dimeusioas ."to'.",=H, so the seIf-enecgy of the regulator field is

233 A. N. REDLICH

2374

-) (MH =

2

e M.1'"

41rIM I

.4".

This self-energy is due to the self-interaction of the fermion ~urrent with the induced electromagnetic field:

(M'#J)=U")M.4"

a"

or

(J")

-~.F"

M-IMI41r

«J") M is the contribution of the regulator field to (J"». While we are unable to determine the constant a in QED without a direct calculation, in the non-Abelian theory a is determined by the quantization condition necessary to maintain gauge invariance. Therefore in the non-Abelian th:ory, we mfty use the above argument to rleducewi:hout any ca!culatiOl's-that N (Jcr'" IM= 8fT

anomaly to higher dimensions, supplementing the evendimensional generalizations of the standard result,13 by establishing the existence of appropriate anomalies in odd dimensions. The "anomaly" in odd dimensions appean as a parity-violating topological term in the ground-state current (J:), rather than as a topological term ( _. FF) in (J s") (there exists no axial-vector current in odd dimensions, i.e., no r s). In both cases, the anomaly causes a physical ground-state current to violate a symmetry of the original action I [.4, ~ ]. Results similar to those presented in this raper have been obtained independently by L. A varez-Gaume and E. Witten [Nuel. Phys. ~ 269 (1984)]. More fe!CeDtly, these results have been derived by a method which makes use of the axial anomaly in two dimensions. 12

M g 2.f! cr,

TMi

ACKNOWLSOOMENTS

lIIhere N is an integer. The violation of parity in odd dimensions is analogous to the nonconservatio!l of the axial-vector curr:nt in two and four dimensions where P~u1i- Villars regularization introduces a mass which violates axial symmetry. We the:efore.. complete the program of generalizing the axial

I would like to thank R. Jackiw for· suggesting this problem and for his continued assistance, as well as A. Guth and G. W. Semenoff for helpful conversations. J am also grateful to E. Witten for communicating to me his results, and emphasizing the necessity of perity violation. This work was supported in part by the U. S. Department of Energy under Contract No. DE-AC02-76ER03069.

IThe Dirac monopole is the oldest example of this. 2S. Deser, R. Jackiw, and S. Templeton, Phys. Re •. Le;,. ~, 975 (1982); Ann. Phys. (N.Y.) HQ, 372 (1982); R. Jackiw, Asymptotic Realms of i"hpia, edited by A. Guth, K. Huang, and R. Jaffe (MIT, Cambridge, Mass., 1983). 3E. Witten, Phys. Lett. .l.l.1B. ,j24 (1982). 4E. Witten, Nucl. Phys. 11m. 422 (I ~83). SA. N. Redlich, Phys. Rev. Lett. 18 (1984). 6R. Jackiw and C. Rebbi, Phys. Rev. Lett. n, 172 (1976). 7S. Templeton, Phys. Lett. JmI, 134 (1981); Phys. Rev. D ~ 3134 (1981); E. I. Ouendelman and Z. M. Radulovic, ibid.ll.

3S7 (1983). aThis argument is due to E. Witten. 91. Affleck, J. Harvey, and E. Witten, Nuel. Phys. ~ 413 (1982). 10J. Schwinger, Phy:;. Rev . 664 (I9~)). "L. S. Brown and W. I. Weisberger, Nucl. Phys. Il.U1. 28S (1979). 12A. T. Niemi and O. W. Semenoff, Phys. Rev. Lett. 11. 20n (1983). IIp. H. Frampton and T. W. Kephart, Phys. Rev. Lett. SQ, 1343 (983); (to be published).

a

.u.

234 International Journal of Modern Physics B Vol. 3. No. I (1989) 117-128 World Scientific Publishing Company

INDUCED QUANTUM NUMBERS IN SOME 2+1 DIMENSIONAL l\-fODELS Yi-HONG CHEN and FRANK WILCZEK' Lyman LaboralO(y of Physics. Harvard UnwerSlly, Cambridge. MA 02138. L:>A

R'!ct;ived 23 Set>tembcr 19&8

The fermion current induced by s!ow variations in b::.ckgrol!nd scalar a~d gauge fields are computed for a class of 2 + I dimensional u-like models. Local current densities proportional to t:lpological currents in the background fieids are found. The coefficient depends discontinuously on certain field ratios. Th~ induced fennion numbers we find. mesh nicely with recent results on induced angubu momentum and induced statistics. In particular. the spin and statistics is intimately related to the global parity anomaly. Lattice realizations are sl'ggested.

1. Introduction Recently there has been renewed interest in 2 + I dimensional physics, and especially tile description of magnetic systems. This interest has been largely stimulated by the properties of various materials containing CuO layers, which are quite remarkable and anomalous even apart from their association in certain cases with high-tempernture superconductivity. Many aspects of the theory of these materials are apparently confused and certainly controversial. There are two essential problems: the identification of an appropriate idealized ftlodel, and the analysis of the model once chosen. Given present uncertainties, it may be wise to explore qualitative features of possible relevant field theory models in some generality, so that we may "know what to look for". This strategy also has the advantage t.hat the produced models may apply to other physical systems, even if not to CuO layers. In this spirit we analyze here some interesting qualitative features of a class of model quantum field theories in 2+ I dimensions. Let us hasten to add that our choice of models is far from random. They commonly fe~ture scalar fields, with internal degrees of fr~edom, interacting with fermions. These are meant to suggest, among other things, the effective interaction of a magnetic order parameter - magnetization or staggered magnetization - with electrons. Some connections with various specific speculation in the literature, and the possibility of concrete lattice realizations of the models will appear in the following. tOn leave from ITP, Santa Barbara. 117

235 II H Y.-JJ. Chen de F. Wilczek

2. Preliminaries, Basic Model It seems appropriate to review briefly ·some properties of fermions in 2 + I dimension, that may be unfamiliar and are r~ther peculiar. For the Dirac matrices ro,r, ,r2 we may take simp!y (a2,ia, ,ia3), where the a;'s are the usual Pauli matrices, since they obey the appropriate anticommutation relations. Notice that these r j are all pure imaginary. Thus it would be consistent to require 'I' to be real; however in the models of interest to us we will be working with complex '1'. The Lagrangian for a free fermion (2.1 )

is invariant under the discrete· pal ity, time-reversal, and charge conjlJgation transformations, under which the fields transform as (2.2)

(2.3) (2.4)

As usual, P and C are implemented by unitary, and T by an anti unitary, transformation 'I' --+ U'I' ut. Note that parity involves reflection about one axis; simultaneous inversion about both spatial axes is just rotation through n. If 'I' is real, it is unchanged under C transformation ano remains real under P and T transformation. An important point is that the mass term

is odd under P and T. We shall be considering models in 2+ 1 dimensions containing a doublet of two-component fermions and a singlet and triplet of pseudoscalar fields ",•. The interaction Lagrangian is (2.5)

where V(",,) contains scalar self-interactions. This interaction is reminiscent of that in the original Gell-Mann-Levy a-model. I We shall also consider models tfi&L

236 Induced Quantum Numbers in 2+ I Dimensional Models

119

contain gauge fields, such that the U( I) and SU(2) symmetries of (2.5) become local symmetries.

3. Induced Currents We now consider the flow of fermion currents in response to slow space-time t,) and cP. due t') th:: i~:~~i'~''''1i:)! t'.; ~\. J"" i:.; f~lmdi1r t"c,nl other (,onlE'x! $, c0nsid~ration of such tiows cJ.n b·-: t~s~fut In understanding the spin and stat i stics of solitons, arnong oth~r t~ j ing$. ! t is straightfcrwarc to Cd Lutali." ih..:' indL1c~~ current, foliowing Ref. 2 (see r\ppr.: udi7:.). The resui t is · .. 3:Iatioo of

(f~) =('P r~'¥)

1

=

Bin 8(lcpl - 1111)e JlU i'e ahL'cb u ap
(3.1 )

where Q Ilcpl, e is the function equal to I, 1/2, or 0, depending on whether its argument is positive, zero, or negative. The interpretation of this result requires some comment. Let us assume, first, that Y/ = 0. Then the form of (jll) is entirely reminiscent of the corresponding results in 3 + I and I + I dimensions. Here, as in those cases, the induced current is proportional to the density of an identically conserved current, and the associated charge has a topological interpretation (see below). The form of the current for 11 #= 0, on the other hand, is rather surprising. After all, (11) contributes to the mass of the fermions. So·what we have found is that as we vary the mass the current dues not change at all until a critical value of the fermion mass is reached, at which point the current suddenly drops to zero. This behavior is quite different from the smooth behavior found in other d:mensions. How can such a discontinuity arise, in what seems to be a totally lion-singular context? The expression (3.1) for the current was derived an the assumption of slow space-time variation - and the criterion of "slowness" is that the 11 and cp fields vary little, as a fraction of their value, over the Com}>ton wavelength of the fermions. But the magnitudes of the masses of the two fermions are 111 ± 4>31· Thus as either of these quantities goes to zero, the criterion for slow variation becomes harder and harder tc meet. In other words, the discontinuity arises from an interchange of limits, and would presumably be smoothed out in a calculation to higher order in gradients. Nevertheless, it is true as a limiting case and is quite significant physically as we shall see soon. It is of interest also to compute the' current in the more general model, where the U(l) fermion num.ber and SU(2) isospin symmetries of the model are gauged. The result is (see Appendix)

237 120 Y.-H. Chen & F. Wilczek

(3.2)

where (3. 3)

i3 the covariant derivative, e is the U( 1) c0upHng constant, and F/J.; and Ffty are re~pectively the SU(2), U( 1) field strengths. The term containing Fp,. is necessary to ensure conservation of (jll). This term, and the final term which is its abelian analogue (and has precisely the same graphical origin), form what is generally called the "parity anomaly". Equation (3.2) presents a remarkably balanced appearance.

4. Results The induced carte::t may be used to calculate the induced fermion numbers of field configurations, assuming that the characteristic length scale of the latter extends over many Compton wavelengths of the fermion. In complete generality, it may be used to calculate the fractional part of these quantum numbers. Two cases of interest are the baby skyrmion and, in the gauged model, the SU(2) vortex. We find, for the baby skyrmion ~t;alar

(4.1)

while for the SU(2) vortex

(4.2) assuming, respectively, I"~ > ,,,, everywhere, or at spatial infinity. Some of the calculational details are presented in the Appendix. These results make contact with some recent suggestions in the condensed matter physics literature. In particular, baby skyrmions carrying unit fennion number should plausibly be quantized as fennions and form an isodoublet. If we Identify isospin in our model with the ordinary spin, the scalar fields as magnetic order parameter, and the interaction of our fermion fields with the scalar

238 Induced Quantum Numbers in 2+ I Dimensional Models

121

background as arising from some filled band whose gap is dominated by the alignment of spin with respect to the order parameter, then we have shown that the solitons, that are electrically neutral, nevertheless carry spin. This is a model for the Anderson's "spinons,,3 or for the objects conjectured to exist in insulating paramagnets by DPW.4 It must be emphasiz~d, however, that the spin and statistics of thr.se solitons depends completely upon the coefficient of the socalled Hopf term. s The assignment a!:>ove is appropriate to zero bare coefficient f':)i" ~tiis ~~;-Ir', that is l\) zero c\~;cffic;cnt tu2f. :'n'C tht heavy fermioft i:) integrated out. The bare coefficient. howev~r, i~ sensitive to the ultraviolet (short-distance) behavior of the theory, as of course the fact that the coefficient can be affected by :!.rnitrari!y heavy fermions exemplifies. Also, the Hopf t~rm, like the vectc!' potentia! in the Aharanov-Bohm effel,;t 6 - to wh!~h in fact it is deeply rl!latedhas nq classical anal·)gue. T~.eref')re, its coe:ficient cannot be deduced from the long-distance Of semiclassical behavior of-an underlying microscopic model, but can only be deduced from a calculation where short-distance fluctuations, both thermal and quantum, are fully taken into account. Also, the "half fermion" quantum number for the vortex is strongly reminiscent of the quasiparticle~ proposed by Laughlin.7 One of us has argued elsewhere that these quantum numbers arise naturally for vortices in an abelian gauge model. s In both these cases, \\<e can restrict and classify the possibilities considerably by demanding that time-reversal symmetry be respected, or that it be broken spontaneously in a structured way.9 Remarkably, for SU(2) vortice~ there are strong a priori restrictions on the allowed spin and statistics. These restrictions can be thought of as arising in either of two ways, that seem very different but as we shall see lead to the same conclusions. First, let us recognize that unlike abelian vortices, that carry an ordinary additive quantum number (their flux), SU(2) vortices carry only a Z2 flux. Thus two vortices can annihilate into a topologically trivial configuration, that should have a sensible interpretation in terms of the ordinary particle conlent of the theory. We have seen that each vortex carries fermion number 1/2, so that the pair c:!:ries fermion number 1. In the topologically trivial sector of the theory, of course, we know that particles carrying fermion number 1 are ... fermions. Now the statistics of a pair of particles each with statistical parameter () is 48. So we must demand 48

=

(4.3)

1l, modulo 21l

or, for the statistics (and spin) of the· vortex

() = :r./4,

3ic/4,

51l/4,

or71C/4~

(4.4)

The second way, is to realize again that the statistics of the vortex is governed

239 122

Y.-H. Chen & F. Wilczek

by the value of the coefficient of an appropriate Chern-Simons term, this time the non-abelian one.1O Now the non-abelian Chern-Simons term is ·only properly globally gauge invariant when its coefficient takes on certain quantized values. A simple calculation shows that these values lead precisely to the possibilities listed above for vortex statistics. In our opinion, the concordance of these methods is quite pretty, and gives an unusually concrete physical interpretation of a global anomaly. The most novel feature of the current (3.2), compared with pre-vious results in other dimensions, is the appearance of discontinuities. Of course the current flow cannot really jus. . disappear abruptly at a surface, since this would be inconsistent with its conservation. To determine the form of the corrections, let uc) consider a general current of the form

where k!' is conC)erved and e a step function. As it stands, this cu~rent is not conserved. However, as we discussed above, the derivation of this form is valid only away from the surface. The true current can differ from JIl on the surface. Can we find surface terms that ensure conservation? In fact, it is not difficult. Anticipating that the correction term will be localized on the surface, write (4.5)

Then it is easy to verify that jll is conserved, if Gil" is antisymmetric and a,,(JIl" = kll • The solution for (Jil" is

(4.6)

a

The physical interpretation of this currtnt is as follows. The jump in the function occurs where some fermion mass goes through zero. The mass will have non-zero magnitude on either side, and we can anticipate there to be fermion zero modes that live on the wall, in the sense that for fermions in these modes, motion tangential to the wall costs little energy. It is motion in these modes that build up the surface current. II

s.

Lattice Formulation

The field theory models discussed above can also be put on a lattice. This is important, in two respects. First, the lattice models may have a dir~t use as models for crystalline solids. Second, one may wonder whether the effects we have discussed above, which are so closely tied to topology, survives latticization. Concretely, one may well be suspicious that some analogue of the fermion

240 Induced Quantum Numbers in 2+ 1 Dimensional Models 123

doubling phenomenon removes the possibility of inducing interesting quantum numbers by fermion vacuum polarization. At one level this sort of suspicion is alleviated by the beautiful work of Frohlich and Marchetti. 12 These authors have constructed anyon lattice field theories in full rigor. Thus there is little doubt that the effective theories we are aiming towards a,e consistent; and we need only wonder whether they result from lattice f'ermion theories, upon i~tegrating out the fermions. This, too, we believe is not a ~2rious ui;flcull.y. T!l.c point 15 lLldl 2-cunlponent fermions in 2 + 1 dimensions, Ij nlike chiral fermions in 3 + 1 di:n:~nsion3, support mass-like terms. Thus we can avoid the doubling problp,ffi, without spoiling any crucial symmetry. by usin.g a versior. of W ilson fermio~s.'3 Let us quickly review the COI!.5truction o!' a lattice formulation. Considei Lagrangian

where a is the spacing of the fermion lattice. The third term is added to eliminate the fermion doubling problem, it goes to zero in the continuum limit (a -. 0). There are many ways in which one can put the scalar fields on a lattice, hert we choose, for the .p fields, a square lattice interlaced with the fermion lattice. We will put the" field on the sites of the fermion lattice. For slowly varying.p and 11, the term ,¥(.p·t + 11)'11 becomes effec~:vely a mass term. The Dirac equation for the fermions is then (5.2)

On the lattice it becomes

A.

+ m'¥mx,my + -('IImx+l,my + u

A.

+ -('II mx my+1 + 'IImx my-l a' . For plane wave solution 'II --

eik.r-iEt

-

'I'mx-l,my -

2'1'mx,my)

2'1'mx my) • ·

(5.3)

the energy spectrum is (5.4)

From this one can see that the possibie fermion doubling at kxQ eliminated due to the presence of the last term.

= 11:, kya = 1t is

24i 124

Y.-H. Chen d F. Wilczek

Appendix. Some Details of Calculation

When the U(I) and SU(2) symmetries ar~ not gauged, the induced current must have the general form (A. I)

+

To determine C, we calculate jll for a configuration of where ~3 ~, « 1. The result for a general configuration follows by symmetry. Then

/"OJ

I,

~2

and

(A.2)

p;

where == (ap~j). Consider the lowest order Feynman graphs that contribute to j" (Fig. I), (A.3a)

(A.3b) where ~ = 1+lr3 + 1171, ~ = -1~lr3 + I'll are both matrices, and a Wick rotation was done to reach (A.3b). Converting to the Feynman integration variables, k = apt - PP2, neglecting

Fig. 1

242 Induced Quantum Numbers in 2+ 1 Dimensional Models

'h respect to A2 -2 PI2 ,P22 WIt and A , performing the trace over the

125

r p matrices and

integrating over k-space, using (A.4)

j"

1 .. 4n:

Q

1

2-

= --e JAIIY p p fr fJ

[1.1 dx-,=='=' x[I~! + Ichl(2x 0

y

I}TJJ

(.J q,2 + ,,2===='=====:--:+ 2:~ii~I(2x _ 1))3

1 --ePfJy pI 0 2 41T' p•. Y'

i'l> 1,,1;·

_

1 llflY lp2 81t e p fJ Y'

1+1 = 1,,1;

0,

1+1 < 1"1·

]

(A.5a)

(A.5b)

where the trace in (A.5a) is over the f) matrix, and (A.5b) is derived by performing the integration in (f... 5r) for each individual element of the f) matrix and then addir.g to get the trace. The square root in (A.5a) means taking the absolute value. Whenever the space-time dimension is odd one encounters a square root after k-space integration. From this result, C =

I

-9(1+1 -1,,1) where 81t I, 8(x) =

2' 0,

x> 0;

x = 0; x
The discontinuity results from the limits taken: Ipd,lp21« 1+1 -1,,1. Without taking the above limit, the asymptotic behavior ofjP at 14>1 > 1,,1 and 14>1 < 1,,1 will still be the same but the tr~nsition will be continuous. The induced fermion number of a baby skyrmion can be found by applying the result for jP on a baby skyrmion configuration: 4» = cos(f(r»

4>2 =

sin(f(r» sinO

4>1 = sin(f(r» cosO

(A.6)

243 126 Y.-H. Chen & F. Wilczek

where

f(r = 0) ='0, f(r = +(0) = 1C,

(A.7)

The fermion number Q is

=

L OO

o

[d

f sinf] 21Crdr 2C--dr r

=

_1',-0-00 41CCd(cos(f(r»)

=

8(1,1 -1,,1) .

(A.e)

When the SU(2) and, U(I) symmetries are gauged, the induced current is determined as follows. First the partial derivatives in (A. 1) become the covariant derivatives in (3.3), then the second term in (3.2) is needed to conserve the current. The coefficient of these two terms are determined by the fact that when g = 0 it should redure to (3.1). There is an additional term due to the coupling between the fermions and the U ( I) gauge field. It is calculated from the Feynman diagram (Fig. 2). (A.9a)

(A.9b) where in (A.9a) m = " ± if». With the SU(2) and U( I) gauge fields, one can have stable vortices. (Strictly speaking, to have vortices one must break the symmetries completely. This requires additional Higgs fields. We have been assuming, implicitly, that these do not couple to the fermions.) A possible configuration of a vortex in , field is (A.6),

244 Induced Quantum Numbers in 2+ J Dimensional Models

127

k-p

A

p

k Fig. 2

f(r

0)

=

0,

=

(A.IO)

f(r =

1£

+ co) = 2: '

and the gauge field has the asymptotic configuration 3

Y

A x = -.. , gr~

A

3 y

X = --2 '

(A. I I )

gr

A 1•2 = 0,

and is smooth at the origin. Applying (3.2) to this configuration, after similar calculation as above we find the induced fe!l11ion number for the vortex to be (A.!2)

Similarly, one finds (A.13)

for the U( I) vortices.

245 128 Y.-H. Chen & F. Wilczek

Acknowledgements

Y. H. Chen thanks Al Shapere for his c0l1l:puter graphics coaching. F. Wilczek would like to thank the Smithsonian Institution for support as a Regents Fellow. We also thank the Wednesday movie club for helpful discussions. This work is supported in part by NSF Grants No. Phy-82-17853 and Phy-87-14654. References 1. M. Gell-Mann and M. Levy, Nuovo Cimento 16 (1960) 705. 2. J. Goldstone and F. Wilczek, Phys. Rev. Leu. 47 (1981) 936. 3. P. W. Anderson, Science 235 (1987) 116. 4. L DZY2loshinskii. A. Polyakov. and P. Wip.gmann, Phys. [('ft. t 17 (1988) 112. 5. F. Wilczek and I~. Zee, Phys. Rl-s.:. Lett. 51 (1983) 2250. 6. Y. Aharonov and D. Bohm, Phys. Ref.'. 115 (1959) 485. 7. V. Kalmeyer and R. E. Laughlin, PhYi. Ret:. Lett. S9 (1987) 2095; R. B. Laughlin, Phys. Rev. Lett. 60 (1988) 2677. 8. A. Goldhaber, R. Mackenzie and F. Wilczek, Harvard preprint, HUTP 88/044. 9. J. March-Russell and F. Wilcz~k, Harvard preprint, HUTP88/045. 10. A. N. Redlich Phys. Rev. D29 (1984) 2366. 11. C. G. Callan and J. A. Harvey, Nucl. Phys. B2SO (1985) 427. 12. J. Frohlich and P. A. Marchetti, "Quantum Field Theories of Vortices and Anyons", preprint, 1988. 13. K. Wilson, and J. Kogut, Phys. Rev. el2 (1974) 75. l

5. Anyons in the Quantized Hall Effect B. I. Halperin, "Statistics of Quasiparticles and the Hierarchy of Fractional Quantized Hall States", Phys. Rev. Lett. 52 (1984) 1583-1586 (Erratum: Phys. Rev. Lett. 52 (1984) 2390) ................................................................. 251 D. Arovas, J. R. SchriefTcr and F. Wilczek, "Fractional Statistics and the Quantum Ha1l Effect", Phys. Reu. Leu. 53 (1984) 722-723 ....................................... 256

S. M. Girvin and A. H. M8c~onaj~ "Off·Diagonal Long-Range Order, Oblique Confinement, and the Fractional Quantum Hall EtT(;ct", Phys. Rev. Lett. 58 (1987) 1252-1255 ........................................................................... 258 R. B. Laughlin, "Fractional Statistics in the Quantum Hall Effect" .......................... 262

249

5. Anyons in the Quantized Hall Effect Perhaps the most clear-cut success so far for anyon ideas has to do with the theoretical understanding of the quantized Hall effect, and especially of the hierarchy of fractional quantized Hall states. In this chapter three brief letters which essentially established and illuminated this connection are collected, together with a beautiful elaboration and exposition of the subject (with characteristic original touches) by Laughlin. In his letter [1] Halperin constructs trial wave functions for the hierarchy of fractional quantized Hall state3 based on physical argument~. In this iterative constrlj~ti0Il one builds up each successive level of the hierarchy by forming effective wave functions for an assembly of quasiparticles 3.t the pre-,.rious level. In these effective wave functions fractional pOWi!rs of the d:stance between quasiparticles appear, and Halperin conjectured that this appearance must be related "to fractional statistics of the quasiparticles. Arovas, Schrieffer, and I demonstrated [2] the fractional statistics of t~le quasiparticles directly, llsing the Berry phase technique, thus verifying Halperin's intuition. In a very important paper (3], Girvin and MacDonald uncovered a form of long r(\nge order in the quantized Hall effect. This order does not exist between bare electron variables, but appears when fictitious flux is attached to the electrons. Girvin and MacDonald also suggest the use of a Chern-Simons theory to describe the quantized Hall effect, a suggestion which has proved quite fruitful. (See, for example, the papers of Lee and Fischer, and of Kane and Lee, in chapter 8.) Finally, Laughlin [4] presents a very clear and explicit discussion of the wh~le including a reconciliation of several apparently different points of view which appeared in the early literr.ture.

situat~on,

250

REFERENCES 1. ~. I. Halperin Phys. Rev. Lett. 52 (1984) 1583. *

2. D. Arovas, J. R. Schrieffer, and F. Wilczek Phys. Rev. Lett. 53 (1984) 722. * 3. S. Girvin, A. H. MacDonald Phys. Rev. Lett. 58 (1987) 1252. • 4 R. B. Laughlin, Fractional Statistics ip. the Quantized Hall Effect, in Two Dimensional Strongly Correlated Electmnic SY3tem3, ed. Zi-Zhao Gar., ZhaoBin Su (Gordon and Breach, Lond'Ju; 1989).

251

VOLUME 52, NUMBER 18

30 APRIL 1984

PHYSICAL REVIEW LETTERS

Statistics of Quasiparticles and the Hierarchy of Fractional Quantized Han States B. I. Halperin Physics Department. Harvard University. Cambridge. Massachusells Ull38 (Received 9 November 1983)

Quasiparticles at the fractional quantized Hall states obey quantization rules appropriate to particles of fractional statistics. Stable states at various rational filling factors may be con"tructed iteratively by adding quasiparticles or holes to lower-order states, and the corre~pondir.g energies have been estimated. P.\CS numbers: OS.lO.-d, 01.65.(1, 71.45.Nt, 1J.4111.Q

Observations of the fractional Qu .. ntized Hall effect l show that there exist special stable states of a two-d~mensionill electron gas, in sirong p{'.rpendi~u· lar m3gnelic field D. o<:ct!rrin~ at ;,;. ~el of rational values of Ii. th~ f:lling fdetor of the Landau level. laughll1l 2 has constructed a:"! explicit \rial wave function (prcduct wave function) to explain the states at v - lim. with m an odd integer, and has argued that the elementary excitations from the stable states are quasi particles with fractional electric charge. Among the propos;:.ls to explain the other observed fractional Hall steps are hierarchical schemes. in which higher-order stable states v J + I are built up by adding quasipa(ticles to a stable state 3 s V of smaller numerator and denominator. J

In the present note, we observe that the quantization rules which determine the allowed quasiparticle spacings are just those that would be expected for a set of identical charged particles that obey fractional statistics-i.e., such that the wave function changes by a complex phase factor when two particles are interchanged. Moreover, by assuming that the dominant interaction between quasiparticles is just the Coulomb interaction between the quasiparticle charges, we are led to a natural set of approximations for the. ground-state energies and energy gaps at all levels of tt.e hierarchy. The appearance of fractional statistics in the present context is stronaIy reminiscent of th~ fractional statistics introduced by Wilczek to describe chilled particles tied to "magnetic flux tubes" in two dimensions. 6 As in Ref. b, the quasiparticles can also be described by wave functions obeying Bose or Fermi statistiCS, the various representations beina related by a "sinlular puge transformation." The boson description was, in fact, used in Refs. 3 and 4 and the fermion description in Ref. S. However, the boson or fermion descriptions require, in effect, a long-range interac:tion between quasiparticles which alters the usual quantization rules. The transformation between representations is analo10US to the well-known transformation between im-

penetrable bosons and fermions in one dimension. As in previous disclissions of 1.1e fractional Quantized Hall eff~cl, we ~onsider a two-dlmtnsionai system of electrons in the lowest LandaU levp!!, with a umfoim positive hackgrou~d. The filling factor II i..; defined b)" I I - ,u'2r. !6. ~:here 1/ i, the dc!o:;jly of ele':trons, and 10 -18('/1."("1,,2 is the magnetic length; hence II is the number of electrons per quantum of nux. Let VJ be a stable ralional filling factor obtained at level s of the hierarchy. I assert that the low-lying energy states for filling factors near to v J can be described by the addition of a small density of quasiparticle excitations to t)-le ground state at VJ' The elemeDlary quasiparticle excitations are of two types-particlelike "p excitations" and holelike "h :xc;tatic ....;"-having charges qJe and - qJe. respectively, according to a sign conventio:l describ~d below. For the present purposes we need only consider states with one type of excitation present. We shall describe these states by a pseudo wave function '1'. which is a function of the coordinates i k of the NJ quasi particles present. I assert that the the allowed pseudo wave functions can be written in the form N

'I'[ik ] -P[Zk]Q,[Zk] nexp( -lqJ llzkF/4'J). k-I

(I)

where Zk - Xk +irk is the position in complex notation, with the sign depending on the sign of the charge of the quasiparticle, P [Zk] is a symmetric polynomial in the variables Zk. and

Q,-

ITlzk-z,Ik

<,

Cl/III ,.

(2)

In Eq. (2), a - ± 1. according to whether we are dealing with particle- or hole-type excitations, and m, is a rational ~ 1. to be specified by an iterative equation below. We may interpret 1'1' [ik ] 12 as the probability density for finding a quasiparticle at each of the positions i •..... iN, at least in the case

© 1984 The American

.

Physical Society

1583

252 VOLUME

52, NUMBER 18

PHYSICAL REVIEW LETTERS

that the Rk are not too close to each other. Since the quasiparticles have a finite size (of order 10)' however, there is no direct significance to the behavior of I'" F when two positions Ric and R, come very close together. The wave function is normalized if fl'i'1 2-1, and two wave functions qt and qt' are orthogonal if ",-qt' -= o. The pseudo wave function (l )-l2) can be derived in different ways, starting from various miCf{lscot>ic descriptions that have been proposed 1 - ~ for lhe elt=clronlc ~late with quasi panicle or .-i.iitsihok e;;.cilations I shall give below a der:vation for p excitations using the pair model proposed in

f

R~f

3.

IS no direCf p"'ysical ~igl"!ificancc tl) phase of the p3eudo wave function, it is perm issl'Jle to redefine :he iactor Q in Eq. (I} by remol'ing :~, .. aosolute va/u;: sig!'! i:1 Eq (1). (This operation !lhlY Je described as a singular gauge transformation.)6 If m, ¢ l, the new wave function is a mllltivalued function of the positions tikI. and one 4ihould consider it as a function defined on the appropriate Riemann :;urface for (Zk)' [Alternatively one could use a single-valued definition and specify discontinuities along cuts in the variable (Zic - Z, ). J Now if we continuously interchange the positions of two quasipartic.les, the wave function will change by a complex phase factor ( - I) ± 11m" witt:. the sign depending on the sense of rotation as the qU8.3iparticles pass by each other. Although the eICtra phase rdctor is perhaps ... complication, the pseudo wave function now has the esthetically pleasing property that it is an eigenstate of the differential operator lV' Ic +iq,e A(RIc )Ilfc J2 with special bo!ndary conditions at the points Zt where A is the vector potential in the symmetric gauge. Then [q. (I) may be described as a general wave function appropriate to a collection of panicles of charge ± qJe obc;Ying fractional statistics, all in the lowest Landau level. Of course, in the Sjlecial case m. - " the qua~iparticles are ordinary fermions. In order to find the ground-state configuration for a given density n, of quasi pal ticles, we must find the symmetric polynomial P[Z] which leads to the minimum expectation value of the repulsive interaction between the quasi particles. Using the same reasoning as Laughlin in Ref. 2, we expect that certain choices of P can lead to specially low energies, namely,

30 APRIL 1984

distribution I'" 12 is then that of a classical onecomponent pla~ma2 with dimensionless inverse temperature r - 2m2+ \. where (4)

and al+ I - 1 or - I as particlelike or :lolelike quasi particles are involved. The density of the plasrna's fixed by a charge neutrality condition, 2 so that the number of qUilsiparticles hl an area 2""'6 is just rI. = i q.I ~ I 11'1. + I.. Since each quasiparticle has charge a~+I';$' we may readily calculate the electron density in '''.e n~w ~tabJ~ state, and we find the filling factor (5)

Se("8use ~here

lh~.

Z,.

P [Zk J -

n<,

(Zk - Z, ) 21, + I,

(3)

k

where p, + I is a positive integer. The probability 1584

If we multiply the

pseudo wave function the factor file Zk, ~or II. - 1•...• N,. we find a deficip.ncy near the orillin o~ l/m,+1 quasi particles of level s. We identify this state as a hole excitation at level s + 1. Similarly, we may construct a p excitation having an excess of II m, + I Qllosipatticles at the origin. The iterative equation for q, is thus de:-;cribe~

3ho\>e

by

q,+I-a,+lq,/m,+I'

(6)

Together with the starting conditions Vo- 0, qo- mo- ai-I. the iterative equations (4)-(61 give a sequence of rational filling factors II, for any choice of the sequence (a"p,). At the lew'el .\ - I, we recover Laughlin's states with vi-I/mi for various choices of PI' If we add holes LO the state "l - !. we find at level s - 2. the complements to the Laughlin states, "2 - i .f . (In order ~o stay in the lowest Laudau level, we impose the restriction a2 - - 1, if "I - I.) From the state "I we achieve such or -IT, with p excitations. and V2" states as "2 or Tr' with h excitations. It can be shown, after some algebra, that the allowed values :>f l', ma~· be expressed as continued fractions in terms of the finite sequellces {a"p,} and that they are identical to those of Haldane. 4 (I have used the opposite sign for a. however, and here p is one-half of Haldane's.) As noted by Haldar.z, every rational value of v with odd denominator, with 0 < v ~ I, is obtained once in this way. There wHl not be a quantized Hall step at every such rational v, however. We know that there exists a maximum allowed value me for the parameter m" such that if at any stage of the hierarchy the calculated m, is greater than me' then the quasiparticles at the density n, will form a Wigner crystal rather than a quantum-liquid state. 2 There is then no stabilization of the electron density at the correspond-

... 1.+.t ....

t. ...

t

t,

t

253 VOLUME

S2,

18

NUMBER

PHYSICAL REvtEW LETTERS

30 APRIL 1984

ing 1'" and there will be no meaning to any further states in the hierarchy constructed from this I' ,. The pseudo wave function (1)-(3) leads to a natural estimate of the potential energy of the system, if we assume that the dependence on the positions of the q~asiparticles can be approximated by the pairwise Coulomb interaction between point particles of charge q, e, in the background dielectric constant E. If E ( I' ) is the energy per quantum of magnetic flux, we have £(1',+1) .£(v, ) +n,E/ +nllq,IS12upl(m,),

(7)

where Elt is the energy to add one particlelike excitation or one holelike excitation, together with neutralizing uniform background, to the state V" and upl is a smooth function of m given (approximately) by " Laughlin's ;nterpolation formulas u pi (m) -

-0.SI41 ----;;;m I-

J .

2

0.23011 e mO•64 ~1··

(S)

We re':'"a!1 that upI(m) is the potential ene:-gy per particle that one would find for a system of e'~ctrons at fitlirtg fa~tor y -11m if one apP"cximates th~ pair correlation func..·~ion g (r) for the elp.ctrons by ttle pair correla!ion function gpl(r) for a one-component plasma at inverse temperature r .. 2m; the factor Iq.I S12 in the last ~erm of (7) reflects the smaller charge and largee magnetic length for our quasiparticles. In order to Lise Eq. {7J, we need an itera:ive formuia for the quasiparticle energies E, ±. It is convenient to write (9) E/ -i.'± ±m,-I[E _I+tlq,_d S12 upI(m.)l. I

The Quantity in square brackets is the energy it would take to add one quasi!,article or quasi hole of level s - I, if one could keep the Laughlin product form (3) for the polynofTlial P, and simply increase the density n, _ I by means of a reduction, of order 1/N, in the magnetic length 10 which controls the distance scale in Eq. (1).7 The term i.,± in (9) mat be called the proper excitation energy; it is relatively small, but is presumably positive for both quasiparticles and holes. For the proper hole energy, we use the approximate formula

i,- _0.313lql_IIS12 m, -9/4(e 2/El o).

(10)

This form has the correct dependence on the charge q,_I; it passes through the exact value O.313(e 2/El o), for qo-I, m I-I, and it yields ii-0.264, Ei-0.OS37, for ml-3 and ml-S, in close agreement with the values o!>tained by LaughIin. S• 7 Unfortunately, there does not exist at the present time any reliable calculation of the quasiparticle excitation energy. Therefore,/or purposes 0/ illustration, I have made the arbitrary approximation i,+ - A.i,- , where A. is a constant independent of m,. The resulting curve for £ ( 1') is plotte:! in Fig. I, for the choice A. - 3. after subtraction of the "plasma approximation" EpI-I'UpI(I'-I), which is a smooth function of 1'. We can see that there are downward pointing cusps in the energy visible at the low-order rational I' with odd denominators. The approximation also gives upward-pointing cusps at all rational I' with even denominators; in fact, I

I

find small discontinuities in E. not visible on the scale of the figure, at all these even points except where continuity is guaranteed by the for V particle-hole symmetry of the cohesive energy, which :s respected exactly by the present approximation. 7 Clearly the upward-pointing cusps are unphysical; the system could always lower its energy by breaking up into small regions of larger and smaller density; alternatively ~here may be a different type of ground state with still lower energy at these values of 1'. The behavior of the approximate energy curve near the low-order rationals of odd denominator should be qualitatively and semiquan-

t.

.03

(~)

•

.01

III l' I '1

(!) (i)

(i),

!9 '7! ,! 5 )6' , °0~~~~~---~O~.5------------------~lD II

FIG. 1. Potential energy per quantum of magnetic nux, in units of e 2/d o• as a function of filling factor II of the first Landau level, from approximate formulas (7)-(10). Smooth function EpI(II)-IIUpI(II- I ) has been subtracted off.

1585

254 PHYSICAL REVIEW LETTERS

VOLUMES2. NUMBER 18

Iq,I-'. which is the denominator of the fraction V" This is in qualitative agreement with reported experimental observations on GaAs samples. I Finally, we derive by induction the starting equation (1). For s - O. the Z" are positions of bare electrons. and Eqs. (I) and (2) are correct, with qo- mo'" at -= I. We assume that the p excitations of level s - 1 can be formed out of puirs of electrons. by a generalization of Eq. (23) of Ref. 3. A system conlaming Nt pairs of electrons, lo~ether 'I.·i~h .\'0-- 2N: u.n;)Clired electrons, is then described by choosing ttle pol),!'Iomial in (1) to have the (schem'ltid form

titatively correct. however. More reliable estimates will be possible when p-excitation energies have been properly calculated. and when corrections are included such as the finite quasiparticle size and effects of virtual excitations of particle-hole pairs. With the approximation described above, the energy gap E,+ + £,- is equal to

~('"

'6) and 00)]. Except

l\)r

m,'/~. the gap is determined by the valu p of

fGf t~e rJ:~e~ w":dk

:-ac

n' . \",- n,1,-', z- )';'~I-' fif'1,2..,-Z,I.

Cl'p-[ Z, 1 I )'Z] .. Ie - J

, <j

4

r

\Z'-"JI

I • .,

.... ~e~e ;:, are the positior.s of the centers of iia·.. ity (.II' the bound pai~, Z.., are the pcsitlolis of the Uff' paired electrons. P is a symmetric polynomial. and./' is an operator which symme!rizes with respect to the positions of all No electrons. I have assumed that the separation between two membt:rs of" pair is small, and have dropped the variables describing these separations. To calculate the probability distribution of the pairs. we ignore the symmetrizer !I', and take the trace of I'" [Z" J j2 over the unpaired electron posit~ns i The result c.!.n be expressed in the rorm I'" lzl 11 ~ lz; J. where '" has again the form of (I) and (2). with P replaced by p, and with ml-2p,-I/mo. a-I. and ql-qOl'ml. while the remaining factor ct» is the partition function of a classical one-component plasma with :wurces of strength 2 - m t- I, located at the positions Zi' Now ~ will be independent of the positions Zi. provided that the sources are sufficiently separated so that thei( screening clouds do nnt overlap. Thus it is consistent to interpret i as a pseudo wave function for the positions of the pairs. Higher levels may be obtained iteratively.

r

1586

30 APRIL 1984

..,

<,

.2"1

(1t)

DenvCit;o~
'See H. L. Stormer et al.. Phys. Rev. Lett. SO, 1953 (983). 2R. B. Laughlin. Phys. Rev. Lell. SO, 1l9S (1983). lB. 1. Halperin. Helv. Phys. Acta S6, 75 (983). 4F. D. M. Haldane. Phys. Rev. Lett. SI. 605 (1983). SR. B. Laughlin, in Prl)c~edings of the Conference on Electronic Properties of Two-Dimensional Systems, Oxf..,rd. 1983 (to be published). 'F. Wilczek. Phr!:. Rev. Lett. 49. 9S7
2SS VOLUME

S2,

NUMBER

26

PHYSICAL REVI£W LETTERS

ERRATA

Statlstl~

of Quaslpartlcles a•• tbe Hlerarcby of Fradlo.a' QuaDtlz" Hall Statel. 8. I. HALPERIN (Phys. Rev. Lett. 52, lS83 (984»). .. Jn the approximate iterative formul~ Eq. (7), for tne encrBics of the quantiU
th,,+ l) • E
The curve in fiB. 1 was calculated with use carre,' fl!xpression.

or the

2S JUNE 1984

256 PHYSICAL REVIEW LETTERS

VOLUMES3, NUMBER 7

13 AUGUST 1984

Fractional Statistics and the Quantum Han Effect Daniel Arovas Department o/Physics, U",,,ersity o/California, Santa Barbara, California 93106

and

J. R. Schrieffer and Frank Wilczek Depa'tment o/Physics and Institute/or Theoretical Physics, UniW!rsity o/California, Santa Barbara, California 93106 (Received 18 May 1984)

The statistics of qUasiparticles entering the Quantum Han etTect are deduced from the adiaoatic lheor~m. These ';',(Cil&lion<.; are [01...10 to iJbe)' \'dCliona\ stalislics, it result closely reiat~d to their fractional charge. PACS numbers: 13.40.!..q.

O~.)O.-d.

n 20.My

Extensive experimental studies have been carried o~H 1 '}n semicooducting helerostruc1 ures tn lh~ quantum limit (uOT » I, where Wo =- eO 01 m is the cyclotron frequency and T is the electronic scatteri!lg tirr.e. It is found that as the chemical potential p. is varied, the Han conductance rr JtY -1.1 E, - lIe 21h shows plateaus at .,- nlm, where n ar.1 m are intelers with m being odd. The ground shte and excitations of a two-dimensional electron las in a ~trong maanetic field Bo have been studied2-4 in relation to these experiments and it has been found that the free energy shows cusps at filling factors II - n I m of the Landau levels. These cusps correspond to the existence of an "incompressible quantum fiuid n for given nlm and an enerlY gap for addillg qU!lSI~ .!rficles which form an interpenetrating Ouid. This quasiparticle fluid in tum condenses to make a new incompressible fluid at the next larger value of n 1m, etc. The charge of the quasi particles was discussed by Laughlin2 by usinl an argument analogous to that used ir deducing the fractional charle of solitons in one-dimensional conductors. 5 He concluded for v - 11m that quasiholes ar.d Quasiparlicl~s have charles ± ee - ± elm. For e~ample, a quasihole IS formed in the incompressible fluid by a twodimensional bubble of a size such that 11m of an electron is reme, ved. Less clear, however, is the:: statistics which the quasi particles satisfy~ Fermi, Bose, and fractional statistics having aii been proposed. In this Letter, we live a direct method for determininl the charle and statistics of the quasiparticles. In the symmetric gauge A( T) T we consider the Laughlin Iround state with filling factor

f1iox

II-11m,

+",- D(zJ-Zt)"'exp(-tI.lz,12). }
722

(I)

where irp.o

+ iy;. A !tate having g: v'!:"

ZI - Xj

·H ZI) i~

III:

%0 -

"Y

Il

quasi hole local(2)

]V + i l l (z; - zo)I$I,..,

while a quasiparticle at ~o is described by I/I~zo_ N _ O,(818z,- zr/al )1/1",.

"0-

(3)

where 21raIBohcle is the flux quantum and N:t are normalizinl factors. To determine the quasipartic!e charge ee, we calas Zo adiabaticulate the change of phase ,. of cally moves arounrl a eirc!: of radius R enclosina flux ,.. To ~etermine ee, ,. is set equal to the Change of phase,

.;'0

(e-IICc)<jA' dl - 21r(e·le),.I,.o.

(4)

that a Quasiparticle of charae e· would gain in movinl around this loop. As emphasized recently by Berry' and by Simon 7 (see also Wilczek and 'Lee' and Schiff'). given a HamHtonian H (zo) which depends on a parameter zoo if Zo slowly transverses a loop, then in addition to the usual phase E ( " ) dt'. where E ( I') is the adiabati: eneru, an extra phase ." occur! in 1/1 (t) which i~ independent of how slowly the path is traversed. ,.(t) satisfies

r

d,.(t}ldl- i <1/1(1) Idtll(t}ldl)

(5)

From Eq. (2),

so that

~; -iN!(+:'ol:' I.ln(z,-zo)~:'o).

(7)

Since the one-electron density in the presence of

© 1984 The American Physical Society

257 PHYSICAL REVIEW LETTERS

VOLUME 53, NUMBER 7

the quasi hole is given by P

+,

I~ I +'0 o(z) .... (e/I.O . . 8(Z/-Z)tIJ'" ), +,

we have dy- i dt

where

f dxdyp +,O(z)-dd In[z-zo(l),] t

(~)

(9)

We

write p +'O(z)-po + 8p +'O(z), with pO-IIB/~o. C:»ncerning the Po term, if Zo is integrated in a clockwise sense around a circle of radius R, values of Iz I < R contribute 21Ti to the integral while Iz I > R con!ri~~tes zero. Therefore, this contribution to 'Y is giv~n by z-x+(y.

Vo. i

rl 1<.R d.x. tly po2Tri JI,

- - 211' (If i R -

-

2.,,;Jq,/q,o.

((OJ

wt.ere (IE) R is lhe mea,} number or electro;"ls an a circle of radius R. Corrections from 8p Vb.lish as (aolR)2, where ao- (tc/eB) 1/2 is the magnetic length. This term corresponds to the finite size of the hole. Comparing with Eq. (4), we find e· -lie, in agreemel1t with Laughlin's result. A similar analysis shows that the charge of the quasiparticle tIl;';, is - e·. To determine the statistics of the quasipartides, we consider the state with quasi holes at z. and Zb,

As above, we adiabatically carry z. aroound a closed loop of radius R. If %6 is outside the circle Iz.l- R by a distance d» ao, the above analysis for y is unchanged, i.e., y - - 2"'1I~/ ~o. If %6 is inside the loop with IZ61-R« -Qo, the change of (n)R is - II and one finds the extra phase Ay - 2", ... Therefore, when a quasiparticle adiabatically encircles another Quasiparticle an extra "statistical phase" (12)

is accumulated. lo For the case II - I, ~y - 2'11, and the phase for interchanging quasiparticles is ~y/2 -", corresponding to Fer.ni statisti<:s. For II noninteger, ~y corresponds to fractional statistics, in aareement with the conclusion of Halperin. II Clearly, when II is noninteger the change of phase ~y when a third quasiparticle is in the vicinity will depend on the adiabatic path taken by the quasiparticles as they are interchanged and the pair perr.lUtation definition used for Fermi and Bose statistics no longer suffices.

13 AUGUST 1984

A convenient method for including the statistical thase ~y is by adding to the actuaLvector potential Ao a "statistical" vectoryotential A. which has no independent dynamics. A. is chosen such that

(~·/tc.·)fA.·dr -~'Y ... 21T1I, when z. encirlces be _ A.(T- T6 ) -

zb'

One finds this fictious

(13)

i\.

to

~oix(T-T6)

1_ - 12

2". r - r 6

(14)

if the quasi particles are treated as bosons and ~o- ~o(l -1/11) if they are treated as fermions.

Thus, the peculiar statistics can be replaced by a more complicated effective lagrangial1 describi:tg panicics with conventional statistics. 12 Finally, we note thaI if ont" pierc~s the plane with a ph,sicai flux tube of magnilude t/J., lhe above arluments sUDesl that a charge IItq,/t/lo is accumulated around the tube, regardless of whether q,/.o is equal to the ratio of intelers. This w.>rk was supported in part by the National Science Foundation through Grant No. DM~8216285 and No. PHY77-27084, supplc:mented by funds from the National Ae!"onautics and Space Admi,listration. One of us (D.AJ is grateful for tile support of an ATAT B~II Laboratories Scholarship.

IK. "on Klitzinl, G. Dorda, and M. Pepper, Phys. Re.v. Lett. 4';, 49'" (980). 2R. B. Laulhlin, Phys. Rev. Lett. SO, 1395 (1983). ~F. D. M. Haldane, Phys. Rev. Lett. 51, 60S (1983). 4B. I. Halperin, Institute of Theoretical Physics, University of California, Santa Barbara, Report No. NSF-ITP-83-34, 1983 (to be publ;shecl). 'w. P. Su and J. R. Schrieffer, Phys. Rev. Lett. 46, 138 (981). 6M. V. Berry, Proc:. Roy. Soc:. London, Ser. A 392, 45-57 (984). 7B. Simon, Phys. Rev. Lett. 51, 2167 (1983). 'f. Wilczek and A. Zoe, Phys. Rev. Lett. 52, :Llll (984). '1.. Schiff, Qut!lItum MechaniCS (McGraw-Hili, Ney.' York, 1955), p. ~90. IOAlthoup. is a variational wave function, rather than the actual adiabatk wave function, the statistical properties of the quuipartic:les are not expected to be sensitive to this inconsistency. We could reprd • to be an exact excited-state wave function for a model Hamiltonian. liB. I. Halperin, Phys. Rev. Lett. 52, 1S83, 2390(E) (1984). 12f. Wilczek and A. Zoe, Institute of Theoretital Physics, University of California, Santa Barbara, Report No. NSF-ITP-84-2S, 1984 (to be published). 723

258 VOLUME

23 MARCH 1987

PHYSICAL REVIEW LETTERS

58, NUMBE:.K 12

Off-Diagonal Long-Range Order, Oblique Confinement, and the Fractional Quantum Hall Effect S. M. Girvin Sur/ace Science Dil'ision. National Bureau

0/ Standards. Gaithersburg. Maryland 20899

and A. H. MacDonald National Research Council. Ottawa. Onlario. Canada /( I A OR6 (Received 24 N(III'e,nhcr 19861 '1emon~trat'!

th:: ex:stcncc 0: a novel type of otf·:.1:ag:):1<:; !!l,lg ·range order in the fractio:1al· This is revealed fu~ Ihe ':JSC \)f r· L:;ul'allilling factor ,. -11m by ai>· p.i.:ation of W.I\,:ZC;K·:; "a~)'l)n" gauge t:ansforrnatlon 10 attach nI ql!anliz«l ftux tubes to each partidc. Tne binding oi the l.er()s of the: wave: function to tr.e particles ir. the fral:tional '~uar.tum Hall effect is a (~ '-I )-o.rnensivIH! ~n !1'Jg 'Jr oblique cOflfintmenl i.~ whi,n oJ. c\lnt1c~:-al ion occurs, !lot of oidinuy pu:i",:c:" but (athel of C
:;"J:\ntIJm·Ha"·:::ff~c: g~'Hnd st;t~\!.

n",,,

PAl'S numLc:r). :2 2v.My. 71.45.Gn •. n.40 Lq

A r~markable amount of progres~ has recently been made in our understanding of the fractional quantum Hall effect (FQHc)' following upon the seminal paper by Laughlin. 2 There rema:ns, however, a major unsolved problem which centers on whether or not there exists an order parameter associated with some type of symmetry breaking. 3- 6 The apparent symmetry breaking associated with the discrete degeneracy of the ground state in the Landau gaugeS is an artifact of the toroidal geometry6,7 and is not an issue here. Rather, the questions that we are addressing have been motivated by the analogies which have been observed to exist 4,8 between the FQHE 3nd superftuidity and by recent progress towards a phe!1omenological Ginsburg-Landau pi':turc of the FQHE. 4 Further motivation has come from the development of the correlated-ring-exchange theory of Kivelson el al. 9 (see also Chui, Hakim, and Ma, '0 and Chui,IO and Baskaran"). The existence of infinitely large rin; exchanges is a signal of broker. gauge symmetry in superft .. id helium 12 and is suggesti -Ie 'Jf something similar in the FQH E. However, the concept of ring exchanges on large length scales has not as yet been fully

p(z,z ') - vg(z,z') -(l'/2~)exp( -

t

reconcilea with Laughlin's tessentially exact 7) variational wave functions which focus en the short-distance behavior of the two-particle correlation funl,;tion. Furthermore it is clear that \"le cannot have an ordinary broken gauge symmetry since the particle density (which is conjugate to the phase) is ever more sharply defined as the length scale increases. The purpose of this Letter is to unify all these points of view by demonstrating the existence of a novel type of ofT-diagllnal long-range o!'der (ODLRO) in the FQHE ground state. In second quantization the one-body density matrix is given by p(; ,z

') - L",."tP!.(z )1P,,(z')(O

Ie:"", 10),

where -v&MI' and thereby obtain the powerful identity:

I z - z'1 2 )exp[t (z*z' - zz'*»,

where g (z ,z') is the ordinary single-particle Grecn's function. 13 We see from (2) that the density matrix is short ranged with a characteristic scale giv~n by the magnetic length, just as occurs in superconducting films in a magnetic field. \4 The same result can be obtained within first quantization via the expression p(z,z') -

~fd 2z 2 • •• d 2zN9*(z,Z2, . .. ,zN)9
(3)

where Z is the nonn of 9" If the lowest Landau level has filling factor 0 -11m and tile interaction is a sbort-ranged repulsion, then in the lowelectron mass Iimit, 7 the exact ground-state wave function is given by Laughlin's expression: 9
n (z,-zj)"'exp (- i- ~IZA: 12)" A:

'<j

1252

259 VOLUME SS, NUMBER 12

PHYSICAL REVIEW LETTERS

2J

:\'lARC~1

19R7

Laughlin's plasma analogy 2.ls proves that the ground state is a liquid of uniform density so that Eq. (2) is valid. The rapid phase oscillations of the integrand in 0) cause p' to be short ranged. There is. nevertheless, a peculiar type of long-range order hidden in the ground state. For reasons which will become clear below. this order is revealed by considering the singular gauge field used in the study of'·anyons" 16• 17 :

markable result that both fermion and boson systems map into bosons in this singular gauge. Substituting (7) into (J) and using Laughlins's plasma analogy, 5 a little algebra shows that the singulargauge density matrix p can be expressed as

~5)

where fj=2/m, and ll/(z,z') is the difference in free energy between two impurities of charge m/2 (located at z and z') and a single impurity of charge m (with arbitrary location). 3ecause of complete screening of the impurities by the plasma. the free-energy difference tJ./(z.z') lapidl}, approaches aconstant as Iz -z'l - 00. This proves the ClC.lf:tence of ODlRO 18 cna!'3cterilc.d by .HI e'pc;Jcnt ,6 - I -m/2 equal \0 the:: J:lasma "temperatUI~" For m -I the asy;nptotic value of llf can be fo:,mc.J exactly~ !Jill_ - -0.01942. For o(her values of m. /J~f(; ,2 ') \:an be estimated ~ither by use of the iondisk approximation 2.lS or the static (linear response) susceptibility of the (classical) plasma calculated from the known static structolre factor 8 (see Fig, I), The rigorous and quantitative results we have obtained above are valid for the case of short-range repulsive interactions for which Laughlin's wave function is exact. We now wish to use these results for a qualitative examination of more general cases and to deepen our understanding of the ODLRO. We begin by noting that p can be rewritten in the ordinary gauge as

2.'

p(z.z')

-(v!2K)exp(-fjtJ./(z,z'»lz-z'l-m/2,

where <1»0 - he/I' is the quantum of flu \ and A is a constant. The addition of this vector potentiiJl to_the Hamil[Onian is not a I rue gauge transformation since a nux tube is attached t,) each particle. If. however. ;.-m ""here :t1 is :tn ir.tcg~r. toe nt t elT';t.:l is j'Jst ;) ctldngc in. the phase of lhe '1..1\':: fU"~'I\Oil: Vpe .. -exp

l(

-,'m L 1m in(:, -:,) l't'o!d· I

<J

)

Application of (6; to the Laughlin wave lunction (4) yields

(7)

which is purely real and is symanetric under particle exchange for both even and ode! m. Hence we have the re-

p(z,z')

-11f d'z,

I

(8)

. t1'zNOXP [-; :c1.·dr' .,(,I.·u,z" ... ,ZN).U',Z,.... ,ZN),

where z and z' are vector representations of z ar.d z'. 'he line integral in (9) is multiple valued but its exponential is single valued because the ftux tubes are q~antized. The additional phases introduced by the singular gauge transformation will cancel the phases in ,., nearly everywhere, and produce ODLRO in p if and only if the zeros of., (whirh must necessarily be present because of the magnetic field '9) are bound to the particles. Thus ODLRO in p always signals a condensatiOG of the zeros onto the particles (independent of whether or not the composite-particle occupation of the lowest momentum state diverges' S ). Because the gauge field .A. depends on the positions of all the particles, p differs not just in the phase but in magnitude from p. This multipar~icle object, which explicitly exhibits ODLRO, is very reminiscent of the topolOSical order parameter in the XY modellO and related gauge modeIs 21 •22 and is intimately connected with the frustrated XY mode" which arises in the correlated-ring-exchange theory. 9 For short-range interactions, the zeros of., are directlyon the particla and the associated phase factors are exactly canceled by the gauge term in (9) Iscc Eq. (7»). As the range of the interaction increases, m - I of the zeros move away from the particles but remain nearby

(9)

1.0 --IDA

-·---Exact

0.5

m=5 -------=I m=3 - - ----~ m= 1--·-·-----------

--'lRA

E

.........

:t D."

~~-

I

-0.5

,,----~--------~ m=5 -1.0 -1

o

1

In (r)

2

3

FIG. I. Plot of -/JA/Cz,z')/", VI ,.Iz -z'l for IDling factor II -1/"" LRA is IiDcar-raponse approximation, IDA is ion-disk approximation
ne

1253

260 VOLUME

58, Nl'MHER 12

PHYSICAL REVIEW LETTERS

and bound to them. 7•2l The gauge and wave function phase factors in (9) now appear in the form of the bound vortex-antivortex pairs. We expect such bound pairs not to destroy the ODLRO and speculate (based on our understanding of the Kosterlitz-Thouless transition 20 ) that the effect is at most to renormalize the eltponent of the power law in (8). As the range of the potential is increased still further, numerical computations 7 indicate that a critical point ;s reached at which· the gap rather suddenly collapses and the overlap between Lauc::h!in's ,tare and t h~ t,Ut: ground 5!ate drops q!Jickly to l.ero. \"e propost' tna! thiS gap collapse corresponds to the unbinding of the vr,rtices ar.d hence to the It)sS oi ODlRO and the onset O! short-range behavior of p(z.z'). Recall t~:\t the dlsfingui!'hi!'lg fe.wJr: of the FQHE s~ate is its iong w .. ·.;:lcll~~r. e'\citalioll gap. ,'I.l least willtin the singk-mode approltimation,' this gap can only eXI~t \I,I~~~ t~e ground state is h0rn~gen,!l)u,; and the tw~-poir.t C')~:·'!!'.i.:IGi' function exhibit.; p.err~d s~~ee'lin~:

In the analog plasnla problem, the zeros of '!' act like point charges seen by each pta, ticle and the M I sum rule implies that electrons see ~ach other as charge-em -Ill') objects; i.e., that m zeros are bound to each electron. Thus (within the single-mode r.pproximation) there is a one-to-one correspondence between the existence of ODLRO and the occurrence of the FQHE. The exact nature of the gap-collapse transition, which occurs when the range of th(. potential is increased,7 is not understood at present. However, it !:lS been proven 8 thot the M 1 sum rule is satisfio:d by every homogeneous and is.otropic state in the lowest Landau level. Hence the vortex unbinding should be a first-order transition to a state which breaks rotation symmetry (like the TaoThouless state 24 ) and/or translation symmetry ((ike the Wigner crj·staI 4•8 ). We know that as a function of tern~rat"re (for fixed interaction potential) there can be no S-

UI) This equation and the parameter 8, which determines the charge carried by an isolated vort~x, originally had to be chosen phenomenologically. 4 Now, however, it can be justifitd by examination of Eq. (5) which shows that the curl of A J is proportional to the density of particles. If

1254

Kosterlitz-Thouless transition 20 since isolated vortices (quasiparticles) cost only a finite energy in this system 4•2S (see, however, Ref. 10). Further insight into the gap collapse can be obtained by considering the exceptional case of Laughlin's wave function with m > 70. In this case the zeros are still rigo&"ously bound to the particles so that the analog plasma contains long-range forces (and p exhibits ODLRO), but the plasma "temperature" has dropped below the freezing puinl. 2.15 If such a statc exhibits (suffki~ntly 10) long-rang~ p')sitional correlatiJns. the FQHE wOlJld be d~stroye<1 by a gapless Goldstone mode .tSSOciaicd with the broken tiansldtion symmetry. Hence in this exceptional case the normal connection between ODLRO ar.d the FQHE \I:auld ~ ~Eok~n b/ a gap r.ollapse due t(l positional ~rd.ef at a till:te wave vectur. The existence of ODLRO in p is the type or infrared prcperty which s;.;gg:;;ts that .l field-theo:-etic approach tc tl:e FQHE ","ouid be viab!e. It is clear from the [¢Sl.Llts presented here that the binding of the zeros of .., to tbe particles can be viewed as a condensation,I8 not of ordinary particles, but rather of composite objects consisting of a particle ami m flux tubes. (We emphasize that these are nOI real ftux tubes, but merely consequences of the singular gauge. The assumption that electrons can bind real ftux tubes 26 is easily shown to be unphysical. 27) The analog of this result for hierarchical daughter states of the Laughlin states 7• IS would be a condensation of composite objects consisting of n particlt!s anrf rn ftux tubes (cf. Halperin's "pair" wave functions I~ J. This seems closely analogous to the phenomenon of oblique confinement 22 and it ought to be possible to derive the appropriate neld theory from lirst principles by use of this idea. Since the singular gauge maps the ;oroblem onto interacting bosons. coherent-state path integration 28 may prove useful. A step in this direction has been taken recently in the form of a Landau-Ginsburg theory which was developed on phenomenological grounds. 4 In the static limit, the action has the "9 vacuum" form

f d1r I (-;V+a)~(r) 12+ i~(r)("'-., - 1) -i(9!8Jr,2H.V)(a+a)(V_),

where a is not the physical vector potential but an effective gauge field 4 representing frustration due to density deviations away from the quantized Laughlin density and , is a scalar potential which couples both to the charge density and the "flux" density. From (10) the equation of motion for a is (in the static case):

23 MARCH 1987

(10)

r we identify a in (10) and (II) as a-A.+A,

(12)

where A is the physical vector potential and we take If- If as the particle density relative to the density in the Laughlin state, then Eq. (II) follows from (5) with the 8 angle being given by 8-21flm. This yields 4 the correct charge of an isolated vortex (Laughlin quasiparticle) of q- -11m. The connection between this result and the Berry phase argument of AroYas el al.29 should be noted (sec also Semenoff and Sodano lO ). To summarize, it is the strong phase fluctuations induced by the frustration

261

VOLUME 58. NUMBER 12

PHYSICAL REVIEW LETTERS

associated with density variations [Eq.
1The Quantum Hall Effect, edited by R. E. Prange and S. M. Girvin (Springer-Verlag. New York. 1986). 2R. B. Laughlin, Phys. Rev. Lett. 50, 1395 (1983). lp. W. Anderson, Phys. Rev. B 21. 2264 (983). "S. M. Girvin, in Chap. 10 of Ref. l. SR. Tao and Yong-Shi WU,-Phys. Rev. B lG, 1097 (984). 6D. J. Thoulcss. Phys. Rev. B 31.8'05 (985). 7F. D. M. Haldane, in Chap. 8 of Ref. l. •s. M. Girvin, A. H. MacDonald, and P. M. Platzman, r-:,ys. Rev. Lett. 54, 581 (985), and Phys. Rev. B 33. 2481 (1986); S. M. Girvin in Chap. 9 of Ref. I. 'S. Kivelson. C. Kallin, D. P. Arovas. and J. R. Schricffer, Phys. Rev. Lett. 56,873 (1986). lOS. T. Chui, T. M. Hakim, and K. B. Ma, Pbys. Rev. B 33,

23 MARCH 1987

7110 (1986); S. T. Chui, unpublished. IIG. Baskaran, Phys. Rev. Lett. 56, 2716 (1986), and unpub-

lished. 12R. P. Feynman, Phys. Rev. 91, 1291 (953). IlS. M. Girvin and T. Jach, Phys. Rev. B 29,5617 (984). 14E. Brezin, O. R. Nelson, and A. Thiaville, Phys. Rev. B 31, 7124 (985).

ISR. B. Laughlin, in Chap. 7 of Ref. I. 16F. Wilczek, Phys. Rev. Leu. "9,957 (982). 17 0. P. Arovas. J. R. Schrieffer. F. Wilczek. and A. Zee, Nucl. Phys. 8151\ 117 (1985). "We refer to this as OOLRO or condensation because of the slo·.v power-law decay even thoulh the largest eigenvalue A=fd 1zp(z.z') of the density mat riA diverges only for m:S 4 (see C. N. Yang. Rev. Mod. Phys. 34. 694 (1162»). 19B. I. Halperin. Helv Phys. Acta 56. 75 (1983). 20 J. V

Jos~. l. P.

Ka
~'n.

Phys. Re\·. B ". 1217 (1~71)_ llJ_ B- Koaut. Re\!. Mod. Phys. 51. 659 (lCn~). 22J. L. Cardy and E. ltabinOYici. NIICf. Pby~. B 1~, I (19~2): J L Cardy. Nucl. Phys. B 10!1. 11 (1ge2). 21 0. J. Yoshiob. Phys. Rev. B 29.6833 (1984) 24R. Tao and D. J. 'houless. Phys. Rev. B 28. 1142 (1983). The :.ymmel:ic guge version of this state eAhibits threefold rotalional symmetry. 2sA. M. Chang. in Chap. 6 in Pef. l. 26M. H. Friedman, J. B. Sokoloff. A. Widom, and Y. N. Srivastava, Phys. Rev. Lett. 51, 1587 (1984), and 53, 2592 (984). 27F. D. M. Halaane and L. Chen, Phys. Rev. Lett. 53, 2591 (984). lIL. S. Schulmall, Techniques and Applications 0/ Path integration (Wiley, New York. 198 U.

290. Arovas, J. R. Schrieffer, and F. Wilczek, Phys. Rev. Lett. 53. 722 (984). 100. Scmenoft" and P. Sodano, Phys. Rev. Lett. 57, 1195 (1986) . liF.

D. M. Haldane and E. H. Rczayi, Phys. Rev. Leu. 54,

237 (985), and Phys. Rev. B 31, 2S29 (985); F. C. Zhanl. V. Z. Vulovic, Y. Guo, and S. Das Sarma, Phys. Rev. B 32, 6920 (1985); G. Fano, F. Ortolani, and E. Colombo, Phys. Rev. B 34. 2670 (986).

Note added: E. H. Rezayi and F. D. M. Haldane have recently succeeded in computing the singular gauge den.:.ity matrix for a small number of particles on a sphere. The results are in complete accord with the discuwion in the penultimate paragraph of this paper. See: E. H. Rezayi and F. D. M. Haldane.Phys. Rn. Lett. Vol. 61 (1988) 1985.

1255

262

FRACfIONAL STATISTICN IN mE QUANTUM HAlL EFFECf

R. B. LAUGHLIN L.~partment

of Physics. Stanford Universi ty

Stanford, California

Lawrence

Liv~rmore

94305

National Laboratory

P. O. Box 808, Livermore, California 94550

Abstract

The

fractional

statistics

reviewed.

physical

and

in

In particular

mathematical

the quantum hall there

t

is shown

origins

of

effect are to exist a

unique Schrodinger representation for quasiparticles of charge e/m in which the equations of motion are the same as

those

except

for

that

the

electrons

in

the original problem.

the wavefunction acquires phase exp(i1T/m)

when particles are interchanged. was proposed by Halperin.

This behavior, which

accounts physically for

the

Haldane hierarchy and is consistent with experiment.

1.

Introduction The purpose of

review

of

the

how

these

lectures

fractional

fractional quantum hall effect. my present view that fractional

is

to give a

statistics

arises

thorough in

the

As many of you know, it is statistics also accurs in

263

doped

antiferromagnets

and

temperature superconductivity.

is

a

likely

cause

of

high

This opinion is based to a

large.extent on the precedents established in studies of the fractional quantum hall problem and is therefore difficult to defend without first establishing the validity of those precedents.

In I ight of this problem. I have decided in

these lectures to concentrate on the quantum hall effect and to make as convincing a

case as

I

can

that

fractional

statistics is real. Part of the conft!3ion many people experience when trying to understand

fractioi.~l

statistics in the context of this

problem is due to the haphazard way it developed. much

else

in

this

field.

fractional

Like so

statistics

discovered accidentally by the experimental is i:S.

was

I refer

specifically to the observation by Stormer et. at. (1983) of the subsidiary fractions 215 and 2/7 and the subsequent observations by other researchers (Chang. 1987) of fractions with denominators of 9 and 11. was made.

Shortly after this discovery

it was pointed out by Haldane (1983) that the

observed fractions could be understood as a hierarchy of quantum hall effects. the ground state at each level of the hierarchy being a condensate of elementary exci tations of the previous one.

The mathematics Haldane used to describe

the hierarchy led him to assert that the quasiparticles were bosons.

Somewhat later, it was pointed out by me (Laughlin.

1984) that this mathematics also applied to ordinary holes in an otherwise filled Landau level and thus should be construed as describing fermions. pointed

out

that

the

only

Halperin (1984)

statistics

physically was that of anyons (Wilczek.

that

made

then sense

1982). particles

whose wavefunction changes by a phase exp(i21r/q). where q is an integer.

when particles are interchanged.

Halperin's

264

theory proved difficult for most people to understand. and thus

to believe,

experiment However.

and

more

because was

it

relied unusually heavily on

somewhat

rigorous

work

imprecise

that

mathematically.

followed

confirmed

essential correctness of Halperin's ideas. Arovas,

Schrieffer,

and

Wilczek

(1984)

Berry phase associa ted wi th adiabatic q'.l~siparticles,

and

two-quasiparticle

found

state

that.

wavefunctions proposed by me.

that expected of connection proposed

particle~

between

in terchange

provided

the

of

two

that

the

approxi~ated pl12S~

the Berry

by

was exactly

obeying fractional statistics.

such

"f:.:actional

In particular, investigated

~ell

was

the

wavefunctions

statistics"

discovered by me (Laughlin.

and

Halperin's

wavefunction

1987).

The

A

was

later

transformation I

found is the main subject of these lectures.

It has now

become clear that the bose. fermi and anyon descriptions are mathemati~ally

equivalent.

Only

the

anyon

description.

however. is physically correct. While

considering

the

mathematics

of

fractional

statistics in this system. it is appropriate to keep in mind the possibili ty that important new physical principles are at work.

The fractional quantum hall effect is the only

known case in which fractional quantization occurs in two or more spatial dimensions.

Since fractional

statistics can

exist only in two dimensions. it is perhaps an accident that this

is

also

the

only

known

occurrence

of

fractional

statistics in nature.

However. another possibility is that

fractional

or its incarnation in one or three

spatial

statistics.

dimensions.

fractionalization. charge

is a

necessary consequence of

It is a

fractionalization

fact

exhibit

charge

that all lmown cases of anomalous

associated with interchange of the particles.

Berry

phases

In two. apd

265

presumably more, dimensions, it is appropriate to think of these

phases

as

constituting

between the particles.

a

long-range

gauge

force

These arise in the quantum hall

problem for the same reason that charge is fractionalized. That is, e/3

to

in order for a particle carrying electric charge exist,

it

is necessary

that

deposited at the sample boundary. the ground

state has a

information about

exactly

to

chara.cterizing

the charge That

boundary. this

2e/3 be

stiffness

that

to be communicated

the

should

force

other

This can only happen if

peculiar kind of

erables

the

the

coupling

equal

the

constant particle's

fractional charge cO'.lld be a coincidence, but this seems unlikely to me. I shall first

These lectures are planned as follows.

review the elementary mathematics of electrons in magnetic fields

and

show

how

the

analyticity

fractional quantum hall effect arises. the

ground

and

exci ted

state

crucial

to

the

I shall then review

wavefu!lctions

for

the

fractional quantum hall problem and discuss some of their properties.

In particular, I shall consider the preferred

nature of the ground state, the fractional quantization of quasiparticle charge. electrons

in

the

overcompleteness

the similari ty of quasiparticles to

lowest Landau of

the

quasiparticle motion.

level,

natural

and

basis

for

the

inherent

describing

I shall then introduce a Schrodinger

representation for quasiparticles and show that it is the only possible such representation.

The general ization of

this representation to two or more particles will then lead in a unique way to a "fractional statistics" representation for quasiparticles in which the equations of motion are the same as Having

those of the electrons in the original problem. shown

that

such a

represent~tion

exists

and

is

266

appropriate.

I shall then review Halperin's theory of the

fractional quantum hall hierarchy. and make the case that the experimental observation of the hierarchy consti tutes proof that fractional statistics occurs in this system. 2.

The Fractional Quantum Hall Hamiltonian Th~

mov i!1g

fractional quantum hall state pertains to electrons plane

in a

~nd

subject€:d to a magnetic field of

streqgth R perpendiculd.r to this plane.

~

where

n = 2 rl --21 179. j

me

1

J

+

~cejl~ n

+

V(r j

)

]

+

The Hamiltonian is

~ j
2 Irj-r e I

• (2.1)

k

V denotes the electrostatic potential generated by a

uniform neutralizing background.

The coulombic form of the

electron-electrorl interactions in this Hamil tonian is not particularly important.

The approximate vmvefunctions with

which we describe this system work even better when the range of the interactions is reduced. 3.

Landau Levels

We shall describe the magnetic field in symmetric gauge. for which the vector potential is B

A=

2"

A

,..

(yx - xy)

(3.1)

In this gauge. the energy eigenstates of the s.ingle-electron Hamiltonian. given by

~o may be written

= -.!. I~ 2m 1 e

+ ~12 c

(3.2)

267

1

"'k.n

=e

2

_ ~x2+y2) ka a] 2 ay) (ax - i ay)n e ' .

2

n4X+Y)[a

(ax +

a

i

(3.3)

with the magnetic length l. given by

(3.4)

set to unity.

The energy eigenvalues are given by

~o ~k In = (n

1 -2)hw c ·~k In .

+

(3.5)

where the cyclotron frequency w is given by c

(,)

c

eB --m c

(3.6)

e

The set of states with the same quantum number n is called the nth Landau level. 4.

Analytici ty of Wavefunctions

We shall consider the limi t

in which there are only

sufficiently many electrons to partially occupy the lowest Landau level.

that these electrons are spin-polarized (by

the Zeeman interaction), and that the cyclotron energy is much greater than the characteristic coulomb energy. i.e.

fu.J

2 »~­ c l

(4.1)

When these conditions are met, it is a good approximation to require the many-body wavefunction to be comprised solely of

268 single-electron wavefunctions

lying in the

lowest Landau

It is convenient to write these in the manner

level.

(4.2)

where z

= x+iy

is a complex number describing the location

of the electron in the x-y plane.

thus

tantamount

to

requirir~

This set of conditions is the

wavefunctions

to

be

2.nalytic. 5.

Fractional Quantum Hall Ground State

The fractional quantum hall wavefunctions 1m), defined by 1m>

=

(5.1)

rr (z.-zk)m e

J·/k '-

J

are approximate variational descriptions of special ground states (Laughlin, 1983a) available to electrons Eqn. (2.1). apart

effectively while allowing that

by

They have the property of keeping the electrons

lowest Landau level. limit

desc~ibed

the

them

to

reside

in

the

They are exact ground states in the electron-electron

repulsions

become

infinitely short-ranged. 6.

Analogy wi th Classical Plasma.

The state 1m> is a quantum liquid of density 1

am - 2mn

(6.1)

One lmows this to be the case because the square of the

269 wavefunction

may

be

interpreted

as

the

probability

distribution function of a classical statistical mechanics problem.

Writing

.~)12 = e

I"'m(zl'

-(34)

(6.2)

with

1

-

(3

(6.3)

m

we obtain

(6.4)

This

is

the potential

Particles

o£

energy of a

"charge"

one-component plasma:

repel I ing

m

each

other

wi th

logarithmic interactions. the natural "coulomb" interaction in two dimensions. and being attracted "neutral tzing background" of

l~g~~ithmically

"charge" densi ty a

to a

= (21T) -1 .

The background ,is "coulombic" because

(6.5)

SincE;

this

"plasma"

rr'ust

be

electrically

particles must arrange

t~emselves

cancel

of

the

However. them

"charge"

the

square

centimeter

background is l/m of a .

the

on the average so as to neutralizing

the particles carry "charge" m,

per

neutral,

required

to

One lmows that

background.

so the number of neutralize

the

the s ta te is a

liquid because the temperature of the equivalent plasma is sufficient to melt it. 7.

Odd-Denominator Fractionc; The

states pertinent

to

the

fractional

quantum hall

270

effect

must

be

odd

under

interchange

of

coordinates. since electrons are fermions. to be odd.

the

electron

This requires m

The first nontrivial value of m. namely m=1.

corresponds to a full Landau level.

This may be seen by

expanding the Vandermonde determinant in the manner

= p2

.IT (z.-zk)

~
J

o

1

sgn(p) z (l)z (2)

P

P

Z

N-l

(7.1)

pen)

wh.ere p denotes a permutation of N things and sgn(p) denotes 1 t3

The state m=l may thus be seen to be a single

sign.

Slater determinant in which the first N states described by Eqn. (4.2) are filled with electrons. state.

m=3.

is

the

quantum hall

The next nontrivial

state at

1/3

discovered by Tsui. Stormer. and Gossard (1982).

filling The next

state in the sequence. m=5. was predicted by me and is now known to exist. is

that

The

c~rrent

belief (Lam and

Girvi~.

1984)

the sequence becomes unstable to crystallization

near m=7. and that the quantum liquid states for m=9 and greater do not exist. B.

Fractionally Charged Quasiparticles

The

quantum

waverunction charge

1m>

± e/m.

liquid

state

contains

approximated

excitations

carrying

by

the

electric

They are described approximately (Laughlin.

19S3a) by the wavefunctions

_ ! }: Iz 12 S z' 1m>

=e

4 I!

I!

,

~ (z i -z ) 1

and

(8.1)

271

where z'

is a complex number locating the center of the

quasiparticle. adjoints

of

The each

S and st are hermitean

operators other.

Eqn.

(8.1)

represents

a

fractionally charged particle because it corresponds to a "charge"

1

screening

cloud

in

the

equivalent

plasma.

Squaring the wavefunction and interpreting the result as a statistical

mechanics

problem.

we

obtain

the

potential

energy

~

=

I -2m22nlz.-zkl + I m, j
IZel2 - ~

2m enlz.-z·

i

I .

(8.3)

1

This is the same as before except the particles now see a phan tom of "charge" 1 I -:lea ted at z·. plasma.

the

particlec;

must

Since this is a

screen

this

phantom

by

accumulating equal and opposi te "charge" in its vicini ty. Since the "charge" carried by each electron is m. however. the number of electrons contained in the screening cloud Is exactly -l/m.

Similar reasoning (Laughlin. 1984b) works for

Eqn. (8.2). 9.

Exactness of Quasiparticle Charge

The

existence

of

fractionally

charged

exci tations

follows generally from the existence of an energy gap.

We

lmow the state has a gap because the experiments make no sense

without

one.

For

those

not

satisfied

by

this

argument. we remark that Haldane and Rezayi (1985) have made a very convincing case that the m=3 state has a gap based on exact

diagonalizations

calculations 1983b).

have

also

of been

small

systems.

reported

by

me

Similar (Laughlin.

If a gap is lmown to exist the excitations may be

generated with the following thOUght experiment (Laughlin.

272 1987) :

One imagines piercing the plane of motion wi th a

thin magnetic

solenoid and adiabatically

forcing

through

this solenoid a quantum of magnetic flux. given by

~

As

f lux

is

continuously its

and

added. a~

eI!ergy

= hee

o

(9.1)

the ground state wavefunction evolves

an eigen5tate of the changing Hamil tonian. eigenvalue

bei!'..g done on the system.

quantum has been added.

increases

in respo:1se

to work

Bowever. after an el'.. tire flux

the Hamiltonian has. up to a gauge

transformation. returned to its original state.

Thus.

state

into which the ground state finally evolves

exact

excited

excitation n~ntrivial

us

state

created

of in

the

original

this

manner

certain

The to

only if the ground state is nondegenerate.

include

in

the

notion of

degenerate

arbitrarily low-lying excited states. example.

is an

Hamiltonian. is

does not possess

the

be Let

the presence of

A Wigner crystal. for

fractionally charged elementary

excitations because its grcund state is cegenerate. We know that the exci tations created in this way are fractioaally charged particles because the addition of flux through

the

solenoid generates gauge forces at

infini ty.

Let us assume for simplicity that the solenoid is located at The addi tion of flux ~ through the solenoid

the origin. then

causes

single-body wavefunctions.

out

of

which

the

many-body wavefunction is constructed. to move over. in the manner

1

r

k

e

ike

e

e

ike

e

-

4' r

2

(9.2)

273

with r and 9 related to z by

z

=r

e

i9

If the ground state possesses

(9.3)

~o

low-lying excitations, it

can only respond at infinity by moving over in the same way. thereby transporting electric charge in or out of the region of

in teres t .

Tbe amount of

charge

transpor ted by

the

z.ddition of arl entire flnx quantum is the average charge ,er

state at infinity. or ± elm. A critically important corollary of this reasoning is that the quasiparticle charge must be exactly ie/m, for any Hamiltonian.

even

one

for

which

the

approximations we normally assume

is

because

depends

the

quasiparticle

charge

hierarchy

invalid.

properties of the system at infini ty.

only

of

This

is

on

the

To illustrate this

idea. let us imagi:le creating a quasiparticle in a system for which Eqn. (4.1) is not valid. so that the approximation of restriction to the appr~priate.

for

which

lowest Landau level

is no longer

We can then embed this system in a larger one. the approximation

accomplished.

£or example.

is

valid.

by making

This the

migh~

electrcn

decrease slowly wi th distance from the origin.

be

~ss

If we now

insert flux through a solenoid placed at the origin,

the

charge of the excitation that results is still the average charge per state at infini ty. namely the ideal. value of

± elm.

regardless

solenoid.

of

what

the Hamiltonian

is

near

the

However. since the charge is accumulated near the

solenoid. it must actually be insensitive to the behavior at infini ty.

Thus.

system is absent.

the charge is exact even if the larger This reasoning is valid so long as an

energy gap exists everywhere in the sample.

274

10. Shift Oper.ators S and Sf The

shift

operators

following manner.

S and st

are

obtained

in

the

We lmow that the action of the thought

experiment far away from the solenoid is to shift the ground state over.

Since we cannot lmow exactly what it doe3 near

the solenoid,

as this depends on the Hamiltonian. firs~

reasonable. as a

in this region too.

preserve

to

gues.s, to shift the ground state over

To accomplish this literally one needs

norm of

the

it is

each single-body crbi. tal.

For

example, if the solenoid is at the origin, we have

1 k ----z

~2k1r k!

~lzl2

e

since

However,

1

k+1

---+ - - - - - -

this

(10.1)

Z

to

difficult

is

implement

mathematically. we substitute the slightly different mapping

- -1- zk+l ~2k1r k

- -1- - zk J2klr k! arguing solenoid

that and

tIle

two are

that

the

mapping

estimate in the first place. substitute st

for

equivalent was

(10.2)

far away only

a

from

variational

This is the action of S.

the action cf

the

the We

reverse evolution.

namely

1

---- z

~2k1r k!

for

the

same

k

e

-

~lzl2

reason.

k

--+ - - - - Z

~2k1r

st has

k-1

11 12 -4'z e

• (10.3)

k!

the

useful

feature

of

275 annihilating any configurations containing any electron in the

state

k=O.

evolution sends effectively

This

is

what

this state

destroying

any

we

into

want.

for

adiabatic

the next Landau

electron

in

level.

Extensive

it.

computer calculations (Haldane. 1987) have now shown that S and S t

create extremely good approximations to the true

quasiparticle wavefunctions for m:.:3. 11.. Simi lari ty to Electrons in Lowest Landau Level

The

quasiparticles

fractionally charged level.

may be

el~ctrons

thought

of

physically as

moving in their lowest Landau

Specifically, the state

_ ~lz'12 S z ,1m> Iz'> = e which describes a

quasihole

(11.1)

located at posi tion z·.

is

analogous to the single-electron wavefunction 1

-pz' (z)

=--e

_~lzI2+lz'12] (11.2)

../lii

This latter is obviously a linear superposition of states of the

form

of

Eqn.

(4.2)~

In

order

to

demonstrate

conclUSively that this analogy is reasonable we must show that the matrix elements of overlap and energy are the same in the two cases.

The former is necessary because the

states are overcomplete.

Part of this task is elementary.

It is clear by inspection that the expected energy of

Iz'>

does not depend on the quantum number z·. just as is the case wi th fJ • (z). This energy is essentially the coulomb z repulsion of the quasiparticle charge with itself. The rest

276 of

the demonstration requires

that we invoke plasma sum

rules and the analytic properties of Iz').

12. Overcompleteness and Analytic COntinuation Both

the

quasiparticle

single-electron states 3..

states

Iz')

and

the

~

.(z) are very overcomplete. It is z anique feature 0f this problem that a basis set of this

kind is easier to use th2,n the just-barely-complete basis

custumarily emplcyed in

qua~tum

The difficulty

mechanics.

of even con8tructing a traditional basis, particularly when

than

~ore

one

fundamental

quasiparticle

p~oblem

is

related

fractional statistics.

present, to

is

the

actually a

quasiparticles'

The penalty we pay for working with

an ove'rcomplete basis is the need to keep track of both the

Hamiltonian and the overlap matrix. two

problems

with

the

For example, even if

F~iltonian,

same

th~y

are

not

equivalent unless their overlap matrices are the same as well.

The advantage t,o using this particular overcomplete

basis is that the off-diagonal matrix elements of both the Hami I tonian aild over lap matrices are related by analytic continuation to the diagonal matrix elements.

This enables

powerful and useful statements to be made about them. This useful

is an appropriate point

property of gaussian

to state the following

integrals:

If F(z)

is any

polynomial and z' is any complex number. then

J

F(z) e

-.1.lzl2 2m

1 . * 2iji'Z z e

regardless of the value of m.

13. Analytic COntinuation of Overlap and Energy

(12.1 )

277

Let us now verify the equivalence of Iz') and ~z ,(z) by working out the matrix elements of overlap and energy. Let Z·

and z'

denote

two different

posi tions.

(12.1). we have for the overlap of

~Z.(z)

and

Using Eqn. ~z.(z)

e

x

~e

-

1

')

~lzl"+lz"

2

] 2'1 z .*Z e

- klz'12+lz'12] 4

=e

e

21

Z'

Z

]

d

2'1 Z' Z* ]

2z

,M

(13.1)

This resul t may. however. be obtained a simpler way.

We

observe that. but for the exponential factors this integral is analyti~ in the variables Z' and zeN.

We also observe

that its w.lue is a constant (unity) when Z' = z'.

This

means that the analytic part must equal exp( Iz'12/2) when

Z'

= z'.

Since this function has the unique continuation of

exp(Z'z' */2). the whole matrix element must have the form of Eqn. (13.1).

Let overlap.

us

now

turn

to

the

case

of

the

quasiparticle

For the diagonal matrix elements we have

x

2 2 rr Iz.-~ 1 d z1

j
J

2 d ~

= (010)

(13.2)

278

10) denotes the state wi th a quasiparticle at the

where origin.

This integral does not depend on the value of z'

because it is the probability to find a plasma particle of

1 at

"charge"

location z'.

allowed to move. any

particular

constant.

given

that

the particle

is

The particle has no a priori reason to be place

in

the

plasma.

so

We now observe that, but for the

the

result

expon~ntials

is in

z' and zIt we added in Eqn. (11.1) to make the norm constant. • t h_e over 1ap matrix' is ana 1ytic in Z ,M and z.

Continuing

this analytically, we obtain

I


= <0 10)

- ~Iz' 12+lz'12] e

t7'*Z ,

2ijiL

(13.3)

e

which is equivalent to Eqn. (13.1).

magn~tic

Note that the

length is effectively larger in Eqn. (13.3) than it is in Eqn. (13.1) by the factor ~. exactly as one would expect of an electron of charge e/m moving in a magnetic field of strength H. The

matri~

elements

of

the

Hamiltonian

obviously

continue in exactly the same way. so there is no need to work them out.

14. Cyclotron Motion of

~iparticles

The similari ty of quasiparticles to electrons

in the

lowest Landau level is exhibited even more dramatically when the quasiparticle is caused to execute cyclotron motion. This is accomplished by combinfng the states Iz') into an eigenstate

of

angular

momentum

analogous

defined by Eqn. (4.2), in the manner

to

the

state

279

Ik>=

1

J(2m)k+2v~!

J

(,*)k z

_ ...!lz'12 Iz '> d 2 z, . e 4m

(14.1)

This is the quasiparticle analog of the expression

. (14.2)

The normalization integral cf the quasiparticle state is

(k Ik>

=

(2m)

k+21 2

v-k!

_ ~lz'12+lz"12]

fI('

z z u*)k e

4m

(14.3) The orbit radius of the state may be practically defined as

(14.4) ~here

p(z) is the electron density operator. given by p(z)

= Ij

6(lz-z.l) oJ

(14.5)

Since. from homogeneity and isotropy of the plasma. we have

280 and since,

from

the

constant-screening sum rule of

the

plasma, we have

(14.7)

the mean square orbit radius is

=

1 _

IT

(Z' *z·)k [Z I Z

l' J

(2 m) k+21i K.

_

1*

+ 2 ]

The analogous value for ar~ electron in the state ~klz) is

J Izl

2

l~k(z)1

Thus, the cyclotron orbit is

2

d

2

Jm

(14.9)

z = 2k + 2

larger for the quasiparticle

than it is for the electron, exactly as would be expected of a pa:ticle of cha'rge e/m moving in a magnetj c

field of

slrength H.

15. SchrOdinger Representation for Quasiparticles In

light

of

the

similarity

of

quasiparticles

to

electrons of charge e/m in the lowest l2uldau level, there is a

convenient

them:

To

and

every

obvious

Schrodinger

"wavefunction" 'iJ(Tl)

superposition of states of the form

representation for that

is

a

linear

281

1

K -

IK) = - - - - - f l

e

~1~12

(15.1)

J(2m)K+l1l' K! we associate a true electron wavefunction in the manner

(15.2)

WP6rc I~> is defined as in Eqn. (11.1). true electron wavefunc t ion

Similarly. to every

Iz') we assign a quasipar t icle

"wavefunction" 'iJ ,(11) in the manner z 1

e

2m z

.*~

. (15.3)

Since ~ maps back to Iz') under the action of Eqn. {15.2}, that is

these operations are inverses of one another. the

totally

ficticious

variable fl as

the

We interpret quasiparticle

"coordinate". 16. Uniqueness of SchrOdinger Representation

This particular Schrodinger representation is preferred because

it

properties.

uniquely

preserves

analytic

continuation

In particular. for any Z' and z' we have

(16.1)

282 This is important because it causes the wavefunction have

the

same

physical

meaning

wavefunction would have.

an

Suppose,

~(~)

to

equivalent

electron

for example,

that we

V to the system and attempt to understand how the quasiparticle moves in the presence of V.

apply an external potential

If the description is physically correct, we will to be able to

find

an

equivalent

potential

in

"V(TJ)

which

the

quasiparticle effe::tively moves. and to correctly describe the motion by the solution of thp, equation

(16.2)

This will be the case provided that Eqn. (16.1) is true and

(16.3) for all Z' and z. a

solution.

reJ,Jresentation. same

analytic

This equation does not. in general. have However,

if

we

use

the

preferred

then both sides of the equation have the continuation properties

and

we

can write

equivalently (16.4) which always has a solution, at least formally.

Thus, the

preferred representation is really the only possibility.

17. Equivalence to Particle-Hole COnjugation When

m=1.

the

transformation

Schrodinger :epresentation is

to

tantamount

the

preferred

to particle-hole

283

conjugation.

implies

This

that

the

operators S create fermions when m=1. this

Since

commute.

central

to

quasihole

shift

even though they the

question

of

quasiparticle statistics. I shall discuss it in detail. Suppose we have N orthonormal orbitals x{r) into which we put spin-aligned electrons. the

state

conjugate

in

which

vacuum.

0,

no

If the vacuum is taken to be

orbital

is

the

is

state

occupied. with

then

every

the

orbital

occupied. i.e.

)(

erN) .(17.1) peN)

with p a permutation of N things and sgn(p) its sign. particle-hole

conjugate of

the

!ttate with electrons

The in

orbitals jl' ...• jM and holes everywhere else. i.e.

is the state

(17.3) However,

since

Eqn.

(17.3)

is

true

for

every

Slater

determinant of the form of Eqn. (17.2). it is also true for any M-particle wavefunction Let us now consider conjugate vacuum is

~.

the

lowest Landau

level.

The

284

(17.4)

where X is a normalization constant.

If we now take

~

to be

a Slater determinant of the form

th IP'rJ(z) defined as in Eqn. (11.2), and evaluate ~ using Eqns. (17.3) and (12.1), we obtain wi

... ,rN)

t(rM+1 ,

x

X IT (11 -11R ) a<~

a

~

=

N fI

I N! (2_)M = ~M!(N-M)! ..

s ... S 1m>

(17.6)

(Z.-11)

(17.7)

111

11M

where S 11

i=M+1

1

and

1

1m>

=

N IT

M<j
-

(z.-zk) e

4"

N

2

}: Iz~ I

~=M+l

(17.8)

J

Thus, the shift operators S are exact in the limit of m=l and generate ordinary holes in an otherwise filled Landau level. 18. More 11lan One Quasiparticle

Let us now consider the case of two quasiparticles.

We

know that the state Iz'> defined in Eqn. (11.1) describes an isolated quasiparticle very well and that the quasiparticle

285

is about a magnetic length in size.

It must therefore be

true that a state containing two quasiparticles. one at z' and one at z". is described very well by the wavefunction

Iz' .z">

= (z'_z,,)l/m

e

_ ~ [lz'12+lz"12] 4m S ,S ,,1m> . (18.1) z z

provided that z' and z" are far apart. good wavE-function when z'

it is a

rea~onable

~d

Whether this is a

z" are close is not clear. but

first guess.

Note that the prefactor in

this expression is a multiply-valued function of the quantum number

(z'-z").

This

makes

sense only

if

(z'-z")

is

understood to .reside on a Riemann sheet. The proafactor (z'_z,,)l/m in Eqn. (18.1) is requi'red because it makes Iz' • z") analytic in z' and z". up to the gaussian factors. ":'~~ile

making

its normalization integral

limit z' and z" are far apart.

constant

in

the

Both of these properties.

analyticity and uniformity. were required of the states Iz') in order that the single-quasiparticle problem admi t of a physically meaningful normalization separations

S~hrodinger

becomes. cons.tant

is a

representation. in

the

limi t

That the of

large

consequence of plasma sum rules.

We

observe that the normalization integral may be written z. z ")
= 11m 2

=

JJ ...

2

2

e -{» d zl· . . d zN

(18.2)

and

}: 2m ln Izj-z.. I - }: 2m In Izi-z' j
I - }:

2m In Izi-z" I

i

(18.3)

286

This is the potential energy of N particles of "charge" m and 2 particles of "charge" I interacting coulombically with each other and wi th a neutralizing background of "charge" densi ty a

= (2v)-I.

proportional

to

Thus.

the

the normalization integral

probabili ty

to

find

the

"charge"

is

1

particles at . locations z' and z". given that they can be anywh~re

in the sample.

This is constant except when the

particlt:::s come wi thin a

screening

which is to say a magnetic The wavefunction

length of

each other,

leng~h.

Iz·. z")

ha.s a natural general ization to

M quasiparticles:

s 1m> nM

.(18.5)

19. Fracticnal Statistics Representation We are now in a

pos i t ion to address the ques t ion of

quasiparticle statistics.

We shall do this by arguing that

the preferred representation for

two quasiparticles

Schrodinger representation in which we "wavefunction"

~(TJ'

a5~0c~ate

is a

with every

.n") in the lowest Landau level a true

electron wavefunction in the manner

wi th of

In' .n"> this

mul tiply

defined as in Eqn. (IS.1).

expression valued.

to

(15.2).

Eqn.

"'{n' ,n")

is

statistics wavefunction. that is.

Note the similari ty Since

necessarily take~

a

the form

11}' .1}")

is

fractional

287

where

F{T}' ,T}")

is

a

symmetric

wavefunctions may be thought of a

polynomial.

Such

representing particles

moving on a Riemann sheet. Let us now consider a particular fractional statistics wavefunction that we will need in making our case:

~ (z'*+z"*){T}'+T}") ~

(I)

[~ (7'*-z"*){T}'-T1',)]2n+l/m

2 I

e

,(19.3) T{2n + l/m + 1)

n::O

where T denotes

the

gamma

function.

The

sum

i-·,

expression merely interpolates between sinh and cosh.

this Thus,

in the I imi t of m ~ 1, we have

=

m-+1

x [ e

1 [z' * T}' +z" *T}"] 2m

1 [z' * T}"+z" * T}'] -2m.

- e

]

, (19.4)

which is readily recognized as the single Slater determinant

with m

~defined

~ 00

we have

as in Eqn. (11.2)

Similarly, in the limit of

288

(19.6)

] y,hich is a synunetric version of Eqn. wavefunction may od~i

thought

b6

ta Is (;entered at z

I

of

as

(19.5).

Thus,

this

putting particles

in

and z" in a way that interpolates

between fermi and bose statistics. The

l,"~)v·t?functi.on

because

it

(19.1).

That is,

1 21T11l

II

Thus.

maps

d~fined

into

.1.* ') 't'z' ,z" (TI' ',TI

to every

assign a

I

under

,ZU)

ITI ' ,TI")

(19.3) the

Since

is

important

action

of

2 2 d TI , d TI " = Iz',z") .

true e lee tron waver .lne t ion

fractional

corresponds.

Iz

in Eqn.

Iz

I

,Z tt

>

£qn.

(19.7)

we may

statistics "wavefunetion" to which it the states

Iz', zit) span the space of

two-quasjparticle states, Eqn. (19.3) defines an inverse to EGn.

(19.1), just as Eqn. (15.3) defines an inverse to £qn.

(15.2).

In

fractional

statistics representation is

other

words,

the

transformation

to

the

the analog of the

transformation to the preferred Schrodinger

repres~ntation

for a single quasiparticle. The fractional statistic$ representation has a natural generalization to M quasiparticles. statistics

"wavefunction"

wavefunction in the manner

~

To every fractional

we associate a

true electron

289

(19.8) with 1~I""'~M> defined as in Eqn. (18.5).

Similarly. to

every true electron wavefunction we assign a quasiparticle "wavefunction" in the manner

Iz 1 ···· JZ~> ;.'1

x a.

.

11.···.lM

~ ~zl'··· ,zM(~l.···~M)

*

(zll1l)

i1

*

... (zMTl..) ,'M

iM

=

1

(2-nn: )

* *

M/2

II [(z .-z.. )(~.-n. )] J K J 'K

1/m

j
(19.9)

x e

with

=

( 19.10)

Since ~ maps back to Izl •...• iM> und~r the action of Eqn. (19.9). these operations are inverses of one another.

As

was the case with a single quasiparticle. we interpret the totally ficticious variables 111 •...• ~M as the quasiparticle "coo"'dinates".

290

20. Uniqueness of Fractional Statistics Representation

The Schrodinger representation we choose for describing the M-quasiparticle problem is preferred because it uniquely preserves analytic continuation properties. with

a

single

M-quasiparticle desc~ibed

quasiparticle. energy

this

is

eigenstates

necessary

are

to

be

if

the

correctly

by solutions of the equation

or a reasonable approximation to it. validi ty

As was the case

of

quasipartic~es

solved.

this

equation.

we

Once we establish the will

have

understood

fully. for this is the equation we have just

Of

course,

first

thing

the

particles

now

obey

fractional

statistics. The

we

integral behaves properly.

must

show

1ba tis.

is

that

the

overlap

we need to rr.ake the

identification


~

(20.2)

as we did with a single quasiparticle in Eqn. (16.1). of

this

identification

has

been

taken

care

of

Part by

Both Iz' .z") and ~, ,,(11' .11") are. up to the z .z gaussian factor. analytic in z' and z". The two overlap construction.

integrals therefore continue in the same way. and thus need only be compared on the diagonal. the identification

That is. we need to make

291

This, however, is relatively easy.

Since both normalization

integrals are functions of the difference coordinate Iz'-z"l solely, we need only find a function 0 such that

lea~t

It Is always possible to do this, at overlap

integral

behaves

"properly

fOimally. If

tv

The

differs

0

significantly from a constant ~ only when 1~'-n"1 is small. This

is

evidently

the

case,

normalization integrals in Eqn.

however,

both

integrals

separations.

0

vanish

is exactly

~

b\)th

(20.3) become constant in

the limit that z' and z" are far apart. that

since

It is also the case

17'-7" l2/m

as

when m=1.

at

small

We mow from

numerical studies (Laughlin, 1987), which are reproduced for convenience in Fig. 1. that it is given approximately by

0(1111)

~ ce [1 ~ 0.5 e

11 .2 - -111 4m

]

(20.5)

when m::3. We must now show that the Hamiltonian behaves properly. However.

since

the

Hamiltonian

has

the

same

analytic

continuation properties as the overlap matrix. as well as the same translational invariance. this reduces to finding a function

~

such that

292 The behavior is "proper" if

(20.7)

where

~

However ,

is

the

t~1 is

energy

is

to

tr i vially

make a

single quasiparticle.

lTd

the ca.se when

is

large.

Numerical studies for m=3. also reproduced in Fig. 1, show it to be accurate to about 3%

fo~

small values of 11 as well

as large ones. This

the

that

stati8tics

fractional

representatton well behaved for two particles. for M quasiparticles proceeds along simil9.r final

resul t

is

that

The proof lines.

The

the M-quasiparticle eigenstates are

correctly described by solutions of the equation

(20.8)

where 0

is a constant

ce

except when two or more

T}'S

are

close together and where

.., ( T} 1 • .••

•T}M)

g:

0 ( T} 1 • ..• ,11M) MA + ce J. "J.
(e/m)2

I11j -llk I

•

(

20.

9)

21. The Hierarchical States: Haldane· s Theory

The primary experimental evidence that Eqns. (20.8) and (20.9)

are

correct

is

the

occurrence

quantum hall "hierarchical" states.

of

the

fractional

Before discussing the

293

hierarchy

in

terms

of

fractional

statistics,

an

idea

originally due to Halperin, it is appropriate and necessary to review Haldane's ideas on this subject. was

the

observed

first in

to

suggest

experiment

quasiparticles,

much

that

were

the way

Haldane (1983)

the subsidiary fractions due

to

condensations

of

the primary fractions were

condensations of electrons.

He outlined in broad terms how

this might occur and

an expression for the observed

deri~ed

In 1 ight of

fractions which we now know' to be correct. this.

the occurrence of the hierarchy ~~~ot be considered

proof that quasiparticles obey fractional statistics unless Haldane's derivation contai'ns an assumption about the way quasiparticles statistics.

move

that

is

tantamount

to

fractional

Let us therefore consider Haldane's derivation I shall take the liberty of converting

of the hierarchy. Haldane's arguments

to

the disc geometry used

in

these

lectures from the spherical geometry used by him. Let us assume

that

the

theory of

the

11m

state

is

correct, and that the effect of increasing or decreasing the occupancy of the lowest Landau level from v=l/m is to add posi tive or negative fractionally charged quasiparticles. Let us further assume that the added quasiparticles behave physically like electrons in the lowest Landau level, and in particular condense into a fractional quantum hall state of densi ty P2 quasiparticles per square centimeter. the radius of this condensate by

~

Denoting

and the radius of the

sample by R , we have for N, the total number of electrons 1 in the system

(21.1) where

the upper

sign

charged quasiparticles.

refers

to addition

of

negatively

Since an electron a distance r from

294

the origin carries angular momentum r 2/2. we also have for the total angular momentum

(21.2) The net charge added to the region r

~

R2 may be expressed

in terms of the number of quasiparticles M in the manner C)

M

Eqns.

= P2

vR2

(21.3)

(21.1) and (21.3) may then be used to express the

total angular momentum in terms of M and N:

L =

Following

-1 2m

Haldane.

(mN T M)2

we

now

± _1_ Jf

(21.4)

4vP2 m

define

"condensation"

of

quasiparticles to mean a wavefunction of the form

~ = J···Ja~~(n:~;)P F(lna-~~I) x S

111

e

1 M - _. }:In...,. I2 4m "Y

2 S 1m> d2111 ... d 11M 11M

(21.5)

or

~ = J. .. Jj~(na~~)P F(lna~~I)

e

1 M 2 --}:I11 I 4m "Y

t ... t I S m> d2111 ... d211M 111 ~

x S

"Y

(21.6)

295

where F is an arbitrary function, with p an even integer, as required by the conmutivity of the S's.

lbat p should be

even is the sense of Haldane's assertion that quasiparticles obey bose

statistics.

The

reason

for

adopting

such a

un~form

in the spherical geometry

and thus in some sense "allowed".

Eqns. (21.5) and (21.6)

wavefunction is that it is

give a secQnd expression for

the angular momentum of

the

form (21.7)

which, when equated with Eqn. (21.4) gives 1 1 P2 = 21r pm f 1

(21.8)

or

v

= iii1 ± 21rP2 = pm

P

(21.9)

f 1

With m=3 and p=2 this gives 215 and 2/7, which are the first states

in

the

wavefunction,

hierarchy.

It

is

the

choice

of

this

or more precisely the angular momentum rule

derived from it,

that is equivalent to the assumption of

fractional statistics.

It should be pointed out that the

evenness of p does not imply that the quasiparticles are bosons in a physical sense.

For example, Eqn. (21.5) with

m=l, p=2, and (21.10) describes the particle-hole conjugate [cf. Eqn. Eqn.

(5.1)

wi th m=3.

ordinary holes particles

Since

this

is a

(17.3)] of

condensation of

in an otherwise filled Landa1 l

in question are obviously fermions.

level,

the

Repeating

296

these

arguments

hierarchically,

Haldane

arrived

at

an

expression for the allowed fractions of the form

v

1

= --------------------m+ a

(21.11)

l PI + a 2

-----------------

------+ a

where

m=1.3,5 .... ,

a.=±l. 1

and

n

Ir n

p.:2.4,6, ... ·1

This

expression is consistent with experiment.

2?.

Hal~rin' s

Theory

Let us now consider Halperin's theory (Halperin, 1984) of the fractional quantum hall hierarchy. based

on

fractional

the

proposition

charge

obeys

that a

This theory is

quasiparticle

fractional

c~irying

statistics.

otherwise described by Eqns. (20.8) and (20.9).

but

is

In light of

the discussion in Section 20, it should be clear that this assu~ption

can be deduced

from properties

of

the

shift

operators S and is quanti tatively correct for quasihoies. It has not been proven correct

f~r

quasielectro~s.

almos t cer tainly correc t for them as we 11 .

but

i~

Let us first

consider the condensation of r.harge 1/m quasiholes with m=3. Given that the equations for quasihole motion are the same as those for electron motion, the quasiholes will condense into a state of the form

_1,(

~ ~l'···'~M

Note

that

the

M ) _ 11 (

-

~ ~n

a
)p+l/m

e

1 M

2

-};I~ I 4m '"r '"r

(22.1)

p

true electron wavefunction

to which

this

297

corresponds via Eqn. (19.8) is Eqn. (21.5) with

(22.2) The assumption of fractional statistics is thus consistent wi th

Haldane' s theory.

Eqn. (21.8) wi th "+".

The corresponding filling factor is

By analogy with the original electron condensation.

v=217.

this

TIle quasihole densi ty is given by

quasihole

condensate

has

a

exci tation

described

approximately by the wavefunction

S

1]

which

is a

missing. [pm+l]

,I",>

-1

(22.3)

region

from

which

[p+l/m]-1

quasiholes are

The electric charge of this particles is thus • or 117 for the case of p=2.

Halperin then assumes

that the statistics obeyed by this particle is its "charge" measured in quasiholes. or [p+l/m]-I= 317.

Note that this

is completely consistent with the discussions in Sections 19 and 20.

These particles therefore condense into a state

described by the'wavefunction

_ lIM (

'i'( 111 • ••. •n..) OM

a
)2+317

e

1 M 2 - 4x7 ~ 111'1

I

(22.4)

which corresponds to the filling factor 21215 v = 7 - (7) 2+3n = 17

(22.5)

There is a minus sign in this expression because we have

298

added "holes of holes". is the following: characterize m

s

At level s of the hierarchy, we shall

the

electric charge q

The generalization of these ideas

state

by

its

filling

factor

the

v, s

of its quasiparticles, and the statistics

s

of these particles.

We shall give q

s

a sign so as to

keep track of the "holes of holes" problem.

The transition

to the next level of the hierarchy is characterized by an

even integer

p

s+ 1 and a sign a si 1"

The equations are

rn s+1 = p 5+1 - a 5+1 1m s

(22.6) (22.7)

and q

Tacit

in

these

statistics

of

(22.8)

s+1 -- a s+1 q s 1ms+1

equations

is

quasielectrons

the is

assumption opposite

quasiholes at any level of the hierarchy.

to

that

the

that

of

These equations

are equivalent to Eqn. (21.11). and are thus consistent with experiment. Even though this theory is built on a number of unproven assumpti~~s.

there

is

little doubt

that

it

is

correct,

particularly in light of the discussions in Sections 19 and 20.

It is the only theory of the hierarchy that ascr i bes

the behavior to a physical property of the quasiparticles. It is also the only theory that makes quantative predictions as

to

which

fractions

should

be

seen.

Since

these

predictions are based on the assumption that an equation of the form of Eqn.

(20.8) exists.

believe they are correct.

23. Berry Phases

there is good reason to

299

Before concluding,

it

is appropriate

that

I briefly

discuss the fractional Berry phases discovered by Arovas, Schrieffer, and Wilczek (1984).

I have deferred this to the

end because I find it difficul t to proceed logically from Berry phases to Halperin's wavefunctions.

Nevertheless, the

fractional Berry phase is a key concept in this subject and is demonstrably exhibited by quasiparticle motion.

let us imagine a Hamiltonian vector R. exterI~l

~(R)

parameterized by some

This might, for example, locate the center of an potential localizing a quasipart:cle. in the nanner

1 ~(R) = IJ [2me I~ij + ~j 12 + V(rj-R) c

r.::-~2 . (23.1)

] + I I j
Let us further assume that for each value of R there exists a unique ground state

Iw(R» such that

~(R)I~(R»

= E(R)I~(R»

(23.2)

and

<w(R)lw(R» Ibis

gr~und

=1

(23.3)

state is arbitrary up to a phase factor exp(i').

The choice of a particular phase at every value of R is effectively a choice of gauge. evolve R along some path P.

Let us now adiabatically As the system evolves,

its

phase changes in a deterministic. manner that is in general different from the phase we picked.

In particular. if we

let I~(t»

then

"Ie

have

=

1~(R[t]»e

-i E(R[tJ) t

n

-i;(t) e

(23.4)

300

~(R[t]}I~(t»

=

in ~tl~(t}>

=

[E(R[t]} +

n ~(t}]I~(t» (23.5)

Taking the matrix element of this expression with <~(t)1 and

observing t:hat

(23.6) we obtain

(23.7) or

A~ = -

JIm[<W(R)lvRI~(R»].~~

(23.8)

.

p

If P is a closed path. then A; has an absolute meaning and is referred to as the Berry phase (Berry. 1984). Let us now locd.lizes posit:on

one

imagine

that

quasiparticle

R. so that

~(R)

at

is a

the

Hamil tonian that

origin

and

one

at

I~(R}> is given approximately by

(23.9) with

R

denoted

as

normalization constant.

the

complex

number

z'

and

H a

If z' is sufficiently far from the

origin. then H does not depend on z' and

301

The Berry phase associated wi th adiabatically evolving R around the origin is thus

A,

= J[-1. + p 2m

1 ] mlRI2

[~ - ~].dR = vlRI2 + m

211'

(23.11)

m

The first contribution is just the phase of a particle of

charge e/m moving in a uniform magnetic field. is

The second

the phase of a particle of charge elm moving around a

solenoid containing a flux quantum he/e. 24. Concludi.ng Remarks

In light of the arguments I have presented it should be clear that the occurrence of fractional statistics in this system is established beyond doubt.

The most troublesome of

some loose ends. apparent

inabili ty

statistics

There are nevertheless

to

construct

representation for

these is our

simple

fractional

quasielectrons.

The most

a

likely source of this problem is the cumbersomeness of the shift operators

st.

Another indication that these operators

may be inappropriate is that in Eqn.

,st

may not be substituted for S

(22.3) wi thout destroying the analytici ty of the One

wavefunction.

very

serious

question

is

whether

fractional statistics, like fractional charge, is robust, in the sense of Hamil tonian question

is

statistics

su~viving

that how near

under any adiabatic change of the

preserves to

the

describe the

phase

energy gap.

the

fractional transition

A related

charge and to

Wigner

crystallization, when the size of the quasiparticles gets very

large.

These and related subjects are appropriate

topics for future research.

302

This work was supported primari ly by the National Science Foundation under Grants No. DMR-85-10062 and DMR·-88-16217. and by the NSF-MRL program through the Center for Materials Research at Stanford Universi ty.

Addi tional support was

provided by the U.

of Energ;]

S.

Depar trr.en t

through the

Lawrence Liv-ermore National Laboratory under Contract No. W-74.05-Eng-4B.

REFERENCES

Arovas D. Schrieffer J R. and Wilczek F 1982 Phys Rev Lett 53 722 Berry lei V 1984 Proc Roy Soc London A392 45 Chang A lei

1987 in The Quantum Hall Effect. edi ted by R E Prange and S M Girvin (Springer. New York) 175 Haldane F D M

1983 Phys Rev Lett 51 605 1987 in The Quantum Hall Effect. edi ted by R E Prange and S M Girvin (Springer. New York) 303 Ealdane F D M and Rezayi E H 1985 Phys Rev Lett 54 237 Halperin B I 1984 Phys Rev Lett 52 1583 Lam P K and Girvin S M 1984 Phys Rev Lett B30 473 Laughlin R B 1983a. Pbys Rev Lett 50 1395 1983b Phys Rev B27 3383 1984a Surf Sci 142 163 1984b in Springer Series in Solid State Sciences 53 eds G Bauer. F Kuchar and H Heinrich (Springer. Heidelberg) 272 1987 in 'lbe Quantum Hall Effect. eds R E Prange and S M Girvin (Springer. New York) 233 Tsui D C. Stormer H L. and Gossard A C 1982 Phys Rev Lett 48 1559 Stormer H L. Chang A M. Tsui D C. Hwang J C M. Gossard A C and Wiegmann W 1983 Phys Rev Lett 50 1953

303

1.4 -; .., 1.2 '2 ~ 1.0

..

.c

-!. 0.8 Q.

ca

0.6

>

0.4

-.: G,) 0

0.2 0

-...... ~

N

..!. ~ Q) c

..

w

0.10 0.09 0.08 0.07 0.06 0.05

0

1

2

3

4

0

1

2

3

4

5

x (arb ui1its)

Fig.' 1:

Explici t evaluation of core corrections in the fractional statistics representation. Left: The function 0 (top) and ~/D (bottom), as defined in Eqns. (20.4) and (20.6). for two quasiholes as a function of the separation x = I~I/~. for m=3. The dashed curve indicates a bare coulomb potential. Right: Diagonal matrix elements evaluated from these compared wi th correct numerical values (solid line on top and dots on bottom) .

6. Chiral Spin States V. Kalmeyer and R. B. Laughlin, "Equivalence of the Resonating-Valence-Bond and Fractional Quantum Hall States", Phys. Rev. Lett. 59 (1987) 2095-2098 .......... 308

X.-G. Wen, F. Wilczek and A. Zee, "Chiral Spin States and Superconductivity", Phys. Rev. B39 (1989) 11413-11423 ........................................... 312

307

6. Chiral Spin States The following two papers indicate a possible mechanism leading to the occurrence of anyon quasi particles around the certain types of ordered spin states. Kalmeyer and Laughlin [1] construct an ingenious approximate mapping of the Heisenberg antiferromagnet on a triangular lattice onto a gas of repulsive bosons in an external magnetic field. For the latter problem one can plausibly adapt the successful trial wave functions used for the ordinary quantized Hall effect. Inverting the mapping t one a class of wave functions representing an interesting sort of spin ordering.

Went Wilczek, and Zee (2] attenlpt to extract the essence Qf the KalmeyerLaughlin idea in a more abstract and versatile form. They characterize the relevant. class of states, chiral ~pin, liquid~ in terms of their symmetry properties, show how they can emerge as the ground states of specific model Hamiltonians in some idealized limits t and discuss some properties of the quasiparticle excitations around them.

REFERENCES 1. V. Kalmeyer and R. B. Laughlin

Phy~.

2. X.-G. Wen, It'. Wilczek, and A. Zee

Rev. Lett. 59 (1987) 2095. *

Ph,l~.

Rev. 839 (1989) 11413. *

308 VOLUME

PHYSICAL REVIEW LETTERS

59, NUMBER 18

2 NOVEMBER 1987

Equivalence of the Resonating-Valence-Bond and Fractional Quantum Hall States V. Kalmeyer Department of PIr.rsics, Stanford Unil'ersity, Stanford, California 94J05

and R. B. Laughlin D('I'arlm('1/I of Phi-sics. Stanford CI/i.'ersill'. Slul/ford. California 94J05. and Unil'er.ril.l" of Califomia. tu .... ·rl'nce U"erfl/or(' ,vat;o",,1 l.tJburutor)·, Lh'ermore. California 94550 (Rc,c.\cJ 24

.Iul~ 191(7)

We pn.:scn! c\"Idenct.: th;a! the grnund s!a!e llf Ihe frustrated Heisenberg antifcrromagnct i:l two dimenis ",<,1' 'l '~L'rihcd hy :t fr.\I:lional quantum lIall Wa\e flln.:tion for b,)sons. This is compalible with lhe rcs\lna!ing-\,.alenee·t>1.lOd concept of Aliderson in being a h4U1d with neutral sp;n- CIlcitatillns. Our ~i()n"

t

rc .. ult .. suggest ::tron~l~ Ihal Ihe rcsonati,,!!·valclic«:-b.,'nd "nJ '-raetio:\..I1 quantum Hal! Slates are the :':lIn.: thil\~. We .IIst. argllC th:lt Ihe ellCllalion 'ipcctrum h;ao; an energy gap. PA.CS IIUA1"o:r~: 7:' 10.J01

It ..... as r';cc!ltly poinl-..:d out to

US

by

L(~

Lind JO&.lnno-

p0l\k.'s' !hal. !hl: physics of the fractional quantum Hall

effect is vcry simit.lr 10 tha! originally hypothesi£ed hy Anderson 2 to be operating in the two-dimensional Heisenberg antifcrromagnet on a triangular laltice. In DOth cases, the ground state is understood to be a nondegenerate quantum liquid 2- 4 with an energy gap. The existence of an energy gap is presently controversial. Anderson el al. S have recently claimed that the gap is zero, contrary to Anderson's original hypothesis, while KivellOll. Rokhsar, and Sethna 6 have argued that a gap exists. In this Leuer we show that t"e frlctic~:,,1 quantum Hall (FQH) slate at ",-2 confined to a triangular lattice has the same \'ariational energy as the Anderson resonating '"alenee-bond (RV8J wave function to within 2'-1,. We see this as strong evidence that the ground state and e,,citation spectrum, in particular the energy gap, survive under adiabatic evolution of the FQH Hamiltonian into that of the antiferromagnel. If this is the case, it necessarily implies I hat the ground state is a nondegenerate singlet, and that the elementary excitat;~ns are spinfcrmions with long-range interactions. The occurrence of FQH-type behavior in a system such as this is impor1;llIt I)ccau,~ il show, thai a magnetic liclJ is not e:.~c:n­ Hal to the physics. It is also a~ indication that su-:h beh.!vjor may be "blquitous in nature. We consider the antiferromagnetic Heisenberg Hamiltonian

t

'HAF-Jr.Sj·Sj,

(t)

(Ii>

where J> 0, the (unrestricted) sum is over all pairs of near-neighbor sites of the 20 triangular lattice, and SJ - t htlj is the spin operator at jth site. Following Lee and Joannopoulos, we now show that this Hamiltonian is equivalent to the FQH Hamiltonian for bosons on a lattice. The first step in this procedure is the Holstein-

I'rimaku/T transr()nn~ltion 7: One interprets the spin prohlem as a lallice gas by imagining an "atom" to be present on every sit~ wi!h an up spin. The atoms are then bosons with creation operat-.>rs a}- h -I (SJ + is!>. W rillen in terms of these, (I) becorr.es (2a)

'H-T+V, where

T-

t J r. (o1al +a/aj). (lj)

and

V-J'La}ala,aj+ tJNs -6Jr,a;tah

(2c)

I

(ij)

where N, is the number of spins or laUice siles. The boson kine::c energy operator T comes from the spinexchange or XY part of the Heisenberg interaction. The potential energy, which is a near-neighbor repulsion of hosons, comes from the Ising part. This Flamiltonianalso contains a de faCIO harti-core repulsion of the form H,

vo-u. r,

t. Qj

aj aja"

,-1

with 'J.- -, due to the fact that configurations with more than one boson on a site do not exist. The Holstein-Primakoff transformation is completed byaltering the Hamiltonian in the manner

'H- 'H+Vo, and treating the lattice particles as ordinary bosons. The second step is the identification in the Hamiltonian of a fictitious magnetic field. The kinetic energy T, as given by (2b), does not have the right free-particle form because the hopping matrix elements hj -J are positive. This makes the boson energy bands disperse down as one moves away from the center of the Brillouin zone. To

@) 1987 The American Physical Society

2095

309 VOLUME 59, NUMBER 18

remedy this problem, we consider JIj to be matrix elements of the right sign, namely negative, in the presence of a fictitious vector potential. To make contact with the fQH problem. we change the signs of the kinetic energy couplings to produce the pattern shown in Fig. I. where solid line segments are positive (wrong) bonds and dashed lines are negative (right) bonds. This is a gauge transformation and has no effect on the physics. In the original gauge all lines would have been solid. We now observe that one obtains just such a bond conflguratiol' by assigning an arbitrary charge e· to the bosons and then roupling them to a fictitious vector potential

0)

A- tS(xY-yi).

with a particular valuo: for B. This coupling introduces pha ..es inlo the horping malrix eic:ments .I" in C~b) :lCcording te

-

J.J IJ.. - IJ

'ex') 12Ki -~ fJI A·d,

'.

I

for Nb bosons: "'gs(z It

... • Ztv.)

- n (Zj-Zk)2expl--=.! ~ Iz;l2). 4/d j
;-1

where zj-xj+iYj is the complex la~tice coordinate of the jth particle. "'15 describes a state of density P2 -4Kld. which corresponds to t boson per unit cell. In other words. the number of up spins is equal to the number of down spins. We must emphasize that the only reason for considering such a wave function is the physical precedent of its success in describing the FQH states. We have.. evaluated the energy of the system in the ground stale (6) by :) semiclassical Monte Carlo method 4 For a half-filled buice. the eqUIValent ia~lice·gas Hamil1oni;1.n is

if -t+V,

•

(7a)

where the kinetic and pot.ential terms are

where ~-hc/e· is the quantum of flux associated with bosons of charge e·. If we fix the magnitude of the fictitious B field in (3) by

~a~ -41r/~.

2 NOVEMBER 1987

PHYSICAL REVIEW LETTERS

t- t !.jij(a}a,+QlQj). V-J!.Q1a,taiQj -1.5JNs '

(7c)

Cij)

(5)

where ao is the lattice constant and 10 - (e· B/ he) -1/2 is the magnetic !ength, then all phase factors in (4) are real, and we get the bond configuration of Fig. I. This choice of B corresponds to one fictitious flux quantum per spin of the original problem. Except for the presence of the lattice. this system is just a bosonic analog of the two-dimensional electron gas with short-range interartions in a real magnetic field. In light of our experience with the FQH effect. we expect this system either to crystallize or to form a nondegenerate quantum liquid .. The former cannot be ruled out categorically, but we feel it is unlikely for the same reasons Anderson· did. 2 If it is a liquid. :t should be describPd approximately by the m-2 FQH wave function)

(7b)

OJ)

We work in th~ cllindrical gauge (J). and so the hopping matrix elements J'j are as shown in fig. I. In fig. 2 we have plotted the kinetic and ~tential terms in the ground-state energy Ep-(~p 1111 ~p>l(~pl~"> for lattices of different sizes. Since the wave function ~p keeps the particles localized within a circular rc;gion of area P2Nb. we use free boundary conditions. The potential energy V15 contains a negative surface term which scales like (Ns ) -1/2, si'lce the bosons on the boundary have

-Q2~--~----~---r----r---~-'

0-Q3

z

..2

•-


•-


....

~ -0.4 0: w Z w w -0.5

E-Q6 Cf)

&.l

z

~ ~

-0.7 -OB~

o

__ ____ __ ____ __ Q05 0.10 0.15 0.20 0.25 ~

~

(Ns FIG. I. Distribution of silns of the couplings i ,} - ± J defined by (3)- (5). Positive (negative) bonds are shown as solid (dashed) line scgments. Open circle denotes the origin. 2096

~

~

~~

r" 2

FIG. 2. Energy of the variationallround state (6). in units of NsJ, as a function of system size. 2N. -N, is the number of lattice sites. Kinetic enerlY (Xy modeO. lozenges; potential enerlY (Ising term). circles.

310 VOLUME 59, NUMBER 18

PHYSICAL REVIEW LETTERS

fewer near neighbors than in the bulk. Our results for the kinetic (XY) and total (Heisenberg) ground-state energies are Tls - -0.62±0.01 and £1'- -0.94 ± 0.02. in units of JNs . These values are 10% higher than extrapolated finite-lattice results of Oguchi. Nishimori. and Taguchi 8 who get Tls - -0.74. £IS- -1.05. in the same units. The potential (Ising) contribution to the energy. VIS - -0.32JNs agrees very well with finitelattice calculations. The agre~ment is siJrprisingly good. in view of the fact that OUi trial wave function ,,~s invl3i"es nc free par:.lmeters. For !he r~:i<Jila;;:1g-vaknce-!:lod (R \' B) siate. Anderson 2 estimated £R\"8 - -0.98 by extrapolation from latii.;cs '"jlh J. 8. 12. and 1(, siles. -"'h;h: Ogtl~hi. Nishimori. and T.Jguchi. working with lallicc!s of up to 20 sites. oblJ,n £IC.\"8 -- --0.95 0.02. in ur.!~.; of /.\"$. !t thus app~ar~ that ':i': R .... B and fra<.:tional GU':n~unl Hall w:we functions arc cl!uiv;.:lent variationally. We: ched the accuracy of Ihe Monle Carlo calcuia!ilm:; by ~om"u';l\g 'he rauiai di$tribulion function ancl the pot;:ntiai cne.-g}, of the ground state (6) using the hyp:rnetted chain (HNC) procedure.· The HNC result for the ground-~tate potential energy is V~NC - -O.30JNs • in excellent agreement with the extrapolated Monte Carlo value shown in Fig. 2. These numerical results strongly support the idea that the RVB and FQH states are in the following sense physically identical: One can imagine adiabatically transfol ming the FQH problem into the antiferromagnet problem. Tnis might be accomplishe<:, for example, by expressing the Hamiltonian in terms of the single-boson basis orbi~als

=

=

9.. (Z) -(II2K) 112 e:
~

1:.11+ t ::zJ, (@)

where z. is a lattice site. and then letting the off-diagonal matrix elements of the overlap matrix S.". given by

S.~- f':(z),~(z)d2z.

(9)

go to zero. On the basis of our experience with the FQH problem for fermions. we can say with some certainty that the: adiabatic evoilition starts from a system with a nondegcnerate ground state described ap~roximately by Eq. (6). an energy gap, and elementary excitations of "charge" t. Charge in this case means excess spin. If, during the evolution, the gap remains intact, which is not clear, the nondegeneracy of the ground state and fractional nature of the excitations almost certainly remain intact as we!!. If the gap col!apses during the evolution, on the other hand, the situation is less certain. The cxcitation spectrum could conceivably survive. A more likely outcome, in our opinion, is that antiferromagnetic order would ::et in. 9 For this reason we believe that the RVB state possesses an energy gap, and that the assertion of Anderson et al. S that the state is gapless is incorrect.

2 NOVEMBER 1987

Our picture of the elementary excitation spectrum is in fact almost identical to that proposed by Kivelson. Rokhsar. and Sethna. 6 We have obtained a numerical estimate of the gap ~ using the FQH quasihole wave function l N,

'11:0 -

n

(Z,. - zo)

"-1

n

(Zj - Zk)2

j>k

r

xexp(_...L 11 11111. 4/d II-I

«(0)

where =0 is a complell nllmber locating the quasihole center. Since the real excitation is necessarily an eigenSLate: of linear momentum. this estimate is for the center of mass of the quasiparticle band. It is a good e~timate rlH the: gap only if ttee band disperses very ~j~tle. which we believe to he the casc. Calculations or thi£ dispe.s;on are under way lind will be rep"rt~ in a tater publication. We find tha. ~ is minimized ·.vhen :0 is 0'1 a lattice sileo The potential energy ".ontribution to l1 is C'.t1culated using the generalized HNC algc,.·ithm," which is insensitive to boundary effects. The value we rind is ~V -0.211. il'ldependent of N s • The kinetic energy cost ~ T to create a quasihole is estimated as t~e difference in kinetic energy between states (6) and (10). Figure 3 shows the results of Montc Carlo calculations of ~T ror lattices of up to 400 sites. Extrapolation to the thermodynamic limit N s - 00 yields ~T- -O.04±O.IO in units of J. Th(. total energy required to create a localized spin- t excitation is thus finite and equal to ~-(O.17 ±O.IO) J. Several remarks are in order. We believe that the true ground state is a nondegenerate singlet even though it is nu. evident that this is the case for the wave fanction of Eq. (6). We note particularly that the true pound state I.O...----r--r---""""T"--r--""'T"-, 0.8

o

Q05

0.10

(Ns

0.15

0.20

Q25

r"2

FIG. 1. Kinetic energy cost 4.T to create .. quuihole excitation, as a function of system size. The HNC rc:salt for 4" is 0.2IJ.

2091

311 VOLUME

S9,

NUMBER. 18

PHYSICAL REVIEW LETTERS

must be real, while the wave function (6) is not. at least for some configurations. This does not mean that the time-reversal symmetry of the Hamiltonian is spontaneously broken. In fact, the correlation functions we have so far been able to calculate for the variational wave function (6) are real in the thermodynamic limit. We also have some preliminary evidence that the groundstate Ansalz (6) is a spin singlet. The identification of the state (6) with the true grou.,d state of the frustrated antiferromagnet is, therefore. physically meaningful. Adopting this picture of the ground state as a quantum spin liquid with an energy gap, we conclude thal its elementary excitations are n~utral spin- objects. Their spin is the charge of the elemeotary excitations of the ~quivalent FQH st 0:. te. which. for a half-filled Landau level. is equal to ± t. The fraC!iona! charge quantization j", exact Civ~r. the uniqul!ne.~ of the; ground st"He and the existence ('f a gap. The underly!ng rotational symmetry of the Heisenberg Hamiltonian also guarantees that the quasiparticle a~d quasihole. which are equivalent to the spin-up and spin-down states of the same excitation. are degenerate. Like the elementary excitations of the FQH system, these excitations may be thought of either as fermion~ or as par:icles obeying fractional statistics. 4•10 The former approach leads to an intriguing situation where the quasiparticles interact via iong-range

t

2 NOVEMBER 1987

forces while all interactions in the original Hamiltonian are short-ranged. This work was supported by the National Science Foundation under Grant No. DMR-85-IOO62 and by the National Science Foundation-Materials Research Laboratories Program through the Center for Materials Research at Stanford University. We are indebted to D. H. Lee for providing the initial motivation for this work, and to P. W. Anderson for stimulating discussions.

H. Lee and J. D. Joannopoulos. private communication. 2p. W. Anderson. Mater. Res. Bull. 8. 153 (1913). JR. B. laughlin. Phys. Rev. lell. SC. 1395 (1983). 4R. B. laughlin. in Quantum Hall Eff~~I. edited by S. M. Gin'in ~nd R ~. Prange (Springcr-Vcrlilg. Nt\,\, Ycr". 1986). p.233. sp. W. Anderson. G. Baskaran. Z. Zoo. and T. Bsu. Phys. Rev. Lett. 58. 2190 (1187). liS. A.. Ki"elson. D. S. Rukhsar. ~'nd J. P. $;:Ihna. Phys. Rev. I D.

B lS. 8865 (1981). 'T. Holstcin and H. Primakoff. Phys. Rev. 58. 1098 (1940). IT. Oguchi, H. Nishimori. and Y. Taguchi. J. Phys. Soc. Jpn. SS, 323 (1986). 'S. M. Girvin, A. H. MacDonald. and P. M. Platzman. Phys. Rcv. B 33.2481 U9!6). IJB.I. Hal~rin. Phys. Rev. lett. S2, 1583 (1984).

312 PHYSICAL REVIEW B

VOLUME 39, NUMBER 16

I ruNE 1989

Chiral spin states and superconductivity X. G. Wen, Frank Wilczek: and A. Zee Institute for Theoretical Physics. University of California, Santa Barbara. California 93106 (Received 9 December 19881 It is shown that several different order parameters can be used to characterize a Iype of P- and Tviolating state for spin systems. that we call chiral-spin states. There is a closely related, precise nolion of chiral-spir.-liljl.iid states. We construct soluble models, based on P- and T-symmetric localspin Hamiltonian~, with chiral-spin ground states. Mean-field theories leading to chiral spin liquids are r-roposed. Frustration is essential in stabilil.ing these states. The quantum numbers of quasipartides around th..: chiral spin liquids are anal}"lt"d. They grneraily obey fractional stalistics. Based on these ideas, it IS speculated that supercondu":ling slates ..... i!!. unusual values of the HUll: quantum may exist.

INTROlJUCflO,,"

,hat play a crucial role ill the dynamics of highSuch speculuion has taken various forms. O••e suggestion is that there is transmulation of the hole statistics, which turn these 1uasipartiC\es into bosons.1J Superconductivity is then pictured as a Bose condensation. Another suggestion is based on approximat~ mappings of spin Hamiltonians onto the Hamiltonian of the quantized Hall effect. 14 Then much of the theory of the latter effect, including fractional statistic!!, carries over to spin systems. This idea, and closely related ideas concerning possible ..ftux phases"IS,16 in the Hubbard model, will be elaborated and sharpened below. Yet another suggestion is that at certain densities a lattice of quasiholes forms, and induces diamagnetic currents whose effect mimics that of a Chern-Simons interaction. \7 A common futule of several of these proposals, is that they escape the constraint mentioned above by invoking (implicity or explicitly) spontaneous macroscopic violation of the discrete symmetries P and T, but in such a way that PT symmetry remains unbroken. \8.19 There are prospects for direct experimental tests of this symmetry pattern. lO In tilis paper we do three thing~. First, we characterize the commol' essence of the proposed P- and T-violating states, which we call generically chiral spin states, in a precise way. We de this, by defining a local order parameter. We shall also be able, within this framework, to give one precise meaning to the notion of a spin Iiquid. 2l ,22 Basically, a spin liquid is a chiral spin state that supports a nonlocal extension of the order parameter. Second, we construct a family of spin Hamiltonians whose ground state may be found exactly, and is a chiral spin state. Our Hamiltonians, although local and simple in structure, are rather contrived. Nevertheless our construction provides an existence proof for chiral spin states. In the course of constructing our model ground states we shall learn some interesting lessons about the sorts of states that support chiral spin order, and derive some intuition about when such states are likely to be entemper3tt!r~ sup~rconductors.11.12

In two sptitial di.nensions new possibilities arise for quantum statistics. Indeed, it has gradually emerged that fractio~al statistics;-l and the related phenomena ofslatistical transmutation, are rather common features of quantum field theories in two space dimensions. They can occur in u models with Hopf terms, 2 or in gauge theories with Chern-Simons terms.) They also appear naturally in a variety of effective-field theories resulting from integrating out massive two
11413

@1989 The American Phyaical SocietY

313

x. O. WEN. FRANK WILCZEK. AND A. ZEE

11414

ergetically favorable. Unfortunately. the simple chiral spin states that diagonalize our toy Hamiltonians are not spin liquids. So. third. we formulate some models that are not exactly soluble but do plausibly seem tu have chiral spin-liquid ground states. These models contain a parameter n. such that for" =2 they are frustrated-spin models. while for large" they are tractable. in the sense that a mean-field theory description is accurate. In the mean-field approximation, we find chiral spin-:iquid states are energetically favorable for a wide range of couplings. We construct an effective·field theory for the low-energy eJtcilatlons around a sp.:cific chiral spin-liquid state, and characterize the charge. spin, and statistics of the quasiparticles. Spin-! neutral particks carrying half-fermion "tatistlcs are fo~nd, in agreement with Lau,hlin's .rgument~.2' W e c~)nclu(:C w:, h ~ome remarks on ,h.. pom,hle relationship btt·.... eeli chirai spin liquids and sl;pcrcQIl(luctivity, and put forward a speculation IhaC Ro.W'S Ilalu:-ally from this circle of ideas, and if true '."'ould h.ve dramatic experimental consequences.

electron occupies each site. so the states are gauge invariant. 24 . 2S Therefore, according to gener2.1 principles. a gauge-invariant object like Xij cannot acquire a nonzero vacuum expectation value. Rather, the simplest gaugeinvariant order parameters we can construct from X are of the general type (3)

or (3')

where the r'lI circle a dosed triangle or plaquette. And indeed. an expectation value of the latter type has been used HI characterize the f\U\ phase. Arc these plaquette order parameters related 10 'he spin ex~cl::li()n values F:' In fa~t simple c.llclJialions show ~ha: (bey ue. ~t'lecifically. we ha.c (4i

and

CHARACTERIZATION OF CHIRAL SPIN SlATES

Part of the reason that the rer.ent literature on possible dynamical realizations of fractional ..t"tistics oRen appears so diffuse and confusing. is that the essential character of the proposed states can be stated in several apparently different ways. Here are four possibilities, appropriate to the context of Hubbard models. 6) As a s:raightforward spin orderiq, consider, in a model of spins t, the expectation value

E l2l == «(1 J .( (12 X (1l»

,

(1)

where 1,1,3 label lattice sites. P symmetry wou!d force Em to vanish (or depend on the poIition of 1.2,3 in tile latice-see below) because it reverses the orientation of the circuit 1-2-3, which changes the lip of'the triple product. (In two-dimen:;ional space, parity is of' course just reflection in one of the two spatial ues.) T symmetry would force E 123 to vanish. But a nonzero value of Em, necessarily real because it is the e,pectatioa value of an Hermitian operator, is consistent with PT symmetry. A nonvanishing, real expectation value E lll , correlated with the siz~ and orientation of the leian21e 123 but not its position on the lattice, is one characterization of chiral spin stales. (ii) Let us introduce electron creation operators CI~ on site i, ...pin (1, and the operaton X'j

==CI~Cjf1

(2)

•

These X operators have proved very cooveoieot in the mean-field theory of Gux phases. and we abaD . . them in this way below. But &nt, we wish to coasider a more abstract use of them, in fonnulating an order parameter. Under a local P!ge transformatioa. whereby an electron at site J acquires the phase e I'} , we have II',-'j) X'j-e

Xij'

In the half-filled Hubbard model at infinite U, exactly one

Thus the chiral spin states are alternatively characterized by their supporting a difference between the expectation values for plaquettes traversed in opposite directions. This of course emphasizes their P-violating nature. An important formal advantage of this second definition of the chiral spin phase, since it uses the electron creation and annihilation operator rather than the spin operator, it may be usc:d aw", from half-filling,large-U limit. It allows us, ill other words, to step outside the framework of Heisenberg spin models. (iii) ~ a Berry phase,26 for transport of spins around a loop. Such phases are known to be a good way to characterize the FQHE. Specifically, consider the operator that transports the spins at 1,2,3 to sites 2,3,1. It is the permutation PlI2l1" Using simple mathematical identities relating cyclic permutations to interchanges. and interchanges to spin oper~tors, we find .fJ 12l

=(Pow) = (P'21l Pml) =~«(l +(12'0'3)0 +0','(7"2)

,

and hence easily

;

1J12l-1J m =i EI2l '

(6)

~ stated, this definition of 1J12l works only for spin models. In scocral, we may take (1 i ==CI~ G(jC11Jo where on the left-hand side (1 is an operator but on the right-hand side it is a numerical matrix, and use the last term of (S) as the definition of 1J12l• In this generality, however, it can no IonRer be interpreted as the expectation of a transport operation. There are other simple relations among the various order parameters E, fJl, and 1J, e.g., ; (7) ImfJl 11l -"4 Elll ,

=

314 CHIRAL SPIN STATES AND SUPERCONDUcrlV1TY

(8)

. (iv) As a st~te, around which the low-energy excitatIons are deSCribed by a field theory with a Chern-Simons tl.:rm. This characterization of course is considerably more vague than the previous ones, Lut it is closely relat~d to the imme~iately preceding phenome!lon that phase I~ accumulaud In transport around loops. We shall make precise connection~ in a specific model below. . In slimmary. we find thaI there are several apparently dlfferenl, but in reality identical, characterizations of '. t-.ir:1! spin staks. The chiral spin order parameter captures some, but not all, of t.he ~ro'perties we would like to postulate of a quantu~ "Pill lt4Uld. It ~as the desirable feature ofleaving rot3fl{'n and translation symm~try unbroken, but unfortunately :~ ducs net cai'ture ~he long-range cuherence we ~:c;p~~t is necc~5ari for an incllmpressi~le Iiquia. Inspired roy allalogy w!th the quantiZed HaU effect. however. we arc.' led to the following preliminary definition. We Sliy :hat W~ b\'1! a I,;.hiraI sP:1A ~i4"id. v.hen not only small triangles or plaqllettes, but also large loops are ordered, in such a way that products around consecutive links en· closing a loop obey (9)

Here f( y) is a positive real function of the geometry of the loop r (in our mean-field models it will be proportional to the length of y), but the crucial (eature is the phase term proportional to the area A (r) enclosed by the loop. Identifying (Xi/X)1c ••• Xli ) y loosely as a sort of Wilson loop. we can think of b A ( y) as the flux enclosed bv the loop y. Although we st!aU not attempt to prove it i~ this paper, we expect that the crucial properties of the :lipin liquid, and specifically the statistics of it~ quasiparticle excitations, are determined by the coefficient b. The mean-field theories we construct below support ground states with order of this type. Before concluding this section. it seems appropriate to address two questions that might cause confusion. First, in what precise sense ca" we distinguish the "macroscopic" violation envisaged in chiral spin Slates from, say. the T violatIon in antiferromagnetism? After all, staggered magnetization is T odd. The crucial difference is. that the combination of T with another symmetry of the Hamiltonian, namely translation through one lattice spacin~, leaves the antiferromagnetic ground-state in\'lriant. Since such a lattice translation is invisible :nacroscopically, the antiferromagnetic ground state is effectively T symmetric macroscopically. (In contrast, a ferromagnetic ground state does of course violate T macroscopi",a Ily.) Second, are flux phases necessarily P and T violating? Let us define this question more precisely. It is no surprise to find P and T violation in the FQHE. since there is a strong external magnetic field applied to the sample. Now we can bosely describe chiral spin states as characterized above, and the closely related flux phase states in the Iiteratur.:, by saying that a sort o( fictitious magnetic field has developed spontaneously. Indeed, the

11415

effect of a magnetic field is precisely to modulate the phase of the wave function as a charged particle is transported around a loop, as in Hii) above. However, we mu&t not be too quick to infer P and T violation from this analogy. In particul2.r, in Refs. IS and 16 flux phases are constructed in which half a fluxoid of magnetic field pierces each plaquette of a square lattice, corresponding to 'P/ 12J4 = negative. But from this alone, we cannot infer P or T violation. Indeed, the action of P or T is to change the half-fluxoid per plaqueUe to minus one half-fluxoid . However, ei"=e -i", and this change is equivalent to adding a full negative fluxoid to the original configuratiun, which is merely a gaug\! transformallon. Hence, these symmetries should be maintained. Yet if we follow the authors of Refs. 14 1!ld 21 by approximating the lattice wave function, in an apparently natural way. by a continuum wave function, th~ etfective flux through a loop becomes proportion~1 to the area of the loop, and does generically show c.xnplcx pt,:l~, indicativf" of P and T violatllln. This ~ge to the continu:Jm anc! te an area law is necessary, if (he sute is to be a spin liquid in our se,\se (and, we suspect, in any reasonable sense.) It is not unreasonable, however, to be suspicious of an approximation that alters symmetry. The constructions which follow, were largely motivated by a desire to clarify this. issue. SOLUBLE MODEL

We have identified, and characterized in an abstract way, what we mean by I chii"al spin state. We will argue that the ground state o( a (rustrated Heisenberg antiferromagnet, treated in a mean-field approximation, m~y ~ a chiral spin state. however. the validity of the meanfield approximation in the present context is far from clear. Ideally, we woulc1lik.e to solve a realistic model exactly, and demonstrate that it possesses a chiral spin ilhase. In practice. this poses formidable problems at two levels-in formulating a realistic Hamiltonian, and in solving it. In this section we tak.e a ditferent~ more modest, approach_ We will presently construct a Hamiltonian, whose ground state can be explicitly identilied. and is a chiral spin state (although not a spin liquid). One purpose of this exercise is to furnish an existence proof: there exists at least one Hamiltonian whose ground state violates T and P. Another is to supply us with concrete wave functions to look at, so that intuitions may be formed and conjectures tried. To begin, consider (our spins_ These may be combined into a singlet in two different ways. Accordingly, the general wave function for a singlet may be written in tbe form (I +,,)\0 +() -,,)111) -21110

,

(10)

where

10=lrru)+lurr) , 11I)=lurO+llllt> , IIII) = Irut> + III U) ,

(11)

315

x. O. WEN, FRANK WILCZEK, AND A. ZEE

11416

containing a parameter Y. The question arises: Is there any intrinsic way to separate the two-dimensional space of singlet states into one-dimensional subspaces? In fact we can think of three ways, all of which lead to the same separation. (i) We may demand, that the different spin configurations each have equal weight; that is, that the squares of their coefficients are all equal. This is the sort of situation we might expect in a liquid, wh~re there are frequent ftuctualions in the spins, but all preserving the overall spin-O character. It is easy to se..e. that equality of amplitude occurs ifand only if \.= ±i\l3. (ii) We may try to impose some symmetry rcquirt'tIlent. Whllc: olle quickly realizes that our tw('-dimensional space IS .rreducible 'Inder T or under the .:omrl"te group of permutalil'Jls. it is easy to check that il reduces under the ~r(\ll\, of ('/ ,.'1 rCrI1llllalio'ls. The '~'arian" c;ub~rdces, arc: spanllcJ by the sta:es with v= ±i \. 3 (iii) We may labd states by chiroli,y. This:r. the most useful for our immediate plJrp05e!o, and we I'OW spell it ,")ut ill detail. Consider again the Hermitian operator, (12) At this stage, we just have a problem of three spins, each with spin t. An easy computation shows that X2= -4(8. +8 2+83 )2+ IS ,

(13)

where 8;=u,12, and;= 1,2, and 3. Out of three t spins, we can form a spin-t multiplet, which we denote 6y Is = t,s,) and two different spin-t multiplets, which we We have, denote by Is = f,s, ) + and Is =

i,S, )-.

x21s =f,S, ) =0 ,

x2IS=t,sz )G= 12,

(14)

a= + or - .

Since IS=t,S,)G must be orthogonal to IS=f,S,), we can write down

(I S)

and by applying the spin-lowering operator IS=t,S,=-p+ = -

~(Ill f) +6111 r 1> +('i!l, 1 1)). \1'3

(16)

Here, tal denolb the ~ubc IOO! oi U'1!lY so that I +w+ol=o. Evidently. the other statcli IS = +.s,)are obtained from (l Sand 16) by replacing w by (112. The operator X rommutes with S and thus X. S, and S, can be simultaneously dialonalized. We find that IS=t,sz=P" is an eigenstate of X with eigenvalue 2i(",-",z)=-2V3. Evidently,IS= is an eigenvalue of X with eigenvalue U(",I_",)=2V3. Note that the time reversal operat('r T takes Is = t,s, = + into IS=t,S,=-t>-. Now we picture the three spins 8 1, 8 2, and 8 3 on three of the comers of a plaquette on a square lattice. Let us couple in the fourth spin 8 .. to form a total-spin singlet. From general principles. we know that two different spin singlets are possible, namely,

.,s,=.p-

t)

(17)

Then two S =0 states are thus, Is =0) + == 1f f 11 ) + III f f) +(&)1 f 1 f 1 ) +",Ilf It ) +(&)2Iu f 1) +",21 flU),

(18)

and Is=o)-, obtained from IS=O)+ by replacing ru'by ",2. By construction, these s:ates are eigenstates of X=UI'(U2Xu3)' where XIS=0)G=2V3aIS=0)G, a=±I.

is best described by referring to Fig. I. We select out of all the plaquettes on a square lattice a subset consisting of nontouching plac;uettes in such a way that the corners of these plaquettes cover all tbe sited on :he lattice. (These plaquettes are shown shaded in the figure.) We label the sites as in the figure. Now,let

H, =J[(8. +82+S)+8.. )z

(19)

+(8s +8,+87 +S.)Z+ ..• ] .

(2))

Thus we have arrived again at the same separation, as promised. Since our two states are invariant under even permutations of the four spins, i.e., under permutations in the classes (12), (34), and (1:), and go into each other under odd permutations, i.e., under permutations in the class (12) and (1234), they are also eigenstates of the other possible chirality operators, such as t=UI·(UZXU .. ), obtained from X by pe~uations. In particular,

Clearly, the ground state of this Hamiltonian IS reached by f.>rming the four spins on each shaded plaquette into a singlet. Namely, the ground state is given by an infinite direct product, denoted schematically,

(20)

Since on each plaqueue, we can take an arbitrary linear combination of Is=o)±, the ground state is infinitely degenerate. Let us now introduce an interaction bt:tween neighboring plaquettes by writing

Now we are ready to construct a Hamiltonian H that has a chiral spin state as ground state. The construction

~

(C+ls=o)++c_ls=o)-).

(22)

pa.q.cle

316 CHIRAL SPIN STATES AND SUPERCONDUCfIVITY

11417

nearest neighbor along a diagonal (NNN) couplings If:...:

+J ~ Sj·Sj +J' ~ S;·Sj , NN

FIG. 1. Tile: ~h:1d~d plaqueltcs art' Hamiltonian in Eq. (211.

~.:k..:l·:,f 1(1 .. ,lll ;(rU':l (h~

H: =K[.t fJZ3lX(S';7l+X(l231\,(9 \0 II!+ ...

The:

1I01.lIil)1l

J.

!~3.

i:c; the obvious on.:; by \:,(5671 we nican

~l·(0'i>XU7J,

and so on. Consider the Hamiltonian H=H.+H 2• Cle:arly. for small K, this describes an Ising system since on each shaded plaqueue the associated "Ising spin" can either;'e up (i.e., the four spins on that plaqueue form IS =0) +) or be down (i.e., the four spins form Is=O) -I. According to whether K is positive or negative, the Ising system is a:ntiferromagnetic or ferromagnetic. Evidently, the ground state of H for K <0 is two-fold degenerate and is a chiral spin state. T and P are spontaneously broken. Notice that at high temperatures, above the usual Ising phase transition, T and P are restored. Clearly, many other choices for the Hamillonian are possible. For instance, in addition to H 2' or instead of H 2' we can add Hi =K'[X( 124)X<S67)+X< 124)X<9 10 11)+ ...

J.

(24)

For K' <0, Hi describes an "antiferromagnetic" Ising interaction. In this case, the expectation value of the order parameter X would have staggered values and macros~op· ically there would be no time reversal vinlation. A:c; we emphasized, our goal in this section ha:c; been to exhilJit speCific P and T invari.. n. s~;n Hamiltonaans who!le t;round state is a c.hiral spin state. The Hamiltonians we exhibited involve six spin interactions and are rather artificial. The ground-state wave functions are rather attractive, however. In forming them we are led to add together many different spin configurations with coefficients that are equal in magnitude. This certainly calls to mind a spin-liquid picture, although to induce the non local ordering necessary for a true spin liquid would require coupling the different squares together in a less trivial way.

where J and J' parametrize the strength of the nearestneighbor and next-nearest-neighbor couplings, respectively. It has been shown by lnui, Donachi, and Gabay2B that the diagonal coupling J' is induced by doping. In absen-:e of the NNN coupling computer simulations suggl'st that the ground state of the Heisenberg model is an an r ife:rr r )lI1agnefic (Neell state which violates neither T Ill)" P. nut 'he NNN coupling (J' > 0 ~ infroduced frusIration~ in Ihe Neel state. For large enough )', the Neel sf:lte j, :1(1 longer favored and the grouf1d state is expected to tl~ a disorderec! state. 29 We ",ill see later that such a ,li~."rdl·,~d state is quite likely to be a chiral spin state. 1 W(' \.:,lnsl.jcrallons suggest thai there is a dose relation ~~I\.eC'n the NNN coupling and the chiral-spln state. First. :1<; di"cus
=~Cjt(ia, )c/-H~ ;

FRUSTRA110N AND A CHIRAL SPIN UQUID IN MEAN FIELD THEORY

Consider a two dimensional spin-t Heisenberg model on a square lattice with both nearest-neighbor and next-

(25)

NNN

~ao(i)(nj-l) ,

(26)

;

where n· =c~c. is the number of electrons on the ith site and the' La~r~nge multiplier term ~;ao(nl-l) is !ntroduced to enforce the constraint nl = 1. H~ in (26) IS obtained by replacing S; in (2S) by

317

x. G. WEN, FRANK WILCZEK, AND A. ZEE

11418

(27)

and reads Hr =

l: 2Jcitaci~IOCja + l: NN

2J'citaci~}OCja

NNN

-2N(J+J') .

complex parameters. The nearest-neighbor hopping amplitudes Xij are p:uametrized by Xi' ; = I •... ,4 in the way described in Ref. 16. The next-nearest-neighbor hopping amplitudes are given by Xi,i+i+Y=XS+( -\

Xto •

)I

(33)

(28)

The constant term - 2NIJt- J') is included for later con· venience. In the path-integral formali<;m the partition fun.:tillo is gh:en by

f Da!)[)(',Dc,·exp (i f L dt ) = f Da,~exp Ii f L.l'!'foold, 1,

liB

Z=

5L~Qol

Sr
--="l:' "'~IJ~ I!-j

/I H can be written as (34)

,

A

(291

where L
~=O.

In momentum

where: and

I.~

is a sumnl:llil'1I over half of the Brillouin zone

h" = 2ReJ/, + 2 RCJ/~r, + Re,,~r., + Im'hTz ' where Ie

k + Ie

71,=X,e • The energy of the approximate ground state, constructed in this way, is given by - Lef'(Qo) at the stationary point. From (26) it is nof difficult to sec that the mean-field energy -L~Qo) is equal to the vacuum energy of the Hamiltonian

(35)

I

1

II

k -l I

+X,:, • ' . (36)

711=XI/k.+Xie -ik'+X1e -'k·+x:e'k, . In (36) we have taken the lattice constant Q = 1. Now we can diagonalize H., The energy spectrum is given by

where the electron operators are now no longer subject to the constraint clci = 1. HI is stiU very difficult to solve. but since the coastraint ft, = I is removed we can easily use the variational method (i.e•• ecrcctively the HartreeFock approllimatioa) to find a state close to the true vacuumofH,. Here we are primarily interested in the spatially homogeneous stationary point Qo =const. The constant Qo acts like a chemical potential in (31). From (30) and (31) it is not hard to see that the stationary value of Qo is such that the total number of electrons in ground state of H, is equal to the number of the lattice sites N. This of course corresponds to our starting point. which was a model with one spin dqree of freedom per site. AJ our trial waV4; function for H I' let us first consider the bond state studied by Aftlcck and Marston. 16 The bond ~::1te is deIned as the ground state of the following quadratic Hamiltonian: H.=1:(t,jcJc, +H.c.)+ NN

1:

(t,jcJcl+H.c.).

(32)

NNN

with a total number of N electrons. We must vary the tlj to minimize the eacrgy the ground state. Following AftIeck and Manton. we win consider the I/J which break the symmetry under translation by one lattice spacing. but are invariant under translations by two lattice spacings. These tlj are parametrized by eight

E. =2Re"1I±[ 4(Re"12)2+ 1"1112]111 .

(37)

In the absence of th.:. diagonal hopping terms (xll,.s •...•• =O) the Fermi "surface" of E. (aC halffilling) consists of just the two isolated points at k =(1T n.1T n.) and k = ( 1T n. -1T n). The low-lying excitations around the Fermi "surface" correspond to two families of massless Dirac fermions in the continuum limit. Each family contains a spin-up an-:l a spin~own electron. However. the nonzero diagonal hopping terms with X6 = - X. = real open a gap at the Fermi "surface". Since genencally it is energetically favorable for fermion systems to open a gap at the Fermi surface, we expect X6 and X. to develop a nonzero value if I' is large enough. In the following we will show. in a mean-field approximation, this is exactly what happens. The ground state 1~. ) of H. can be obtained by fiUing aU neptive energy levels with electrons. Using I~.) we can obtain (cltCj) = X'j for nearest-neighbor and nextare again nearest-neighbor bonds. Those X'j (l8l'UDetrized by eight complex parameters X" i = I •.••• 8 corresponding one by one to the X. In fact the X, can be expressed as derivatives of the ground-state energy of H. with respect to the X,. In terms of the the expectation value of HI [see (31)] in the state I~.) can be written as

X,.

318 11419

CHlllAL SPIN STATES AND SUPBR.CONDucnvlTY

Since the Xi are functions of the fi' we can adjust the fi to minimize the ~nergy E B • By computer search we find that there are two local minima which are potential ground states. One is a chiral spin state. characterized by

x, :'3 ........ =!e i ,,/4 • \',=X7=0.

X,,=

(39)

-·r~.:. =g~O.

wher,. ! a~ld g are real constants. One ~;In easily check the flux lhrough the triangles is +17"12 (-17"/2) for r~= -X~= -t'g (-'g I. In this ~'at~, P'lnd T a:'e br:lbn. T~e chira! spin state exis:s (j;tly for J' /J ~ O. 5. When J' / J ~ C. 5, we lind g = () and the chiral spin state is bettered hy tl~e flux phase discussed in Refs. 15 and l6. P 3nj r are (lot brr.k::n in th!s flu\ rhase. Stili \\ :lh:n thl! framework of bond state:>. an(di:er local minimum is the dimer phase characterized by (40)

xil; ... =o.

The energy (per site) of the dimer phase is equal to -I. The energies of the chiral spin state, the flux phase, and the dimer state are plotted in Fig. 2. In addition to the bond states, another obvio;.Js meanfield state to consider is the Neel state, characterized by (c;tulc;)

=( -I); , (41)

and compute corrections to the corresponding saddle point, we shall find them to be huge. In order to have a context in which we can use the relatively tractable mean-field method, and yet have a con_rolled approximation, we can go to an appropriate large n limit, so that there are many indivduals contributing to the mean field. As is well known in ma'\y other contexts, in this limit the case of the mean-field approximation is at least selfconsistent, order by order in lin, A large-n limit appropriate to our problem can be achieved by considering the Hamiltonian, NN

J'C;tllc1bC]"Cjt.

NNN

-n\J+I')N,

'4-3)

\\:heu: a, b =~, ... ,n, In the ground state, Nfl contains liN 12 fermions For n = 2, H If reduc~ to HI in (31). We may repeat our previous calculations for the energies of bond states. The. energies of the chiral spin state and the dimer state are the same as before ex.cept for an overall factor ,.2/4. Now let us consider the staggered phase (corresponding to the Neel phase for,. = 2), characterized by

<4G1CI6)=ta:+(-1)'T: ,

(44)

XIJ=O,

T:

where is traceless Hermitian r.l8t. ;x. "!":. ~ minimum or the energy is at

X,:=O.

I

I

The energy of the Neel state is given by -2(/-1'). A second spin-ordered state, characterized by (ctu 3c;) =(

I

H. =21 JC,toC;bC?Cjll +2

I

T

T=

-I );. ,

for n =even , -

I

T

(42)

X;j=O, I

also has low energy when J' is large. We will call this state the stripe Slate. The mean-field energy of the stripe slate il; - 2J'. Thus, in the Olean-field approximation either the Neel state (for 0 < J'I J < 0.5) or the stripe state (for 0.5 <J'IJ < 1) always has the lowest energy. But near J' 11=0.5, the chiral spin sta~e comes very close. Actually at I' IJ=0.5, Echinl= -G.918 and

T

(45) I

T I

T

T=

o

for n =odd , -1. 1

E Neel = E stripe = E dimer = -I .

Because of the expected large quantum fluctuations, the saddle-point or mean-field approximation is inadequate to determine which, if any, of the mean-field trial states actually describes the true ground state. However, the above calculation at least indicates that the chiral spin state is a serious candidate for the true ground state. As we have mentioned repeatedly, the mean-field approach to the spin-t Heisenberg model is not reliable. Specifically, if we take any of our trial states and 10 back

-t The minimal energy is given by (H) .. =-2NIJ-J'1

If I·

(46)

where [.x] is the integer part of .x. For the stripe state (and other "spin"-ordered states) the mean-field energy is also of order -O(n). Thus in the large-n limit the bond states are always favored, because their enerlies are of or-

319

x. O. WEN, FRANK WILCZEK, AND A. ZEE

11420

E/N

Chirsl Spin

FIUl( -0.921 ...............•

Dimer

-I-----------=~--~~

Dimer

Chiral Spin

o FIG. 3. Tht mean·field

1'/1

FIG. 2 The: me:an·tic:ld

c:l~rgies

of the chiral "rin slate. the

1'0 rhas~

diagram or Ihe Hamillonian

(431 wilh an added lerm 1471.

nUl( stal~. alld the: diOlc:r stale.

~er - 01 n: /. Tlh:re are then only tWt) possit-Ie phast"S.

the chir:!1 !lpin ~,a(~ and 'he <.Iilncr slate, in th .. range 0< J'IJ <: l. eJlergy llf the chi~al spin stale i\ slightly highe~ tl:::!, 'ha~ of the dim'!r s:a:e. At J' If .~~ I we: gel Ec:hiral = -O.Q94J Ilnd Edimer = -J -the two energies arc

n,,"

extremely close. Thus the chiral spin state is very likely to Ix a locally stable state. It is hard to imagine a path connecting the two states without encountering a potential barrier. There is an argument, relying on a result we will show later, that strongly suggests that such paths do not exist. That is, the effective action for low-energy excitation around the chiral spin contains a Chern-Simons tenn with int~ger coefficient while the efl'ective action for excitations around the dimer phase ('.ontains no ChernSimons tenn. The integer coefficient of the Cherr.Simons te!1l1 CIIn jump to another value only when the gap in the electron spectrum is closed. Therefore, for any path connecting the chiral spin state and the dimer state, there must be a state along the path for which the energy gap in the electrun spectrum closes. However, such a sapless state very likely has higher energy than the chiral spin state (with its gap). The gapless states, we conjecture, constitute a potential barrier between the chiral spin state and the dimer state. The chiral spin phase will definitely become the energetically favored possibility once we consider Hamiltonians containing an addiiional term of the form

- -; ~ G;G} . n

:47)

;.j

In (32) G; is the order parameter discussed before [see and (4)] and is given by

(I)

G; =i[(c;tcl+.i )(cjt+jc1+.i+ i )(c;t+.i+;c;)

the ground Slale. It lihould therefore be sensible to study the qU3siparlick cli..:itatillns arou'ld this ~latc. We also I~")ulld thai (he meJn·fidd encrgy of lhe "hirai-spin state is very close to thaI uf (i.h('r ordered ~ta:es. i.e., ,he Neel, slrirt'. a"d dim(:r ~'''It'! even for the original case of " = 2. This su~~e'its that Ille quanrunl ftuclual ion may "'ell melt the ordered pha.o;es, resulting in a chiral spin ground state. Even if this does not happen, the chiral spin state may appear as ground state of .. modified Heisehberg model. Hopping terms around the plaquette [see (47)] favor the chiral spin state. Furthermore, although ''Ie will not review it here, hopping terms of the simpler sort

..

(49)

also favor the chiral spin state. Altogether, we are encouraged to take seriously the possibility that order of this kind develops under rather general circumstances, in frustrated-spin ",odels. The chiral spin states defined hue are spin liquids, according to our definition. Indeed, in the larse-n meanfield theory the expectation value of products of X's around arbitrarily larse closed paths is merely the p-roduct of their nominal values on single links. Since the number of elementary triangular plaquettes enclosed by a closed path is proportional to the area enclosed, and each contributes the same constant to the imaginary part, the area law (9) is manifestly satisfied. Finally, let us remark that although we have made life easy for ourselves by going to mean-fi~1d theory, we have probably meode it hard for the spin liquid. After all, we expect the liquid to be stabilized, relative to say the dimer, precisely by ftuctu:ltions, and mean-field theory systematically minimizes ftuctuations. QUANTUM NUMBERS OF QUASIPARTICLES

(48)

Such a tenn does not change the enersy of the dimer phase because G; =0 in the dimer phase. But the added tenn lowen the energy of the chiral spin state if T> O. The mean-field phase di&gram is plotted in Fig. 3. In this section we have argued that the chiral spin state is very likely a locally stable state of the frustrated Heiser,~rg model, in the large-n mean-field approximation. For slightly modified Hamiltonians, it is plausibly

We now tum to discuss the quantum numbers of the quasiparticles, first qualitatively and then more fonnally. As we have seen the chiral spin phase is stabilized, relative to the dimer, by hopping. Very roughly, we may say that the dimer melts as the electrons become even ~Iightly free to wander. Presumably, this effect of including plaqueue i.enns or of increasing I oy hand also would be induced dynamically as a by-product of doping. Indeed, as one moves away from half-filling, vacant sites

320

CHIRAL SPIN STATES AND SUPERCONDUcnVITY become available, so the electrons begin to move. In any case, it seems sensible to think of the chiral spin pha')e as a quantum liquid. We expect it to be incompressible, due to strong Coulomb repulsion. In this context, the postulated-area law (9) acquires a simple physical interpretation. It means that the effective magnetic flux, introduced above as a Berry phase associated above with transport around fixed loops in phys:cal spac::, can instead be ascribed-much more rea!>onahly-to the transport of particles around one anu,I:·~r. [This ascription i~ possible, if (and only iO the :.pin ftlliJ is incompressible. For only then. are part ide (lUlw:r \\:u1in a loop. alid thl' are" of the loop, intert:h:J:ig ..·:lbL:.] In other words. tktitious fluxes and charges a;'~' l,\ be atlached to cadI particle, in slich a '.\ ;1\' that 'he n~rry phase is rcali7l'd a~ the phase accumulated accordi~e !,\ :11·- Ahar01l0\'-Boh", etfc~l for loan._port of litec;e fic!l!i\~us ch"rr,cs a~d fluxes :lrounJ "ll1e another. SQrhl~ti('a:l:d readers will recognil.e .here the appearall~e of 'it:tlisrj('al transmlJl:lti'.'r.. hd.:ed. th'! ana!:, ;;5 h<.'r'! i'i :!lHir'~I:- parallel t~1 a sim;\u t)lIe f()I' thl' FQiiE. l ef II~ d~t':rmine, fc!low:n& a slight:)' Ji1TcC1!1Il palh from the one laiJ down in that analysis, the relevant numbers. A defect in our fealure:ess singlet spin liquid can be introduced by constraining the spin on one site to be, say, up. The density of the liquid is then reduced, because the site in question can only be reached by neighboring electrons already spinning up. In effect, one-half a site-and therefore, by incompressibility, one-half an electronhas been removed. Sit:'ce the phase was e;fI per encircled electron, it becomes e "r/2 for encircling the defect_ Now we can expect that, upon our slowly delocalizing the constraint, the system will relax to an energy eigenstate with spin t· As long a~ there is a gap neither the total spin nor the phase accompanying transport around a loop far from the aefect can be altered by the relaxation, which is a local process. Thus we expect that the defect relaxes into a .:.pin-t neutral, half-fermion quasiparticle. The conclusion of the preceding highly heuri~tic argument can ~ illustrated concretely in the continuum limit of our chiral srin phase. In the flux phase the Fermi "surface" consists of twe' isolated points. The low-t..lergy e.\.:itauons ,:orr~pond to t,~·o families of fermlons in continuum limit, wl . .)se propagation is described by the effective I.agr311~ian

~

,,:sI,2

i7aa r UU (}I£+al£+Al£h/'ao,

(50)

a-t

where a = ± labels spin up and spin down, A1£ is the electromagnetic gauge potential and y" is given by (5 !)

a" is the dynamically generated gauge potential discussed in Refs. 24 and 25_ ao comes from the Lagrange multiplier term used to enforce the constraint ";=1. a;I;_1.2 comes from the phase of the hopping amplitude Xmil ,

exp-if~ A·dxXmll=~lIexp

(if:.·dx) _

(52)

Here ~II denotes the ground-state expectation value of

11421

c!c"

in the chiral spin state. For exampl.e, in the meanfield approximation, X~" is specified as an Eq. (39). In other words, the effective statistical gauge potential a represents the fluctuation of Xmil away from its groundstate value. The electromagnetic gauge potential is included to make the right-hand side of (52) an electromagnetic gauge-invariant object. In the chiral spin phase the electron spectrum opens a gap at the Fermi surface. This corresponds to the fermion fields "'tla obtaining a mass term. We find that the mass terms obtained by and 1/!2u h\·c the same sign

"'10

(53)

SUl.!h a mass tel m breaks T a.ld P, which just retlecls the properties of the chiral spin pha~. PUlling :50i :'IIHt !~Jj l<"'
symm~try

L--=

L Q~'

iP.,,,Y"UJ,.+a,..+A,,hllt1,.+m¢tlclll<:n'

(54)

1.2

At half-lilling we can safely integrate out the massive fermions and obtain the following effective Lagrangian:

LcI"=41: 1 g~ (a" + A" )a..(al + Ad~ ..l

.

(55)

The factor of 4 results from the four fendions. Using the effective Lagrangian we may obtain the lowenergy properties of the chiral spin phase. First we would like to show that there is no zer~magnetic-field Hall effect in th: chiral spin phase (i.e., the conductance u =0), even though it is no longer forbidden by the (bi'okeni symmetries P and T. The electrical current is defined by

aL eal I m ...M....a..(al + JIl=--=--'-IC' t

aA"

11"

m

~ )•

1111

(56)

TIle equation of motion for a" reads

aL etr 1 m Aa ..(a .. + Ill" ~ ) 0=--=--1-1<1:' .,M ..

aa"

.".

m

(57)

This implies that the electrical current vanish~ for a~y background electromagnetic field and the chlral span phase at half-filling is an insulator. This is.hardly shocking, since our effective theory !54) or (55) IS su~ to describe the Iow~nergy properties of the Hasenberg model (25), which contains no charge fluctuations. However, it was not completely otivious a priori in our meanfield approximation, which does allow charge fluctuations. Now let us consider the excitations in the chiral spin phase. The simplest excitation to consider is an exci~ed electron in the conduction band. We must emphasize that the IIppearance of an electron in the coOduction band does not correspond to introducing an electron into our system, because integrating out ao stiD enf?rce5 the constraint = 1. We will see that such an eXCited electron corresponds to a neutral spin-t particle. At low en-

"i

321 X. G. WEN, FRANK WILCZEK, AND A. ZEE

11422

ergy the excited electron can be regarded as a test particle. The effective Lagrangian in presence of such a particle can be written as

.L cff =

2~ I: 1E""A(a" + A" )a,.(aA + AA) (58)

where j" is the current of the (est particle. Now the electrical current and the equation of motion become

1..=

~

~lIar"(ia,,+a,,+A")"'lIa+m~lIa"'lIa'

(61)

II'" I ..... q It'" ~

After integrating out the fermions we obtain a ChernSimon term as in (58) but with the factor of 4 replaced by 2q. The neutral spin-t excitations then acquire fractional statistics given by eil.,+mlJ/lmlql. This is exactly the same result, as would follow from our heuristic argument. CONCLUDING REMARKS

(59)

and

7he first tel'm ;n !SQI car. h\. reg;~..JcJ ~" Ih~ l'o.Itrit-;:ljnn to ekctrical charge lrising (mm the vacuum ,olari7..atior.. The equation of mOlicn implies thai Ih~ eiectflcal charge of the: excited ele~1 ron is .:ompklc.-l~· s..:rccncd by \';tCUUnl polarization. The screened electron behaves like g II~U­ tral particle. Because the chiral spin vacuum is a spin sin,let even wben a,. and A" are nonzero the vacuum p0larization cannot chanlJe the spin quantum numbel of the excited electron. Therefore, the screened electron is really a spin-! neutral particle. Due to the Chern-Simons term in (58). the statistics of the excited electron is also changed. From Refs. 5 and 31 we find the statistics are pven by a phase factor +-./2hll ll. Thus, the screened electron behaves like a half-fermion. The quantum numbers and the statistics of the screened electron are exactly the same as the spinon in tne JC ',uaneyer-Laughlin state. and of course the same as we obtained heuristically before. It may weD teem that there is no connection whatsoever. or even a mismatch, between oar heuristic araument and our formal argument. According to the former. the quantum statistics of the quasiparticle is determined by the ratio of fictitious lux density to particle density. According to the latter, it is determined by the number of points at which the energy gap closes, if the lux is turned 01'. Most remarkably, however. these two quantities are related by an index theorem. 32 We shall illustrate how this works, by considering a generalized f1uJ( phase. The construction of the chiral spin state given above was based OIl the particular flux phase such that the flux through each plaquette is equal to.. A similar constructioa of the cbinI spin state can be also done for ,eneralized flux phue where the lux throuah each plaquette is equal to 2.,/9, with q an even integer. It bas been shown that the Fermi ·"urface" of such a lux phase (at half-filling) consists of 9 isolated points, and that each point corresponds to a two-component massless Dirac: fermion in the continuum limit. Including propel' nonnearest-neighbor-hopping terms, we give each of the q pain of fenilions, C:OI"'ected by a perturbation at the appropriate wave vector, a mass. The mass for each family can be shown to have the same sign. Thus the generalized chiral spin state is described, in the continuum limit, by

e" ..

Now let us briefty discuss what all this might have to do with high-temperature superconductivity. A spinon of the type described above, carrying half-fermion statistics. plausibly binds to any introduced hoI'!, creating a ;pillles!> char~ed haIr-fermion ~omposilc. Two halffnllli",no; ~;III pair 10 ma!.:!;' a h}son. and s,;.:h b.)son pairs .If:.: ~ •.•od ~·andid;,;t:~. far J 'Url"I~(1I:dm:(1I1~ t:llllt~ensa!e. The p3iring IS t':lergetu.:ally d.: ..irablt. hecall~t: Ii pair of introduced lade ... gcn('ralc a tklliiulls Ihn whi.:h is all in-

tegral multiple vf Ih~ iUlid:irnentlil ftuxoicl. In other words a pair can pucefully cOC!lI.ist with the chiral spin phase background. and therefore need not carry spinons along.J) The qualitative idea here is not altogether unlike that underlying "spin-bag,,14 or "spin-polaron" mechanisms. According to these pictures too, holes are associated with disordered patches, and so it is advantageous to minimize their effect by clumping them together. There is a significant diff~rence, however; the present mechanism does not require an antiferromagnetic Neel or spinwave background tf) play against. Another related argument for superconductivity in doped chiral-spin liquids. given by Laughlin,l2,lS is the following. It is known that fermions with arbitrarily weak attraction become superconducting at zero tcmperature. Now half-fermions can be considered as fermions with a special sort of long-range attraction. lbus, they mUSl condense at low temperature. We conclude with some philosophy and a speculatIon. The message of this paper, and of several others in the recent literature might be phrased roughly as follows. The success of the Laughlin wavefunctions in describing the incompressible quantum-liquid phases of the FQHE, shows that they ,>rovide an excellent way to reconcile the desire to order (in that context, order in real space is desired, to minimize Coulomb repubion) with the oifficulties introduced by frustration (in that case, by an external magnetic field). Roughty speaking. in the FQHE the electron by collective correlatioas, manufactures an el'ective magnetic field to cancel the raJ one. Now .. frustrated-spin system faces similar problems. ut us imagine attempting to find the ground state in the usual way, by evolving the system in imaginary time. We c::.n think of spin sampling various loops as it "decides" how to point, and in general getting confticting instructions_ By condensing into a chiral spin liq.uid, the spins introduce collective phases, that partially ameliorate the frustration. It is DO accident, then, that the sorts of dective-field theories and order parameters we find for chiral spin liquids, are so reminiscent of those Camilie" in the quantized Hall dect. Concretely, spinon excitations

aas,

322 CHIRAL SPIN STATES AND SUPERCONDUcnvlTY

11423

around the chiral spin state near half filling, analyzed above, have the same statistics as one finds for the quasiparticles around the m = 2 Laughlin state. We believe this conclusion, originally derived by Laughlin from an approximate mapping of the Heisenberg antiferromagnet on a triangular lattice into a quantum Hall system, is much more robust; it follows generally for chiral spin liquid states having b = 1T"~ in the area law (9), where is the density of electrons. This circle of ideas strongly suggests a conjecture, that if truo! leads to a drama.tic consequence. It is quile concei', able that in difT.:rl'nt parameter regimes. or in the real mataials at diff~ren: doping IC':els, other pos~ibilities tluo In =2 occur Indeed. we ha"e bril"fty dic;cu'is"'Ci such possibilities before, ill iliustraling the consistency of our qualitative 'lnJ quar.ril3iive 2I'guments for frac[~onal ~Ialislic~ of SpiIlOHS. In mean-field theory, thes~ different pussibilities lead 10 gaps opening at dift'crer,1 plltc~. It sh{'uk1 be fa'o':")rahlc '.0 open a gap at the Fermi surfa~e. so t!11! <,ystcn; might swireh from nile phase hI .. noihl'r I!S the position of .his surface c!langes. And. of course. a w,",ole

menagerie of states has been observed in the FQHE. Now if, for instance, and m=4 chiral spin state were formed, holes doped into it might be expected to CORdense in quadruples, thus producing a ftuxoid unit h /4e. (This time, we are speaking of genuine magnetic ftux.) All this suggests that it would be worthwhile checking the unit of ftux 'luantization carefully in the new materials under various circumstances, not prejudging the uni':ersality of pairing. There may be surprises lurking at dilferent doping levels, pressures-or even simply at lower temperatures.

·Present address: Institute for Advanced Study, Princeton, NJ 08S44. IF. Wilczek, Phys. Rev. Leu. 49, 957 (1982). 2F. Wilczek and A. Zee, Phys. Rev. Lett. 51, 2250 (1980). )0. P. Arova:., R. Schrielfe r , F. Wilczek, and A. lee, Nuel. Phys. B251, 117 (1985). 4N. Redlich, Phys. Rev. 0 29,2366 (1984). SA. Goldhaber, R. MacKenzie. and F. Wilczek (unpublished). 60. W. Semo:nolf, P. Sodano. and Y. S. Wu. Phys. Rev. I.e{t. 62, 71S 1I9811). IJ. Frohlich and P. Marchetti, Commun. Math. Phys. 116, III (1988); Ill, 117 (11)89). 'The Quantum Hall Effect. edited by R. E. Prange and S. M. Girvin (Springer, New York. 1967). 90. Arovas, R. Schrierfcr, and F. Wilczek, Phys. Rev. Lett. 53, 722(984). lOS. I. Haperin. Phys. Re\,. B 15.2185 (1982): R. B. Laughlin, ibid. 13. 5632 d<Jt:IJ; F.D.M. Haldane, Phys. Re\. Leu. 51, 60~ (1983). lip. W. Andersor.. ill F"ontic:"s and Borderlinc's if! .\funy Particle Phps;c:s, edued by R. Schriefl'er and R. A. BroClia (NorthHolhlnd. Amslerdam, to be published). '~R. B. Laughlin, Science 242, 5H (l9g8). uS. A. Xivelson, D. S. Rokhsar. and J. P. Selhna. Phys. Rev. B 35,8865 1I987); see also I. Ozyaloshinkii, A. Polyakov, alld P. Wiegmann, Phys. Leu. 127, 112 (lQ88). 14y. Kalmeyer and R. B. Laughlin. Phys. Rev. Lett. 59, 2095 !1987). ISG. Kotliar, Phys. Rev. B 37,3664 (1988). 16 1. Aftleck and B. J. Marston, Phys. Rev. B 37, 3774 (1988).

17p. Wiegmann (unpublished). 18J. March-Russell and F. Wilczek, Phys. P.ev. I.e:t. 61, 2066

"t

ACKNOWLEDGMENTS We would like to thank B. I. Halperin, R. B. Laughlin, R. Schriefter, Q. Niu, S C. Zhang. and Z. Zou for helpfu~ di~u!'sions. Toil' rt'Search was S'lpported in part bv the National Science Foundation under Grant No. PHY82t 7853, sU[lpleme:l!ed by fund:; frem the National Acronautic~ anJ Space Administrar;on, at the Unive~:ty of California 8t Santa Barbara.

(1988). 19X. O. Wen and A. Zee (unpublished).

lOB. Halperin, J. March-Russell, and F. Wilczek (unpublished). lip. W. Andenoa, Science 235, 1196 (1987).

ZlS. Chakrrvaro/, B. :. 'tfalperin. and O. R. Nelson, !»hys. Rev. Lett. 60, 1051 (l988); Phys. Rev. 039,2344 (1989). 21R. B. Laughlin, Pbys. Rev. Lett. 60, 2677 (1988); and Ref. 12. 240. Baskaran and P. W. Anderson, Phys. Rev. B 37. 580 (1988). 151. AfIIeck. Z. Zou, T. Hsu, and P. W. Anderson, Phys. R.ev. B 38, 745 (l988)~ E. Dagotto, E. Fradkin, and A. Moree' ibid. 38, 2926 (1988). 26M. y. Berry, Proc. R. Soc. London, Ser. A 392, 45 (1984); A. Shapere and F. Wilczek, Geometric Phases in Physics (World-Scientific, Singapore, 1989). HZ. Zou, B. Doucot. and 8. S. Shastry, Phys. Rev. B (to be published). 11M. Inui, S. Donaehi, and M. Gabay, Phys. Rev. B 38, 6631 (lqSSI. l~p.

Chandra and B. DOUCOI, Phys. Rev. B 38, 9335 (1988). JOQ. Baskaran, Z. Zuu, and P. W. Anderson, Solid Slate Commun. 6J, 973 (1987). )1 X. G. Wen and A. Zee, J. Phys. (Paris) (to be published). 12X. G. Wen and A. Zee, Nuel. Phys. B (to be published). llX. G. Wen, Phys. Rev. B 39, 7223 (1989). . 14R. Schrieffer, X. G. Wen, and S. C. Zhang, Phys. Rev. Lett.

60, 944 (1988). lSA. L. Fetter, C. B. Hanna, and R. B. Laughlin (unpublished).

7. Anyon Superconductivity R. B. Laughlin, "The Relationship between High-Temperature Superconductivity and the Fractional Quantum Hall EfTect", Sc~nce 242 (1988) 525-533 .......................................................................................... 326 R. B. Laughlin, "Superconducting Ground State of Noninteracting Particles Obeying Fractional Statistics", Phys. Rev. utt. 60 (1938) 2677-2680 ....................................................................................................................... 335 A. L. Fatter, C. B. Hanna and R. B. Laughlin, -Random-Phase ApDroximation in the Fractional-Statistics Gas", Phys..Rev. B39 (1989) 9679-9681 ........................ 339 Y.-H. Chen, F. Wilczek, E. Witter and B. I. Halperin, "On Anyon Superconductivity", Int. J. Mod. Phys. B3 (1989) 1001-1067 ..................................... 342

325

7. Anyon Superconductivity Surely the most dramatic result of the study of anyon statistics mechanics has been the demonstration a new mechanism of superfluidity (and, for charged anyons, superconductivity). This superfluidity is quite a robust consequence of fractional quantum statistics at appropriate values of the fraction. It occurs even in the presence of other repulsive intera\:tions, and gives quite a large energy gap. For these reasons, together with the facts that it is essentially two-dimensional and plausibly associated with exotic spin ordering: it is terrlpting to speculate that thp. anyon mechanism of superconductivity will shed light on the copper oxide high temperature superconductor::. 'Nhether or not this s:->eculatioh works out, the mechanisnl is of considerable theoretical interest. and will undoubtedly play an important role in physics in the fut'..lre. The intuitions which led to the proposal of the mechanism are forcefully explained in a long paper [1] by Laughlin. Som~ formal arguments leading to the conclusion that the mechanism does indeed work were presented in brief papers by Laughlin [2] and by Fetter, Hanna, and Laughlin [3]. In their long paper [4] Chen and collaborators further distilled out the simple essence of the ideas, demonstrated the superfiuidity in a controlled approximation, and extracted the effective Lagrangian governing the low-energy phenomenology of anyon superconductivity.

REFERENCES 1. R. B. Laughlin Science 242 (1988) 525. *

2. R. B. Laughlin Phy,. Rev. Lett. SO (1988) 2677. *

:J. A. Fetter, C. Hanna, ;}.nd R. B. Laughlin.Phy,. Rev. B39 (1989) 9676. * 4. Y.-H. Chen, F. Wilczek, E. Witten, and B. I. Halperin Int. J. Mod. Phy,. B3 (1989) 1001. *

326 Reprinted with permission from: R B Laughlin, Science 242 (28 October 1988) 525 - 533. © 1988 by the AAAS

The Relationship Between High-Temperature Superconductivity and the Fractional Quantum Hall Effect R. B.

LAUGHLIN

(OIlCept

'111: case is made that the ~pin-liquid state of ~ Mott insuhtor. hypothesized to exist by Anderson and ickntined by him as tf1C' correct context fo:, dUcussing temperature 5Uperconducto~ occurs in thCS(' matcna!s and exhibits t;le principles of fractionaJ quantization identified in the fractional quannun Hall eft"ect.lhe most irr.p.:".... (,[ uf m~ is that particles carrying a fracuon of an elementary quantum number, in this case spin, attract one another by a powerful ga~ force, which can ~d to a new k.!nd of superconductiVity. The tempa:aturc scale for the supercm.cf'lctivitr is set by an energy gap in the spin-wave spcc:trum, which is also the fundamental measure of how "Jiquid" the spins are.

hiP.-

to our understandirg of the solid state thal it is implicit in

th~ voahul:f" W~ U~ :md prciudk~ the questions we ask. Th~ high·tanper~rurc superconductivity lireranu·c i~ fillec1 wilt. aprcs-

tcm~lt"ranlre supcr~ondul."tivity

sions such » ~fermi su:face," "density of ~~tes,·· "Pauli st.SCCprihil. itV." and "'c:learon·!'h0non inn:rxrion." aU of which require the ((\on:u nf a f~~i hquid even to mak.e sense. In ligh:: of !hc: O'·C"n\- helming empirical cvidcnl·e that aU mctais ar~ ~enni iiquiJs. it is 11< >t surprising dlat the resonating vaicnce bond concept. although widely respected. is not \\idely b--Jicvcd. This is unfOrtunate. for Allderson's reasons for thinking it to be "'t the bottom of high transition temperature (Te) superconductivity <'re compeUing (.~). Let me parapruasc these as I undctstand man: 1) High Te superconductivity occurs in a dass of systeaN, the Matt insuJators (4), that we have never understood. It is hard to understand how this could be a coincidence. There is probably some prcvtOusly unknown Pf(\perty of these systems that causes the effect. 2) The systems in question arc inherently magnetic. Stoichiometric LazCu04 is an ordered spin-1/2 antiferromagnct (5) and also an insulator. Doping the material (6) by substituting Sr for about 3% of the La destrOys the magnetic order cUld makes the material a "mew" in the 5ense of conducting el..:ctricity at 7.A:ro temperature. It i~ hard to understand how dopin~ at this Ic-;('/ cmdd rum: destroyed atl the spins. A morc r~3'iOnable gut·55 is that r:l~ c..'ttta holes make ordering more difficult. and that the spins art' still prcscm in some sort of "quantum spin liquid" sta~..:. 3) The only rotationally invariant spin-1I2 system for which we have an aact solution (7), the linear Heisenberg chain with nearneighbor in~ractions. possesses a disordered ground state that might well be termed a quantum spin liquid. It is reasonable to adopt this state as a paradigm for the putative spin-liquid $We in higher dimension. No one has proved that such states exist. but surely some Hamiltonians can be found in which quanaun ftuaua· ril.lll.~ prc\'CI~t ordering. After all, magnetic ordering is physicilly similar to crystalli7.:nion. and helium ha... both crysWlinc and ftuid

~=;~U::::~=:'~~5ClM305,

4) The elementary excitations of the Hels.:nberg chain are known (8) to be neuaal spin-1/2 particIcs poacssing a linear cacrgy. momentum relation. If the form of these acitations wae generic to spin-liquid states, one would c:xpc:ct the charpl excitluioas induced by doping to be very strange, at least within the conteD: fX mcr.aIs as we know them. For aampIe, one possible fate of a hole doped into the material would be to become anachcd to a nc:auaI spin-112 excitation to form a charged spin1css particle. Andmoo rGas to this object, which was invented by Kivdson, Rokhsar, and Scdma (9). as a "hoIon." He calls the neutral partic1e a "spinon." Bcause the hoIon is spinIcss. one would guess it to be a boson, in which case it might cause superconductivity by Bose condensing.

w.

N THE FALL OF 1986, P. ANDERSON (1) MADE THE BOLD suggestion that superconductivity in La2CU04 and rdated materials might be caused by the occurrence in these rmterials of the "resonating ,·alcnce bond" state, a hypotheticaJ magnct;~ liquid state proposed by him (2) in the early 197vs. ~"hile ncidler the resonating v.lIen('e bond state nor the theory of high·t'('mpera· rure sapcn-unduah·lty it engenders is very weU defined at pr-:scnt. I am persuaded that the core of the idea, that the Fermi liquid principle fails in high-temperature superconductOrs, is ri~. This has led me to some: new pcrspectiv~ on mis subjel.'t, which it is the purpose of this .lnicle to discuss.

I

Fenni Liquids and the Resonating Valence Bond 111e most appropriate place to begin any discusmXl of high· is the Fenni liquid concept. A Fc:nni ;iquld is ~ definition any system with low-encrgy excitarions srmilar to mose of a nonintcracting Fermi sea. It is an empirical fact that all known substances, except" perhaps high-tanpcratwe superconductors, which arc metals in the sense of conducting eIcaricity at zero temperature, arc Fermi liquids. For this reason there is a deep-seated belief among solid-state physicisb that mcraIs should be Fami liquids tIS /I mIIlkr of priltcipl~, even though there is no prima facie thcomicaI evidcn:e for this. It is almost impossible to dcmonsttatc &om first principles that a given mataial is a Fermi liquid. The equations arc simply too complicatuf. So central is the Fami liquid

28 OCTOBEB. 1988

phases.

327 Experimental Properties of High-Temperature Superconductors There arc several significant ~torS reinforcing Ihc iUpticism toWard the resonating valence bond idea. One of them is the reluctance of most scientists to abandon thinking mat has served them well in the past unless it fails spectacularly. This mayor may not have occurred, depending on which Cltperimcnts one believes. It is regrcnably the case that all but a h;'\Rdfui of experimental properties of high Te superconductors can be understood qual itath'ely in terms of the traditional Fermi liquid meory of superconductors. Let me mention a few of thc.sc: (10): I) The spin susceptibility (It) is roughly \.-omistcnt with Pauli p3r:l1Ngnerism of a Fermi sea containil1[t the number of electronS bcliC"al from stoichi~' to ha\'e ~ dc.1JlC'd into thc material. This carrier density. in rum, is roughly consistent both with the DrucIe-like conduch\'ity (11) obscn'ed in the far infnred and the pl;l.~m.l oscillation induced by it at higher c:ncrg)'.Xbtcsnsisrenq' is onl\' rol'ldt bccall~ (he "'band" etfccth-c n~ of the eh"trullS is noc: knc;wn. -E.~timatc., t>a~d on c~rimcnt rll1~ from one to ten dC(Tro:1 .!' ,-\,'>Cs. ,\!temj'IS b' "ak:.alnc rhi:l m.a.~ \'-:; 1C1< .1'1' of st;m·I"rd h.ln<1 ~'~Nrl" '(:d\Oiqu("~ an:' di ....:uh II:> i:lfert'm ; I,':. Sr()~hi()mctri, Ll~CU04 conk' OUI to a met":. which it is Of", 2 i The ei«tn..'.11 n:~i ..'t.\"lt\; " ~41 aM'c 1~ L\ the ~;1.c opCl.'t~-d of a scmK:}f'\,·u~'tor d,,~ etf :0 ~hc approprilfc J:-."Ct. 3 ,I The traIlsiriun to supcn:onJuai\'ir)' is ~"Uted with ... spt:cit: ic heat anomaly (tl, I.il rcLued in approxinwdy the right way to thc Pauli suscrpribiliry. Extra Row

1

,.-,

S::: 1/2

Fractional Quantum HallSt~te

r
CA

~

Fr:lCtionaily Charged Quasiparticle

Spin Liquid State

·.·,.:t!. ...·..· ...Spinon

FIg. 1. Ifa spin liquid exists, 10 should spinons. This is because a S)'SIaI1 with 'lit CYa1 number «spins (top 11ft) musr be a sinp:t. while a syIIan with an odd nwnbcr 0( spins (top right) annex be. Beaule me two syaems cannot be di~ ~ Ihcy ~ IarF. hoftvcr, Ihe Iaaa-IIlUIt be viewed as a ncuaaI spin-Ill excitation of the former. Idcmific:aaioa 0( the haIf-intqral Spin as a &actionaI quantum number sugaa that the spin-liquid pound state (oenIIIr Mel bottom right) is anafosous ID Ihc hctioriaI quannun Hall swe (center Mel bottom 11ft), and mar the spinan is maJosous to ill &aaionaUy cIwpI quasiparticle. SUI

4) Energy gaps have been observed in bach Ihc optical absorption (16) and runneling spectra (17). These gaps are roughly cq~ and related in roughly the right way to Te. Both gaps disappear

above Te. 5) The critical field (tB) He collapses continuously as Te is approached from below. There· is evidence (19) that the tunneling gap also coUapses continuously. 6) The zero tcmperatw"C London pmctration depth ). is roughly consistent (20) with the density of carriers induced by doping. Both ). and the coherence length ~ diverge (18, 21) at Te. 7) The ac Josephson efl'cct (22) observed in granular samples indicates the presence of a conventional chargc-2 order parameter. This is consistent with Cooper pairs but not: consistatt with Bose condrauarion of hoIans. lbc latta' would produce a charge-I order parameter. A related and somewhat distressing &aor is Ihc inability of the idea to credibly predict anything or, fOe that matter, even to account for tm supcrconducri\'ity. The most: concrete prediction of me approach, Iwndy, that the ord..r paramcttt should ha\'C charge 1, "'as rro\·aj fal5C within a few wcds ofia pn>posirinn. The siruation h.:I~ ~'I:()ni~ so grim thar nrcrimcnt2ii:;ts hJ\'c larp;~~' ..tl.'Ppcd hSlc!IIn~ to theorists.

The Mott Insulator Problem Br tar the most sttious impediment. however, is the unwillingof most solid-san: physi:ists to acttpt the fundamc-r1tal intcllccru;d problem a disordc:red Moo :nsuJator (4) presents. The m..>flOXides of iron, cobalt, and nickel arc insular'lfS with sintilar properties (23). However, COba!: oxide cannor possibly be an ordinary insulator bccau.K it has an odd number of cIcctrons per unit ccU. All three oxides are, in faq, Matt insulators, materials that insulate solely as a ~s\11t of Coulomb repulsions bctwccn electrOns, but only in cobalt oxide is this conclusion inescapable. Mort insulators are very poorly understood. One frustrating consequence of this is that there are no agreed-upon criteria for iderttifying than. Thus, evert though it is obvious to me thar hightemperature supcrcor.ducrors are Mort insulators, it is intpossible tu convince my disbelieving colleagues that this is the cast' on purely phcnomcnoIogical grounds. Lacking an understanding of Mott insulators, we usually assume thtm tn be saniconduaors, a weUunderstood class of maaiaIs that bcc:omc metals when doped, and wait for the cxpcrimcnts to tdI us otherwise. This would be a pcrfa:dy reasonable way to proceed if the c:xpcrimcntal results were more clear-cut, for good c:xpcrimcnts gcncraUy lead to Ihc bUth whcIhcr ideas motivating thcm are correct or not. However, it has been Ihc case historica.lly with Mott insulators, and is Ihc case prcscndy with high-tcmpcnturc supcrcooducrors, that the experirncnn ~ plagued \'lith materials problems and interpretation ambiguities. so that this strategy docs noc: wort well. There is, of course, no reason wha~'U to ~xf""CI a disordered Mutt insl1lator to be a semiconductor. It is certainly noc: true that Mort insulators arc dtmotulnJbly scmicondUClDrS (24) tkscribcd well by a commensurate spin density wave ground state. rtnlI

Analog,' with the Quantum HaD BWect Let us now ask why Ihc n:sonatiog valc:ncc bond idea, if it is so insightful, is having so much di8icuIty accounting for Ihc supcrconduaivity in these marcriaIs. The nDt obvious possibility is that there is a minor error somewhere in its logical dcvdopmcnt which has led us down a blind alley. Where could it be? The notion of a SCIBNCB, VOL 2+2.

328 spin-liquid state seems sound enough. It is an experimental fact that high-tempcrarurc supcrconducton have no magr.ctic order. It is hard to Wlcierstand how the spins could simply have vanished. There is certainly no reason to believe that quantum mechanical melting should occur only in one tfunension. The most likely source of the problem, thcrcforc, is the identification of the one-dimensional Heisenberg model as an appropriate paradigm. There arc a number of reasons to be suspicious of the Heisenberg chain as a model spin liquid. The most obvious one is that it cannot be ordered, just as a matter of principle, whereas higher dimensional S}'stems can. It is conceh'able, for example, mat the gaplessness of me spinon spccaum in one dimension simply reflects the system's kndc:ncy to be ordered and thus to ha\'e a gapless spin "'~\'e. Also, it is uniquch' the case in one dimension that bosons can be con\'ctted to tennions and vice \'crsa b\' means of canonical transfonnatioll ,b"). ThliS, C\'en if the excitations of the hig."er dime~ional spin iiquids arc analogous to thos.- of the Heisenberg ciuin, it is not clear ,nat ~tatistics to as~lgn t:acm. Even while suspecting the paradigm of me Heisenberg chain, one should probably belle\'! in srinor.s. :\..:cording to Dzyalosh~r.~!rji 1.?6} Landau -:onsidered the notion of a liquid $[;tte with spin· 1:2 exci~tions so ob\ious that he dicJ not bdin'e ordered antitCrrnmagrom existed. While it is not com~ICttly cbr wh~' h.: mought chi,. an oh:C'u~ rOSo<,bilicy is ilIllstr:mo:d ir Fig. I.. S':P:X>$C it is esrabli~L::d that the ground. State or some H.uniltol\ia.1 is a nondegeneratc spin Ii'lllid. rnen the ground state must ~ a singlet when the numt-er of spins is even and a doublet •...·hen the number of spins is odd. Since mere is no long-range order, however, me two systems must be physica1ly equivalent. Therefore the doublet must actually be a spinlJl excitation of me singlet ground state. The existence of spinons is

£

£ Magnetoroton

~

~ ~~~~:,?

\~'1 ...~~ Soft Mode

Y

q

q

Liquid

G/2 I I

£

I' I' I

1

:

\

E. \

N

Crystal

I I

\:

'.' I

q

I

I

1

I

q

fig. a. Whahcr or DOl: fraaional quantizatioo oc:cun in the quantum HaD problem is dctamincd by me prcscocc or absence of an energy gap in ill coIIcaivc mode apccaum (top 11ft). This mode is a compressional sound waft in the &aaiOnal quantum Hall sate and praumabIy a spin wave in the ipin-Iiquid Ja.1hc Hamiltonian is c:~ II) induce a)'~ (top right) the ~ minimum of Ibis mode IOftaaa, leading II) a divapnt ~a-~ ra:iproaI Iaaia vcaor G of the crystal VICMd in the rcduc:cd zone schc:mc of me ayaal (bottom 11ft) this spearum is qu* similar II) mar of a ayDI (boaom right) c::m:pt fix the ,.acnc:c of a pp at the zone cam:r and the abecna: of a pp at me zone

--=.

CIdF· :II OCI'OBBB. 1988

on much sounder footing than the analogy with the Heisenberg chain would suggest. This brings me (0 what I believe very strongly to be the key idea missing from Andenon's vision of the resonating valence bond. If a spin-Itquid state exists, which it probably docs, it is expected to have ncuttal spin-1I1 excitations. But how could this be~ In the absence of interactions between the spins, the elementary excitations of the system consist of the aa of flipping a spin from down to up. These obviously have spin 1 and are bosolls. How could it be that these "elementary particles" of me problem could combine quantum mechanicallr to make spin-1I2 jtrm;t."IS? In one dimension this point is moot because bosons aud fennions cannot be disringuished. In higher dimension. however, me paradox is real. There arc (\':0 esrablished precedC'nts (27) for rurning bosons into fmnions in higher dimension. roth of which arc improbabll' in the conteXt of thi~ prublem. Olle in\"oh·es borrowing the half-imcgral spin repre' sentation from isospin degrees of freedom. The other in\'Olvn addin.; topological terms to th.: boson Lagrangian. The Ian.."!" amoun~ to changing the la\\'s of phrsi~ in a fundamental W2)'. Thus, given that spulOns occur in dincnloion ~rc3tCT than 1. ~ir cxiscence is propn-Iy con~idcrc,i m;racultlus. lc implin mM the elementary spin-l e~ciurions h3\'r been split In t\\'o. wim half the exc.rark,,' .lprcarirt~ in the ~aml"e in~or and half 1t the bounduJ'. Thl"re is on!" on~ i~.:nciti,j else ofbt-ha\'ior ofmis kili.J in nature of v.111ch I am aware: the fraaionalization of ele.:tric charge m~t ()(01B I~ me fractional quantum Hall effect (28). This beha\'ior :s so unusual that I find it hard to understand how there could be rwo distin~:t ph1SCS of matter ext:ibiring it. I therefore beliC\'e mat rhe fraaiO!1a1 quantum Hall state is me only possible correct paradigm for the spin-liquid state.

Properties of Incompressible Quantum Fluids Let us now explore ctte possibiUt'f that ~e spin-liquid state and me fractional quantum Hall state arr one and rhe same. Here arc the generic characteristics of the state as I see than (29): 1) A fearurdess liquid-like ground St3te that is nl)t degenerate. 2; ~emelltar)' ex(,i.tations above mis ground 5tate mat carry fra,rional charge. This ch.u"e is qU.lllrizcd to a particular \'alue d\a~"terist:i, of the state. 3) An energy 5ap for making either fractionally charged excitations or collective modes. The coUC\."ti\·e mode may be thought of either as a pair of fractionally charged particles or as a density Ruauation. Cltanges to the system's Hamiltonian that pmerve this gap also prescr\'e the fractional charge txactly. 4) Long-range gauge forces between the fractionally charged J,atticles. The~ appear in the fraaional quantum HaU literature as fractional statistics of the particles. They arc also prcsct'\'ed C"~acrly by changes in the Hamiltonian rhat preserve m(' gap. Ttw dlCSC arc acru.uly properties of the n=a1 fractional quantum Hill state is indi~~table. We !~now such a state exists ~I" because a H..uniItOnian has been found (30) for which the variational solurioo I proposed for the problem is exaa. We ~J1OW that excitations out of the state arc fractionally charged because we have wave functions fix these excitations that arc c:xact (30) in this limit and can prove that their charge is unaB'ccted by changes in +.e Hamiltonian that preserve the energy gap (29). We also know that the Hall conductance, which is a specaoscopic mea5lft of dais charge (Jl), is cuct to within c:xperimcntal unccnainty (J2). We know that these quasiparticles obey fraaional statistics bach because it can be dcduad &om their wave fimctions (29, JJ) and because me Hall conducanccs of the &actional quanaun Hall hierarchical stateS arc the correct w1ucs (34) to within cxpcrimcntal uncertainty. Al.nCUl$

sri

329 Like any liquid, the fractional quantum Hall sta~ is dilicuJt to distinguish from a crystal. In particular, it posse.sscs a colle,tive mode (36) that may be thought of either as a compressional sound wave or as an exciton (37) formed when two charged quasiparticles bind. The dispersion relation of this mode is known to have the general form illustrated in Fig. 2. Its deep minimum, named the "magnetoroton" by Girvin, MacDonald, and Platzman (36), who discovered it, occurs at a wave vector corresponding to the interparticle spacing. This gap is a "measure" of how liquid the state is. If the Hamiltonian is tuned so as to induce crystallization, which we know must occur in certain limits, this gap should coUapse continuously (38), as appropriate for a second-order phase transition. This has not been proved to be the case, but it is very reasonabIe. and it has certainly not been c..>ntradicted by an~' experiments (39). Three things happen as the gap gets smaller: 1) The state gets increasingly susceptible at the crystallization wa\·e vector. This susceptibility diverges at the crystallization point. 2i The size of the quasiparticlcs grows. An apf"\J\Mogy Would be the growth of a Cooper pair that results from diminishing the gap of a 8a;"dccr;-Coopcr-Sdulcffi:r (BCS) superconductor. 3: ·;"he ene~· <:\ l';t to make 3 quasipan:de dinini~ltes. The ,""0$( is zero J[ the tr;anslri"l"I. 1be qu.ulpartide durgc l1\.1inra;.b its inte"iry ~ these t.'linp occur as k)n~ a~ the ~ap is non;xm (291. Let me emphasize again th.lt dlis hc:ha,·ior I~ expected ()lJ .\~. gencr.ai gwund\ ami, in particulu, is comptttcly IftSt.uitil1t t(l what tltt HllrniitOflWrI is. The large susceptibility, "ilich is pankularIy significant for systems on the verge of crystallizing, must lead to intense belastic scattering at the magn~roton wave vector. & iUusrrated in Fig. 2, this could easily be mistaken for B~ scaacring. Since the Goldstone mode of the crystal and its ground-sta~ degeneracy are really the same thing, the

di8"erence between liquid and crystal is simply the pracnce or absence of the energy gap. It is reasonable to expect all of these features of the fractional quantum Hall state to have analogs in the spin-liquid state. Thus, the analog of the fractionally charged quasiparticle is the spino", the anaIog of the compressional sound wave is an antiferromagnetic spits wavt, the analog ofWigner crystallization is atttifmom4gra«ic orrItri"g, and the analog of the magrietoroton gap is a m4glltticJlwctuatio" gap. The most obvious implication of the connection between these two states is that the spin-wave spectrum of the magnet must have a gap, as this is the measure of how "liquid" it is. Above this gap the ftuauations ~hould be indistinguishable &om those of an ordered antiferromagnct. Although the energy gap has not yet been seen experimentally, magnetic ftuauations similar to those in the ordered phase have been detectcJ in supen:onducting samples with magnetic Raman scattering by Lyons «al. (40). Similar magneDc 8uauations in insulating samples have been seen with indastic neutron scattering by Endoh tt at. (6). Since existing experiments do not have the r:solution to se": this gap. let us guess that it is comparable in size to me one mcasured in tunndang. or roughly 30 me\'. Raman cxpcri.."ents show dl{' maximum spin-wa,·c energy to bt- roughly 200 rile\'. The possibili~' that the Andcnon l"Cl'onaring valence bond stan: might oonsriNtc anotht:r c.ulllplc of fractionaJ quantum Hall bd\,·:ior was sugg:srcd to m!: ~. D. H. J cc an..i J. D. Joanoopoulos about a year before high-rempcnture superconductivity was disrovcred. V. Ka1meycr and I (JS) investigated this idea numerically and succeeded in making a "ery strong case that it makes 1CnSC. I must emphasize that we did not prove it to be true. Indeed, no one has conclusively proved that a spin-liquid state even c:D:1S in any dimension higher than 1. In light of the present experimental situation, however, it seems a bit silly to worry about this. For reasons I have ·w.ady stated I find the mere consistency of the idea adequate reason ror believing it true.

Fractional Statistics of HoioDS £

The most important conscqu.:ncc of !"be analogy between the &actior\al quantum Hall and resonating valcncc bond swa is the prediction of a powerful gauge force between the spinons. In the tiactiooaI quantum Hall c&a:t, this force, which CWSCI the quasiparticles to obey &aaionaI statistics (34), is known (29) to be a natural concomitant to the presence of fraaional charge. A particle carrying decaic charge lie mows dynamicaUy as though it carried with it a lOlenoid containing magnetic 8m WWt, flO NIter tItt Hamiltolli4tt is, proyided that this Hamiltonian can be adiabatically evolve.! into the "ideal'" Hamiltonian without dcsttoying the energy gap. The only thing aiFeaed by a change to the Hamiltonian is the size of the solenoid, or equivalently the size of the quasiparticle. Thus, given that the mechan:sm for quantwn number fractionalization in the magnet is the same as that in the haioaal quantum Hall e&"cct, such forces are MCaSfJrily an amibl.m of spinoos. FurtherID(ft., the fraction of the statistics must be III because the "charge" of the spinon (J5) is III A. FraaiooaI statistics only make sense in two dimcoaioos. It is DOt dear to me what would happen in a threc-dimeusioa spin liquid, assuming one exists, but a good guess is that the forces become 10 IttOIIg that they confine. Let us now consider the dwgcd dcpa oIhcdom (9). Whahcr or DOt ac:itations anaIogous 10 the holoo CIist in the &aa:iooaI quantum Hall dFa:t is not yet dear. The c:xperimcntaI discoYay (41) of the "S/l" state and the subecqucnt coafinnatioa (4Z) 01. ita magnetic characta' do danon..'":r.ttc that charge fraajonaljzarjon an

III""

330 occur in systems possessing bodt dc:ctric and magnetic degrees of freedom. The quantization of the Hall conductancc shows that one of the excitations is a charge-Ill, spin-O quasipanicle (43). I believe that a charge-O, spin-I/2 quasiparticle also exists, but this has not ~:en demonstrated experimentally. The main reason to believe that holons exist, however, is that they make so much sense. In aying to conceptualize a charged excitation of a spin liquid, which we know exists because high Tc superconductors can be doped, one immediately runs into the following problem: In order to placc an additio... ~1 electron (hole) on a site it is rim nec·:ssary to make sure that thc electron (hole) already there has me opposite spin. Ho'\·cver. because the elC\."trOIl already there is ftu\.nl3ting quanmm mechanicallr between thc up and down statcs, this requires that Olle reach in and stop it trom ftul..-tuating. This could hc al:e:ompli!'hcd. fur example, by proiecting the ~und state om,! rh~ S~~ of state:s ',vith a t;iwn electron down. Ho\\,c\'cr, because d>i~ c!'Catl'S a 1J.!'gc disntrbancc in thc "vacuum," it should be more la',or.l'lk cnerg.:tically tu crc.ue a spinon, K.lImcyer .and I (35~ f(lwld this (0 rhe: Clse when we I'(sttd Mth ~t3tC!' using variarional wave: fll!l...ion!> ooml\\'ed trom the' fra.:[ional 'luanrw.l Hall pmblC'lll. So Ie:. u" m.lkc a spinv: .. With the" !-pill of the c:IC\.'t:,'UI\ in que.,rion t!ms

"'c

Sq 10>

PqIO>

cqlO>

Spin Wave

Density Wave

Hole

Fig. 5. In the presence of a holon condenme, the "elementary particlcs" of the: theory. spinons and holons, ".annoc be" isolated. The physically observable panidcs m\L~t therefore consist of pairs of mem. The: th~ pos.~ible pairings may be- thought of as the excitations S"lo>. p"IO), and c~IOi.

defined. it is possible to rc:mo"c the C'!cctron in an unambiguous way. thus crcatm~ a holon. The holon is spinless bccalL~ holding dIe clC'CtrOn down and [hen removing i[ is c:qui\'a!cm [0 holding it up :md then remo\'ing i[. The: particle: creatc:U in [his \':ay ob\'iously oL:\'s fra.crional'itatisti~ bc:cau.~ it is consml(:ted from a hole: and a ~pi'~)fl, Howc\·cr. on: .:al~ argtic nlor: ~cnerall~ Jut the ahs<'rpOon of the sj>ill of the' h..;1: b}' the vacuum C3'.'\ onl\' ha\': or.(.~n"Cd through (he fra"'l.io,,aliurion of the lIpin quantum number, and rh~ musr h:we gJ\'cn rise to :\ Iong-~ foxc.

Superconductivity from Gauge Forces The fractional sta[isrics obeyed by nolons has the capacity to cause superconductivity. Unlike the pairing forces in an ordinary superconductor, which arc by most measures weak and which have no cft'ect unless they are sufficiently strong to overcome Coulomb repulsions between e1ccaons, the gauge force to which fractional statistics corresponds is strong and leads to charge-2 supcrfluidity under very gqtcral circumstanCCS. 'The conclusion that fractional scarisrics causes superconduaivity is based on a n:cent theoretical study by me (44) of a gas of holons obeying 1/2 fractional statistics and described by the: frc:c:-particle H3I1liltnman (1)

where V is a pair potential, nominally a Coulomb repulsion. The starcmmt that an energy eigenstate 'iT of this Hamiltonian obeys v fractional statistics means that it takes the: form (45)

.
"'(Zit ... , z.v) =

v

n Iz) - z,11 -.

cI»(Zh""

!'N)

(.2)

J< ~

fig. 4. H<'Ic:s (top 111ft) and panidcs (tap right) in me &aaiona.I statistics gas may be Ihought of as chargaI vonica. The velocity Iidd of me vortt:Il (lftiddle left) faUa oJf as I/r 31: IarJc cIi.sanca., as ~ lOr a quanaun ofcnndarion (I - 11)4. The size ofme VOlta core u t - [2w(1 - lI)pr"Z, when: p is me dcnsiry of me fluid. The aaion of chc density opc:ntor p.,. on me ground . . (middle right) l1li)' be daought ofcithcr. a~ IOUnd wave or as an aciton (boaDIn 11ft) tixmcd from two YOrUCa ofme opposite sign. The scpanbon of die vortica is ~ to me aciton momentum q and perpcndicuIar to it. The dispcnion rdaIion (bottom rtght) crosses over from linear to Ioprithmic behavior when me vorta separation becomes comparable to t. 28 OCI'OBEB. 1988

where 4» is a Fenni wave function and z = " + iy is the position of a partidc cxprc:s...:d .1.\ a complex number. When written in tmnS of 4», t!le c:quatiOAS of motion bccomc: dlOSC of fermions 1tlO\1ng in the " - y plane and carrying with them a magnetic solenoid containing (1 - v)lrel, of magnetic flux. A Hanrec-Fock solution of these equations (44) reduces the problem to a gas of nonintt:raeting fcrmions moving in a uniform magnetic field of strength hcp B= (1- v)7

(3)

where p denotes the particle density. As illustrated in Fig. 3, such a system possesses an mngy taP in its fmnionic excitation ~ whenever the particle density is an integral mulriple of the quanaty lre/tB, which occurs in this case when (1 - v) -I is an intega'. In the fractional statistics gas, this gap rums out (0 be logarithmically ARllCLBS

SZ9

331 divergent with the sample size and thus dI'cctivdy infinite. The

energy grows logarithmically because adding a particle polarizes the surrounding fluid in a IIDIkX of circulation (1 - v)l. The condition that a gap exist is prccisdy the condition that an integral number of particles be bosons. The Hartrec-Fock solution thus indicates that the ground state of particles obeying v fractional statistics is a superftuid with a Charge-(1 - v)-I order parameter. This charge is 2, as appropriate for a superconductor, for v = III holons. The brokt:n symmetry characteristic of a superfluid is not manife:sdy present in the: Hartree-Fock solution. This is a well-known pathology of variational solutions, which is remedied (46) by h~'bridizing longitudinal collectivc modes inoo the ground state. This collective mode, which is physically tht same: as :0 compressional sound wave, appears formally in the HattrcC-Fod description as -Ul rxci,,,,, (37j funne:d. as iUu.urated in Fig. 4, truro a ho6e in the 'X(upicd Landau 1C\'cI ~nd a hl,lc in the lowest \lIlOQ:upicd one, II: is ph~'sic3I1y similar to the rurdy magnetic colkcti,'c mocJc ~n in t-=i~, 2. II: n\3~' ~ thought of a.- a vortc:x·anumattx" pair ~ a di~l:anu: (IUltiJ)., anJ possessing momentum "'1. The dispersion reution of this at"itat"iun i~ linear at kHlLl wavc/,·ngtit." The: fra'""tjon4l·sutl~~-s tta) will cca.\C to he.l "ircrlful.i ,,"hen tilt:nl:rrp.. rtidc ~U.'1(1'1 :' bc,-"UmCli t(lO SfRJf'@. tor thell ttl( !.xJ:tOr05 muo;( form a Wiper crysul, For C..:M1Iomb inrer:k"UOI\5 this is :bought to oc,-"Ur lor, "I: JcnNti~ (oJ!' ,,'h...h

III;

A2('IrP)'1: ~ 100

(4)

It has been nottd by Peters and Alder (47) maL this expression is \"ely nearly an equWty at w minimwn doping density (31M») required to make LaZCU04 mctaIIic.

Gap Collapse and Confinemt;nt Having identified fractional statistics as a possible cause of superconductivity, one is placed in the awkward ?OSition ofhaving explained too much. The pairing of holons » a n:suIt of &actionaI statistics is inevitable provided that the spinon gap is nonzero. Thus, in a universe containing only hoIons, superfluidity would be lost through thct ..nal fluctuations the order parameter, the transition would look 5()I1'ICthing hke dk A-point of liquid hcliwn. and superconducring runocIing would be imposAbIc.. Since chis is ckarly not the c.uc expc:rimcntally, the dtcory can be axn:ct only if spinons are central to the spectroscopy and thermodynamics of dIcse materials. Let me now argue that this is expcaa1 to be the case. One of the most suiking fcatwa of the hip Tc cxpcrimcntaI phenomenology is how well it fia the BCS theory. It is hard not to be comforted by this, for it indicates &irIy saoogIy that much «the physics of hign-temperature superconductors is the same as that of ordinary superconductors. A momcnt's rd1caion. however. m~ mat this does not teU one very mudl, Most of the impurt.an! propcrtic:s of a superconductor, such as I'hc: Meissner dIi:a. the Josepluo.l effect, and the relation between He and Te. ~ dirccL c:onscquc:nccs of the occurrcnce of spontanClOUS brokm synmcuy. Thus, aperimena which measure all these things are in some sense the same cxpcrimcnt. Similarly. the fact that the ~ Tc and the energy gap measured in tunnding or by opcicaI speaaoscopy are companhie in size and the faa: that He coUapses in a roupty mQu-fidd way as Tc is approached from below merely indicate that the dcstruaion «superconductivity is a gap-dosing transition. ~ is nodUng inconsistent in this. Even in ordinary superconduaon, IhcnnaI ftuauations «the order parameta" wiD destroy supcrmnductivit unIcss something else destroys it first. It is just an aa:idenr that the gaps ordinary superconduetors are so smaU that they

collapse at a relatively low temperatures. Now, given that the holon liquid has the properties I have asaibed to it, the only way to destroy ia superftuidity, other than by the thennal ftuctuation mechanism, is by destroying the fractional statistics. This, however, can only occur if one destroys the spino" gap. Thus the question we need to ask is whc:dter raising the temperature destroys this gap. Before addressing this question it is necessary to comider the physical meaning of the spinon gap in the presence of a holon fluid. An isolated spinon is expected to induce a vortex in the condensate exactly the wayan isolated holon docs, because a spinon is simply a holon with an electron added to it, center. This means that the energy to make an isolated spinon diverges logarithmically with the sample size, and chis means that making an isolated. spinon is impossible, One thcrerore has the sttange situation, similar to that occurring in baryoos, in which the fractionally charged particIcs of the Iheory. the: holons and spinons, cannot be isolated. The frcdy I"urag.ating, and thus spo.troscopically significant, excitations of die sysrC'm ctlflSist of pair( of them. As iUusttatcd in Fig. 5, there 5hould be three of these: J) A ~pin ""a'·c. consisting of two spinons. This is dlC aciution i1h~tr.ttat in fig. 2. 2) .\ ch:\~ C\lrrcnr. a»mistillf; of twu ilolon). My ~t undcntanJing j5 dut thIS should be analogous bl dK Ktion of me d-ns;ty ClIJICntC'l"oo an cllliillary supc;rconduc:tor. ,,'hich is a Iongiru" dan:u eXl'il~(:(JO th.u: Ioscs its id.:nti~· by h),bridi%ing lotn.mgly with

me GoldstOllC mode.

3) An elcctton, consisting of a halon "hole" and a spinon. This is thc excitation created in a runllelin~ experimcm, Each of these panicles should be characterized at low cnc:a-eY b)' a spin and total momentum, as approp~ fOr a tighdy bowld state of two panicles. Because the impossibility of isolating spinons is a property of the hoIon fluid and not of the underlying spin system, it docs not invalidate the concept of a spinon gap. It merely requires that one detect coUapse of the spinon gap through coUapse of the spin-wallt gap which is apcctcd to occur simultaneously.

0'

«

no

«naDa'C

s:

1,0

c;;- 0.8

.1::

C

::>

~

~

0.6

.!! >-

e»

G)

0.4

c

w 0.2

0.0 0

r

4

2 X

8

6

10

r

M

FIg. .. Hypothesized behavior «rnagnI:Dc CKimion s.iO> in a real high Tc supcrmnduaior. N«* the IimiIarity to Fis. 1. The CI1CIJY II r is aJIIIfIInbIc to dtc opcic:aI ....... «the IIII:ifcm:xnapa ordaaI--. The lIP 4. II r is aJmPIIII* to the auper-'JIIduains pp 14.. . SCIBNCB, VOL

~

332 Let us now consider the question of gap collapse. It is known that disorder, and thus pmwnably thcrmal1y excited coUective modes, can continuously collapse the magnctoroton gap in the fra:tional quantum HaD state to zero. This is known both from theoretical CXKlSiderations (38. 48) and from the experimental observation (39, 49) that increasing disorder lowers the activation energy for au continuously to zero and then destroys the fractional quantum Hal! effect entirely. Because of the na~ of self-consistent gap collapse, namely, that thermal excitation of particles across the gap lowers the gap, which aDows even more particles to be excired, which lowers the gap sriU further, and so forth. one can say without knowing any c:k.tails that the tempera~ of gap collapse must be comparable to the gap itself. Thus, for the superconductOrs we would have

(5)

where ~, denotes the spin-wave gap. The \'alue of 30 meV that I escim:lteu for .1s gives a \'3Iue f.'r this ~;uil.' of 3, which is qUite re.lsonable. It is therefore the.:ase that self·consisrent gap collapse of the t:l'lC n.:cessary to de;tro\. surcrtlulJil~ IS expected at a tanpcra· rufC mal i) about :-ighL

on me single-mode approximation. However, it is wdI known (SO) that quanmm fluctuations tend to linearize a mean-field dispersion relation of this kind in the limit that the gap is small. Let w thcrcforc guess that the dispersion relation ncar the minimwn is roughly of the form (6)

where v is an asymptotic spin-wave velocity. This dispersion relation and the density of states to which it corresponds arc shown in Fig. 7. Let us now consider the behavior of the "'electron." Whatever this excitation is, it should have a significant projection onto the srate ~$!O), where 10) denoteS the true ground stale of the system. Thus, to calculate its properties. one could either axnpute the timeordered Green's function Gu(T) = -;(OIT{4u(T')(~(O)}IO) or use the state ,:'!O) 1$ a variational anutt. Un1~ "elecrrons" and "holes" intera(.= \'c:rr anomahus!y, the Green') functions for spin and densin' 8uctuarions. namely, -i(OiI'{Sft.TiSk(O)}lO} and -,(O;T{PI&(T)PIo;(OrHO), with (7)

and Conseqnc:n~es

of Fractional Statistics Pairing

[n "Hier to p:('ceed further it \,,;ii IJe n(,cr"5~al'}' for me to make

,~

Pq

,

=L'II. q.J(k.s

(8)

lu

some Nucated guesses about the precise nature of me excitation spectra in a system of this kind. These "gedanken" calcul300n.s arc a poor substitute for real ones, particularly because most of the significant Ci'..1cstions about these materials arc quantitative, but it is the best any of us can do at present. Undopcd La1Cu04 is known (5) to be antifcrromagnctica1ly ordered along the rHOJ direction in the Cu-O planes. Accordingly, the spin-wave spcctrwn of the magnetic liquid state shouid look som.:thing like Fig. 6, with the gap As occurring at the M point in the BriUouih zone. This gap is physi.:ally analogous to the magnetorotOR minimwn shown !n Fig. 2. It is shown with a parabolic shape because this would be the OUtcol11C "If a variational calculation based

must have large spectral weight at the cnc:rgies of free ~earon hole pairs, 3S th~· do in ordinary superconductors. n.us the pre!'CIlce of a soft spin wave at M implies that there arc also soft electron hole pair excitations with this momcDtum. Let US thcr-forc guess that the electron spectrum looks something like that depicted in Fig. 8, with a small gap 1l1e at the X point. This gap should be sIighdy greater than ~I or equal to it, according to whether the electron hole interaction is attractive or repulsive. The &ct that this gap is direct implies that it would produce a strong signa1 in optical reftcctivity, as is cxpcrimcntally the casco It should also be the gap observed in mODeling. I h3\'e a.~rumed the electron spa:num to ha··e its minimum ar a p(linf in the Brillouin zone because this is the I1lO5t likely ourcome of .1 \°:triJ.t!or:.al cqim.at~ based on the "'a\'e function .:~!O) oc a t.rojc:
t

28 OCTOBIa. 1988

Aa.T1CLES

531

333 minimum depicted in Fig. 2 is negative. & illusttmd in Pig. 8, however, this excitation may also be viewed as a pair of spinons bound together. In the presence of the holon gas, the energy of this excitation, and thus ~, wiD be raised. If the separation of the spinons associa(ed with this excitation is assuf11CCl to be pGll. where p.is the holon density and G is a reciprocal lattice vector, then the vortices in the hoJon fluid arc farther apart than their core size, and we may write, up to an unimportant logarithmic term, (9)

where 11~ is the (possibly negative) value of the gap at zero doping and a is a coefficient of o:der unity. This is precisely the type of doping dependence of the gap proposed by Anderson (3) and found phenomenoJogicaUy by Uemura tt Ill. (54). Assuming 2 value of 10 - 2 A-2 for p and a bare electron mass, one oIxains 40 meV for this increase, which is of the correct order. I would like finally to make a remark about gaplessness. It is a fact that the energy gap or gaps of high-temperature superconduaors arc extremely difficult to measure. It is commonly the case, for examp!e, that a tunnel junction (17) exhibits a smooth transfer -:haraaeristic with a small lImooth bwnp where the ;ap ~t to be. ~ a result. runncling s~"Opists cannot agree on the value of the tunneling gap ro within a factOr of 2. Some c!:ny that there CVOl is it gap. Similar!y. the value of the S3P determined from infrared absorption (16) varies from sample t(, sample and docs not agree \-ery weU with the value: determined from tunneling. It is not yet clear why these: difficulties occur. However. since gapless supcxon· duaors are known to exist and to be caused by magnetic impurities, and since defects in the stn'oure of materials with such low carrier densities are bound to be magnetic. it is quite conceiv:able that high· temperature superconduaors are C:UOIllCaUy gapless. In light of this possibility, it is important to make clear that gap1essnc:ss docs not invalidate the fractional statistics concept. Fractional quantum Hall systems arc always dirty and thus always in some sense "gap1c:ss."

Nevathdcss we know cxperimc:otaUy chat the quantum of Hall conductance, and thus the quasipartide charge. is exactly quantized so long as the dFcct is not destroyed entirely.

Conclusion The purpose of considering the cxperimcntal implications of the equivalence of the fraaionaI quantum Hall efFect and high-temperature superconductivity in this cursory way is not to prove it correct but rather to show that it is not obviously wrong. I am in agreement with Anderson that the mathematical tools required to accurately calculate propertic:s of this state probably do not yet exist. Before making the effort to invent them it is obviously a good idea to find out if the approach makeS sense. The cxistt:ncc of a spin-liquid state and the occurrence of charge fractionaIizari in such a state arc, in my opinion., on firm ground. The ability offraaional statistics or its thlTC-dimc:nsional anaJog to calUC supc:rconduaivi is less clear-cut, but probably right. Whether or DOt such things occur in real hightemperature superconductors is probIematia1. I am persuaded that ther do. but this mnains to be demonstrated

llEPBllENCES AND NOT£S 1.

r. w. Andc:non., ~ 23". 1196 (1987).

Z. _ - . MtIur. Rts. Y. 8, 153 (1973). ill",.",..,. AI,.., J.lL SchricI"c:r and lL A. Bropa, U. (Nanh·~. Amstcnbm, mprat). 4. N. F. Moa, PttIc. R. 5«. ,..... Scr. A 6l, 414 (1919,; B. H. Brandow, AW. ""rs. leS, 651 (1977); I. Yu. romc:randudl.) AIrs. ( " - ) " 356 (I~I). 5. D. Vaknin II III., ""rs. Rn. Un. 58, ZI02 (1987); Y. J. Ucmun filii., ibW. &9, 1045 (1987); J. M. Tnnquada filii., w. 60. 156 ,i988). 6. G. Shirmc fI III., iIIiI. 59, 1613 (1987'1; Y. Endob II III.• AIrs. Rw. B. 37,7443 3. _ - . ill

FMmm,,""iIta

(1988). 7. H. A. Babe, Z. ""rs. 71, lOS (1931). 8. L. D. FadeeY and LA. TalrJlajan, LffI. 85A, 315 (1981). 9. s. A. Kivdson, D. S. Ilokhsar, J. P. SedIna, PfIrs. Rw. B. 35,8865 (1987). 10. A good ~ ol* apaimcnaIliIaauR be bind T. H. GcbaBe and J. K. HuIm, ~ 239, 367 (1988). 11. B. Badog, A. P. Ramin:z, lL J. Can, lL B. VIII Doftr, E. A.1t.icanan, Rn. B 35, 5340 (1987). 12. J. Ormaan" III., iIIiJ. 36, 129 (1987); w. II III.,"'., p. 733. 13. L F. Mmbcisa, Pftrs. Rw. LffI. 58, 1028 (1987); J. Yu, A. J. FIUIIWl, J.·H. XU, ibW., p.l035. 14. M. C.urvitdllllld A. T. Fiery, ibW. 59, 1337 (1987). IS. S. E. lndat.ca filii., iWd. 60, 1178 (1988). 16. z. ScbIcainpr,lL T. CoIIinI, D. L lCaiIcr, F. HaIabq, W. 59, 1958 ('987).

""rs.

mar

m

""rs.

r.-u

1.0

J!l

17.!::,~~~=:..;..~

0.5

unrdiabIc. Sec M. D. Kilt" III.• PIrp. Rw. 835.8850 (1987).

~:: ~·I~-;:~'r~s.!.1~ ffrrsim 140, 322 (1987).

'c

::> ~

g

I

0.0

>. ~

Q)

c: W

-0.5

20. w. J. KoaIcr II III., PfIrs. Rw. 8 35, 7133 (1987); G. AqIpIi" III., iWd., p. 7129. ll. B.·D. Oh II III., i6i-f. 3i, 7861 (1988). ll. J. S. Tui, Y. Kubo, J. Tabuchi, Pftp. Rw. LffI. II. 1919 (1987};},...). AppI. ""rs· 26,C701 (1987};J. Nianqcr .. III.• Z. PfIrs. 869,1 (1987}; D.&Icft~III., l:iInpIIrs. LffI. 3, 1237 (1987). 23. B. KoiIlc:r and L M. FaIicov,]. ""rs. C 7,199 (1974). 24. K. Tenbn, A. lL Williams, T. Oguchi, J. Kubler, AIrs. Rw. LtIt. 52, 1830 (1984). 25. E. H. Ucb IIIId D. C. Mattis, ~ Pfrt*s ill 0. DIiINIuNn: s-./r SoIII6It MOIItIs of".,.,." lWrida (Aadcmic Pres, New Ytd, 1966). Z6. I. DzyaloIhinsIW, A. PoIyUov, P. w~ Pftyr. LIII. 127, 112 (1988). 27. F. Wdc::zdt and A. Zec, ""rs. Rw. LIII. 51, l250 (1913),_ rdaaIca thaan. Addiban olal1lpOlasicalllCml is bMiI lOr IIIIIdl recall SCIrict wort. Sec A. M. p..1yakov (MM. AIrs. Un. A3, 325 (1988)J aDd P. 8. WIC~ (AIrs. Rw. Lm. 60,821 (1988)]. 28. lL B. LaupIin, Pftp. Rw. LffI. 50, 139S (1913). 29. _ - . m710r ~ HIIII E#rt,lL F PnIIF _ S. M. Girvin, Eds. (Sprinp,

*

-1.0 ~----~~~~

o

r

2

x

4

____~____~____-L~

6 M

8

10

r

fig. 8. Hypothesized behavior of "c:Jcctronic" aciwions ~) and c..lO). The gap ~ is the one measured in runneling. The energy at r is axnparablc to the Fermi energy of a gas of dcarons at the hoIon density. Bcausc the Vp is din:ct, it may be obscrwd optically.

SP-

~.1~ ':,?3:;O1;...

30. 51,605 (1983). 31. ri~l~ Pftp. iff'. B lS, 211S (1912); lL I. I.IupIin. W. 23, 5632 32. G. S. Bocbinpr, A. M. Clang. H. L ~, D. C. TIIIi, W. 31, 4168 (1985). Alp. Rw. u.. 53, 7ll (1984).

:!:~. ~lLiWd~~~·

('i:t'

35. V. ~ aDd lL B. Laughlin, W. &9, lOPS (1987). 36. S. M. GUm, A. H. MacDonald, P. M.l'IIam., W. M, 581 (1985). SCIBNCB, VOL 24-2

334 31. YIL A. Bycbkov, S. V.1ordInIIdi, G. M. ~", .... Zh. Sap. T_. Rz. 33, 151 (1981) UETP lAo 33, loU (1981)]; c. ICaIIin aad B. L HaIpcrin, PIIys. Rw. B 30. 5655 (19M); IL B. Laughlin. PIIysitlJ 1268, 154 (1985). 38. R. B. Laughlin tf /I'., PIIys. Rw. B 31, 1311 (1985). 39. I. Kukuahkin, V. Timofcev. K. von lCJjaing, K. PIoog. FcsddIrprrpru6ltrM (Ad".

(19116); E. L PaIIDdt IIICI D. M. CcpcrIcy. PIrp. Rw. B 36, 8M3 (1987).
Solid S,,* PIIys.) lB. 11 (1988\. 40. K. B. Lyon&, P. A. FIcury. L. F. Schncancycr. J. V. Waszaak, PIIys. Rw. lAo 60. 731 (1988). 41. R. Willet «." .• i~. 59. 1776 (1987). 41. J. Eisenstein tf ."•• ihid. 61.997 (1988). 43. F. D. M. Haldane and E. H. Rc:zayi. i~. 60.956(1988). 44. IL B. Laughlin. ibid.• p. 1677. 45. F. WiIc:zck, iIIi4. "9.957 (1981); D. P. Arovu.IL SchricII'u. F. Wilczek, A. Zce, NMcI, PIIys. B lSI. 11i' (1985). -'6. E. Fcenbc:rg, n-,. oJo-- R,,;.u (Aadanic I'laJ, New York, 1969). p. 107. 47. O. PClI:n and B. Alder, in C"""""" S~ SfNMa ill CDfIIIttuol "1411" PIIylia: Rttnll DwtIoprtIffIu. D. P. Landau and H. B. SdlGdI:r. EdI. (Sprinp. Berlin, Hctdr-ibcrg, in pn:.csl; D. M. CepcrIcy a.'lCf 1::. L. Pollock, PIll'" RftI. L:tr. 56,351

SO.

c.1Ciftd, QuMfum n-y oJSoII4s (Wdcy, New York, 1963), p. 58.

51. 51. 53. 54. 55.

G. 1Codiar, Pfrys. Rw. B 37. 3664 (1988). P. W. ~ and Z. Zou, PIIys. R,... UtI. 60, 137 (1988). J. IL Schric:II'cr. '17Itory oJ~i..uy (Benjamin, New York, 1983). p. 116. Y. J. Ucmun d "'" PIIy•. Rw. B 38.909 (1988). I ~..udiilly acIcnowIaIge numerous helpful discussions wi!h S. ~, J. ScduIa, V. KaImeya, C. Hanna, L. SUIIkind, A. L. FctIa'. P. W. Andcnon, F. Wilczek, B. I. HaIpcrin. J. IL SduiefIU. T. H. GebaIIc, M. R. BcuIcy. and A. ICapW1nik.. 'Ibis wad; _ supporud primari1y by !he National ScicnR Poundatioa UDder DMR·85·10061 and by !he NSP·MlU. pmgnm !hr. die: CaIa:r b- Materials Raean:h ar Sanford Universiq·. Additional support was pcoI'idcd by !he U.S. DqIartmcnt of ~. dvough !he LaWmKe Uvcrmon: NItioaal Labomory IDda contnCt W·i405·Eng·48.

1606 (1985).

anne

335 VOLUME 60, NUMBER

2S

PHYSICAL REVIEW LETTERS

20 JUNE 1988

Superconducting Ground State of Noninteracting Particles Obeying Fractional Statistics R. B. Laughlin DeptJrllllenl of Ph)·sics. Sianford Unit'ersil),. Sianford. California 94305. and Unil'ersily of California. Lawrence Lil'ermore Nalional Laboralor)" Litoermore. California 94550 (Received 28 January 1988)

In a previous pap.:r, Kalmeyer and Laughlin argued that the elementary excitations of the original Anderson resonating-valence-bond model might obey fractional statistics. In this paper, it is shown that an ideal gas of such parlicle~ is a new kind of sUPt"rconductor. P:\CS numbers. 7-1.65.+n.

(l~ ~O.-d.1J7."O-",. :~.IO

Jm

In a recent Letter, I Kahnc:yer and I proposed that the ground state of the frustrated Heisenberg anliferromagn(\ in two diml."nsions and the ir;)c~i~r.31 qu.:rnlum Hal! Silt.: tor ht)~ons might b.: th( .. :>;lPle., .. in ,he sense: that ttle two S\'SlemS could be adlanaricalh' ev,'tvcd into one another ~ithQut c!,l}!;<;ing a phase ~o:J~da~' Whether or nOI (hi~ is the case is not presently clear. Indeed, the existence of a spin-liquid )latc of on)' spin- t antiferromagnet in two dimensions has not been demonstrated. Howeyer, the case for a phase boundary's not being crossed is sufficiently strong that it is app:ooriate to ask what the consequences would be if this occurred. Adiabatic evolution is a particularly useful concept in the study of fractional quantum Hadl "maUer." So long as the energy gap remains intact, the "charge" of its fractionally chargeJ excitations remains exact and the concomitant long-range forces between them, their fractiooal statistics. remain operative. This is why the fractional quantum Hall effect is so stable and reproducible. The ~r­ siste,lce of the gap under evolution of the fractional quantum Hall problem into the magnet problem would allow us to make exact statements about the magne~ without knowing allythillg about its Hamiltonian. In particular, the excitation spectrum of the magnet would be almost identical to that proposed by Kivelsoa. Rolthsar, and Sethna, 2 and completely within the spirit of the Anderson resonating-valence-bond idea, 1.4 except for one crucial detail: Both the chargeless spin- t excitations, the "spinons." and the charged spinless excilations, the ·'holons," would obey t fractional st:,~istl\"s, ~.6 The purpose of this Letter is to point out lhat tbis overlooked property may well account for high-temperature superconductivity. Kalmeyer and I found the mapetic analoa of the charge- t quasiparticle of the fractiooal quantum Hall effect to be a spin- t excitation, wdI described qualitatively a a spin~wn electron on lite J surrounded by an otherwise featureless spin liquid. 'Ibis particle is our version of the "spinon_" Like the quasiparticle of the fractional quantum Hall state. it carries a "charge." that is. its spin, that is in a deep and fuadameatal sease fractional. In the limit that the antifenomagnetic interactions are turned ~ff, the excitation spectrum of the magnet is

purely bosonic. Spin- t particles occur because these ··elementary" excitations arc fracti~naliled: Half the !>')SO!l is ciepolSited in toe ~ample interior and half at th: boundary. It was first pointed out by Halperin 6 thl:. in the fractior.al quantum Hall effect. the fractionalization of tl·e electron charge ~ inlll the quasiparticle charge j e causes the quasiparticle to obey t fractional statics. That is, each quasiparticle acts as though it were a boson carrying a magnetic solenoid containing magnetic ftux t xhc/t. This fact, deduced by Halperin from the experimentally observed fractional quantum Hall hierarchical states, was later shown by me 1 to follow from the analytic properties of the quasiparticle wave functions. It arises physically be-:ause the states available to the multiquasiparticle system must be enumerated differently from those available to fermions or bosons. In other words. it comes from counting. Now, it is clea[ by inspection th&~ tne preferred nature of this representation does not care about the existence of a lattice. Thus the validity of our identification clearly predicts that spinons obey t statistics. Let us now imagine dopi", this lattice witl! boles. The most natural way to do this, in my opinion, is first to make a spinon, thus fixing the spin on sit~ j,.and then remove the electron possessing that spin. It is necessary to make the spinon first because an electron cannot be removed before its spin state is known. If one simply rii'S an "up" electron from site j, one tacitly projects the ground state onte the set of states with the jth spin up, thus creating an excitation with spin I. This may be lhoug;u of as a pair of 5pinons in close pro.(imity. Unless the interaction between spinons is attractive and suffici~ntly large (Kalmeyer aad I found it to be repulsive I), to make this "spin wave" will be more expensive energetically than to make an isolated spinon. Given that this occurs, the resulting spinless particle, the ""bolon," should also exhibit t fractional statistics because it is a composite of a spinon and a fermion. Assume now that we have a ga of such boIons obeying fractional statistics. What are its properties expected tc. be? This qUCltion was addressed to some extent by Arovas et aI., • who comPuted the second virial coefficient of an ideal gas of particles obeying fractional statistics as

C 1988 The American Physical Society

2677

336 VOLUME 60, NUMBER

PHYSICAL REVIEW LETTERS

2S

a function of the fraction v. Not surprisingly, they found a smooth interPolation between the case of fermions, which acts like a classical gas with repulsive interactions, and that of bosons, which acts like a classical gas with allraclit'e interactions. Thus, if we insist on thinking of these particles as fermions, we must conclude that there is an enormous attractive force between them. This is also evident when one considers the low-temperature rroperties. Fermions at density p have a large degeneracy p!"essure. and thus a large internal energy. while bosons have neither. Since fraclional-statistics particles are in between, they have, l'is-iI-vis fermlolls. attrac'ive forces comparal>le in scale to the Fermi energy. It is also important that spin less particles obeying f ract ional stali<;tics car.not undergo Bose condensation. T'l\'y arc not bosolls. tlowever. if the fraction is !. t"'cn pair.1 of partidcs are bosons. The,.: is thi!refore ~ood reJson to SUS~Cl that a gas of ra.r~iclc:s obeyi:lS I statistics mighL aCluJliy he a sUI'erconduct':', \\::h a chargc-2 order parameter. Let us :nvestigate this possibility by considering a gas of fractional-statistics particles described by the free-particle Hamiltonian N

~

1I-"E~.

(I)

J 2m

Any eigenstate of this Hamiltonian may be written in the manner

-[n (,ZJ-ZIJ,:jCZ>(zI, ... J
Z,-Zk

,IN),

"

where Z j denotes the position of the jth particle in the x-y plane expressed as a complex number, v - t, and ~ is a Fermi wave function. This is the singular gauge transformation first discussed by Wilczek. S If we have an eigenstate'" satisfying 1(,.,-£., then 4> satisfi,..s

with (7)

in units of the equivalent cyclotron rrequency h{J)c - 21Cv(A 21m )P. where n. denotes the projector onto the 11th landau level, and a is a regulation parameter. cft'ectively the inverse of the sample radius. Since 1( HF preserves Landau-level index. the state we guessed is a true variational minimum. Note. however, the logarithmic divergence in the Lagrange-multiplier spectrum, im2678

20 JUNE 1988

11 ' - E where JY

11' -

I

~ 2m 'Pj + Aj '2,

0)

and

Thus, ir. the Fermi representation, each particle appears to carry a magnetic solenoid with it as it moves around in the sample. The vector po!ential felt by a particle is then the ~um of the v~tor potentials generated by all the other PJrtides. Because particles obeying t stalistics behave like fermions. in the sense that they possess degeneracy pressure. let us att~mplt·) ~lve this prob;(m in the Hartrcc-Fock appn)ximaftoo: We make a variational wa\'e function thaI is a single Slater determinant con;tructed of orbital:, (lj{z) and minimize the e'
where 1( HF is the first "ariation of (1(') and Ai is a Lagrange multiplier. The latter t:3~ the physical sense of a partial derivati~e of the total energy with respect to occupanc~ of the jth orbital. Since, in the mean-field sense. each particle mu~t see a uniform density of magnetic solenoids ca;rying flux vhc I e, it is reasonable to guess the solution to be Landau levels, with the magnetic length ao related to the particle density p by al - (2Irvp) -I. Self-consistency is achieved when the lowest Ilv Laneau levels are filled. Thus, the fractioll~ v -I. t . t , ... are special ':'a5CS in which a gap opens up In the fermionic spectrum. Let us now test these equatiutlS in a case for which we know the answer, namely v-I, the noninteracting Bose gas. If the variational procedt!re describes this limit correctly, there is good reason to trust its predictions for v - t. Evaluating the self-consistent field with one Landau level filled, I obtain

plying that the cos~ to inject either a "particle" or an "antiparticle" is arbitrarily large. This is absolutely the co~ !'esult. The noninteracting Bose gas has no -Iowlying fermionic excitations. The fact that these divergences are logarithmic suggests that the relevant excitations are actually quantum vortices. nat this is. in fact, the case may be seen by our imagining an extra particle to be placed at the origin and calculating the expected current density (J(,». 1be current-denltity operator may be written J(r) -m -1(p+Aoacs+4A), where ~ is the vector potential in the absenCe of the extra partICle

337 VOLUME 60, NUMBER

25

PHYSICAL REVIEW LETTERS

20 JUNE 1988

c1e repulsion and the presence in the "unperturbed" Bose and !J.A is the vector potential generated by a solenoid at gas of a collective mooe dispersing quadratically with the the origin. Since (p+ Aold) -0, the current density must mass of the bare particles. In the present case, it is easy just be the particle density at r times !J.A, or a vortex of to see that the variational solution possesses a collective magnetic strength he/ t. mode that disperses quadratically. Since (1f')/N is proThe expected energy of the ground state state is N /4 portional to the particle density, the pressure is constant, in these units. This is considerably higher than the and thus the bulk modulus is zero. It is a straightforcorre~t answer or lero. This discrepancy is due to the ward matter to calculate the mass of this mode by the fa~t that tile wave runction is forced by its construction magne:toexciton procedure of Kallin and Halperin. 10 My to go to zero when th~ particle:;. come together. It is thus preliminary results give a value of approximately t the m~)re appropriate ror the description of real helium than nonintcr..acting hosons. It should also be noted that this bare mass. ~he precise value of this mass 'is not so imhcha\"ior is actu
+

L

N~2

n+ [

L fnl (_I)k [ I-k Lk 1.--1- 1+_3_ _ _1_ 1 k-llkJ 4 I 4k S(n+l) S(n+2) nil' II

Thus, we again have a true variational solutiof1 with vortexlike fermionic excitations. Repeating the arguments for v -I, I find that the ftux quantum to which the vortices correspond is he/2e, exactly as expected of a charge-2 superftuid. Once again, a soft collective mode will mix ir.IO the ground state to break the symmetry when repulsive interactions are introduced. Thus, the ground state is a superftuid very similar to liquid helium except that the charge of its order parameter is 2. While considerable work needs to be done to quantify this picture, some of its implications may be se:n at a glance. By far the most important is that a normalmetal state, in the ~nse of Fermi-liquid theory, does not exis~, just as Anderson 4 suggested. A corollary is that the occurrence of superconductivity does n(\t have any· thing to do with self-consistent opening of an energy gap in the tunneling spectrum, as occurs in the DeS theory. Indeed, I find th31 tunneling cannot even be understood outside the context of the creation of spinons by the tunneling event. It should be noted that this is also consistent with Anderson's views. II A critical prediction is that an energy gap must occur in the spin-wave spectrum, the spin analog of the: collective mode t2 of the fractional quantum Hall state. This is because the presence or absence of this gap is precisely the difference between the disordered and ordered states. In summary, it l:; possible that high-Tc superconductivity can be accounted for by the following simple idea: The force mediated by the spins of the Mott insulator is

I-I

(S)

not an attractive potential, but rather an attractive utepotential. I gratefdly acknowledge numerous helpful conversations with S. Kivelsoo, J. Sethna, \'. Kalmeyer, L. Susskind, A. L. Fetter, p, W. AnderS
t~r

IV. Kalmcycr and R. B. Laughlin, Phys. Rr.v. Lett. 59, 2095

(1987). 2S. Kivclson, D. Rokhsar, and J. Sethna, Phys. Rev. B 35, 8865 (1987). lp. w. Anderson, Mater. Res. Bull. I, IS) (197). 4p. W. Anderson, Science 136, 1196 (1987); P. W. Anderson, G. Bukcran, Z. Zou, and T. Hsu, Phys. Rev. Lett. 58,

2790 (1987). SF. Wilczek, Phys. Rev. Lett. 4', 957 (1982); F. Wilczek and A. Zce, Phys. Rev. Lett. 51, 2250 (1983). 6B. I. Halperin, Phys. Rev. Lett. 51, 1583 (1984). 7R. B. Laughlin, in The QUQntum Hall Effect, edited by

2679

338 VOLUME 60, NUMBER 25

PHYSICAL REVIEW LETTERS

R. E. Prange and S. M. Girvin (Springer-Verlag, New York, 1987), p. 233. 10. P. Arovas, R. Schrielrer, F. Wilczek, and A. Zee, Nucl. Phys. B15IIFSIJI, 117 (1985). 9N. N. Bogoliubov, J. Phys. (Moscow) 11,23 (947); a good discussion of this may be found in A. L. Fetter and J. D.

2680

20 JUNE 1988

Walecka, Quantum Theory of Many-Particle Systems (McGraw-Hili, New York, 1971), p. 313. 10C. Kallin and B. I. Halperin, Phys. Rev. B 30, S6SS (984). lip. W. Anderson, private communication. I2S. M. Girvin, A. H. MacDonald, and P. M. Platzman, Phys. Rev. Lett. 54, S81 (J 985).

339 VOLUME 39, NUMBER 13

PHYSICAL REVIEW B

I MAY 1989

RandoIII-pbase approxbaatlon Ia the fnetloaal-statisdcs gas A. L. Fetter and C. B. Hanna Department of Physics. Stanford Un/uersity. Stanford, California 94305

R. B. Laughlin Department of Physics. Stanford UniL-ersity. Stanford. Call/ornla 94305 and Unil'ersity of California. Lawrence Lil.'f!rnlore National Laboratory, P.O. Box 808, Lll.'f!rmore. (Aliferma 94550 (Received 24 October 1988; revised manuscript received 13 March 1989)

The random-phase approximation for a gas of particles obeying t fractional statistics. in the contexi of Feynman perturbation theory performed in the fermion representation, is shown to yield II gauge-invariant Meissner effect with full screening in the ground state, a coherence length comparable ""jth the in interpartide spacing. and a linearly dispersing undamped collective mode.

It was recerltly rroposed by one of us I that tlte chrre carriers in bish-tempcraturc supc;-conductors might obey v - t fractional statistics.. Z and that this mil'ht be the cat..se of the c~argc:-2 superftuidity. In this paper, we strengthen this point of view by explicitly calculating the lii.ear response of such a system to an applied external electromagnetic potential. The key itep in this calculation is the use of random-phase approximation (RPA) to account for lite long-range gauge potentials associated with the fractional statistics. The resulting response function exhibits II Meissner effect and also closes the gap in the unperturbed collcctive-mode spectrum, yielding a linear spectrum in the long-wavelength limit. This latter effect is the inverse of the "plasmonization" of low-lying collective modes in an electron gas. These results imply that the quantum-mechanical ground stale implicit in the random-phase approximation is a true supcrftuid, and in particular exhibits brok:n symmetry. In a first-quantizcd fermion representation, the manybody Hamiltonian takes the form 11

1 -1:1pj+Aj(rj)! 2. j 2m

length a-(h/B)112 and cyel:>tf.:.n frequency OJ,.-6/~. We usc this mean tield to defict. an UI:pertUrbed one-body

Hamiltonian

12 , 'No -~...!...I ~ 2 pj+A(rj) m

j

the eigenfunctions , jll (r ) and eigenvaluCi Bjll - (n + t )1& QJ" of which are those associated with the Landau levels in the field B.3 With this definition of 110. the analysis becomes an expansion in the perturbation Hamiltonian

11,-11-110 -I:.L[(Pj+Aj ). (A j -Aj)+ t IAj-Aj 121. j m

(5)

Note that the interaction:. implicit in 11, couple to the particles through the mean-field density and curren:density operators, defined by }0(r)-I:8(r-rj),

(I)

(6)

j

and

where r denotes a two-dimensional vcctor in the x-y plane and where ~ zxrj'

AJ(rj) -1&(1- v) ~

4_j

The physical density operator Jo is t.he same as jo, but tht physical current density

-1-12 • rjlr.

Here v characterizes the specific form of the fractional statistics: " -0 corresPQnds to a fermion representation of noninteracting bosons and v - t is the case of current interest. The syste!'!1 may be thought of physically as spinless fermions interacting throu.h long-range magnetic vector potentials, including three-bodr contributions ass0ciated with the terms proportional to LAj 12. We first consider the mean field A generated by the average density p of the particles. Replacing the sum in Eq. (2) by an integral. we find A(r)-p~l&(l-v)(zxr)-tBxr.

J(r)-I: j

(8)

differs from j(r) by an internal diamagnetic contribution. The problem of interest is the linear response to an external electromagnetic field, described by a potential A;Io'(r,t), where p runs over 0, x, and y for the time and space components. The perturbation Hamiltonian associated with this field is l!J.1I'J(r,t>

(3)

Here B - 2~ph (I - v)i is an equivalent uniform mean magnetic field that defines the corresponding magnetic

t (pj+Aj(rj),8(r-rj)} ,

-Ar'(r,t)Jo(r,t»dr.

(9)

The linear response has two contributions,4.s a diamagnetic part' proportional to the density and a paramagnetic 9679

340 A. L FErrEll. C. B. HANNA, AND R. B. LAUGHLIN

9680

part proportional to the retarded correlation function of Ip and Iv: t\pv O,2) - -/(LipO )';v(2)])9(tl-t2).

(10)

In this expression, the angular brackets denote an average in the exact ground state, the caret denotes Heisenberg representation, and I denotes a space-time point rlt I. The linear response in Fourier space, defined by

Il'

4,r(Jp(q,CI)) - - K"v(q,CI)A: (q,CI)

K",,-pB... O -cS"o)+A"v(q,CI). The first step in obtaining all approximate expression for .1 is to introduce the unperturbed correlation function

-;(T(j,,(l)jv(2»)}c,

(14)

using the usual Feynman rules of field theory. 6 The interaction HaMiltonian contains long-range potentials similar to those familiar from the electron gas. As in tha! C8U, the leading contributions at long wavelengths arise from the repeated ""bubble" diagrams (the RPA) in which the samo momentum transfer q appears on each interaction !ine. One of these lsee Fig. 1(a» arises from the ' (p+ A)' A term in 11 I. S~ this part of the interaction involves aU three components of A,., it couples the various components of 2>f". For example, the fim-order contribution to nOll involves both 2>8, and (we take q along i). The t:lree-body interactions lead to three RPA-like diagrams (Figs. • (b)-I (d», but only the first of these is divergent. The significant RPA-Iike contributions reduce to an expression for 2>" v of tbe fom.

n:o

1) -no+nOcyn,

(1-~)2Jr q

ated with 8J - J - j. In the long-wavelength limit, we obtain

where

i]

0 0 [000

tI- (I - v)2" 0 0 0 . q

\

(18)

It"

The final linear-response hrn...1 K/~ follows by combining Eqs. (12) anc! (1"). Given these expressions, it remains only to detennine the unperturbed matrix 1)0(q,Q). Specializing to the case of v and taking length and energy units for which II and hCl)r are both unity, we obtain

t.

q 21:o

I

iqI.1

where

-Iq 0 0

and %-q2/2. We nute that the resulting Kf.,PA. is manifestly gauge invariant" because the three-compc~ent vector with elements (-m,q,O) is an eigenvector with zero eigenvalue. 1bc Meissner effect foUows from the static limit of the response function K(q) -K"(q,CI)-O). A direct expansion for q - 0 yields the relation

i q 2+0(q"»).

The fiDal step in the caic'llat;on is t~ correct the matrix is defined in terms of the meln-field currents j, by adding the "internal" diamagnetic contribution associ-

qCl)l:Q

nO(q,Q) - ; qCl)1:o 0)21:0-1

[! ~ ~].

KRPA.(q) -pH -

(eI)

1), which

(IS)

where CV is the 3 x 3 Hermitian potential matrix

CV-

Ie:)

FlG. 1. First-order diagrams relevant ror RPA description. \a) Two-bt'Ciy te""" (b) three-body term that mUllt be retained, and (c:) and (d) tItRe-body terms that are oegJigible in compari&Oll witn that in (b).

(Il)

where the subscript 0 denotes an expectathn -,alue in tbe unperturbed ground state. We then perform Eo perturb 11tic.n expansion fQr the mean-field correlation function

1)"v(t,2) - -;(Tlj,.(1 )j,,(2»),

(b)

(l1)

is given specifically in terms of A by

1)2. (1,2) -

(8)

(2 I)

Here, the 1 is the diamagnetic contributiOn, and the remainder arises from the paramagnetic part. As in the usual BCS theory," the paramagnetic contribution van-

iCl)I.1

isha for long v.·avelengths, leaving a full Meissner effect. with atl the particles contributing to the effective superconducting density. Comparing the form of Eq. (21) with the corresponding result for the phenomenological Pi~ pard1kemel

KI'-pl1_(q~O)2/S+0(q"»,

(22)

we obtain a ZCIO-temperature cOherence length ~ of (1 S/8) 112a. which is comparable with the interparticle spacing. The collective modes usociated with density ftuctuations occur at the poles of the response function _ In

341 RANDOM-PHASE APPROXIMAll0N IN ...

the present RPA. these arise from the zeros of the determinant, since the sinplarities at 0)-" cancel identically. Expanding for small q and 0), we obtain

where the sound speed v, is J2 in units of O)eG. This value agrees with that calculated from the total energy per particle 1£ - h 01(' of the unperturbed .ystem with two filled Landau levels, in the manner

The correspondiDg Hartrce-Fock energy is sa.aller by a factor 29/32. which implies that the Hartree-Fack. sound speed is sere smaUer tha" this value. Note that the pok in 400 is !lJarp. with no background Cl')nttlluum of tbe IOrt found in a Fermi liquid. Not~ also that the structure facto.. S(q) vanisbes linearly foc smaD q, as in the calC both in a Fenni liquid ilnd a Bose:: !''Jperftuid, ill cuntrast to the: quadratic behavior of the unperturbed structure factor. This diffcrcoc:e reflects the presence of superlluid density fiuctuatlons in the ground state iulplicit in the RP A. We note finally that the RPA Hall conductance, given

IR. B. Lau"hlin, Phys. Rev. Lett. 60, 2677 (988). e, 957 (1982); B. I. Halperin, Ibid. 52, 1583 (1983). JR. B. Laushlin, in The QuQlltum Hall Effect. edited by R. E. Prange and S. M. Girvin (Sprinpr-Verlas, New Yo:k, 1987), p.233. ~J. R. Schrielfer, Theory of SupercoNlue.I"lty (Benjamin,

2F. Wilczek, Phys. Rev. Lett.

9681

at small q and .. by

I K RPA

RPA (S1l,

-

10)

Il,

-

4w

(v"q )2 0)2 _ (v"q)2 '

(

)

25

is almost certainly an artifact of the calculation, attributable to neglect of nonsinplar diagrams. A Hall conductance of this form also results for the case of v -0, which is a Bose gas. The PrelCllt paper has shown how Fcynman diagrams for the coupled density and current corrdation fcnctions of the fractional-statistics glS can be summed to yield physically seDlible results, and that these include the Meissner effect and presence of a sharp Goldstone mode. The same techniques should prove valuable in considering other ISpects of the problem, .uch as the dloct interparticle repulaions. This me.!Cb ba... been aupporttd in pa..'1 by the National Scieace Foundatioa under Granta No. DMR 84-18855 and No. DMR-8S-I0062 and by the Nationaf Scicn~ Fouodati~ Materiali RClClrch Laboratories Prusram through the Center for Materials RCIeIICb at Staniord. Additional support wu provicw.l by the U.s. Department of Eaeqy through the Lawrence LivermoR National Laboratory under Contract No. W -7405-&g-48.

RcadiDJ, MA, 1964), pp. 203-213. 5A. L Fetter and J. D. Walecka, QIIQlltum Theory of Mtllly· Particle SY8.ewu (McGraw-Hill, New Yark, 1971), Sec. 52. 6Fetter and Walecb, Ref. S, Sees. 7-9 and 12-16. 7Felter and Walecb, Ref. 5, Sec. 30. 'felter and W.lecb. Ref. 5, Sec. 49.

342 International Journal of Modern Physics 8 Vol. 3, No.7 (1989) 1001-1067 C World Scientific Publishing Company

ON ANYON SUPERCONDUCTIVITY VI-HONG CHEN FRANK WILCZEK· EDWARD WITTEN~ School of Natural Sci~nc~s, Institute for Adva~~d Study, Olden Lan2, PrinCelofl, N108540. USA

and

Physics

D~partm~nt,

BERTRAND I. HALPERIN* Harvard Un;v~rsity.. Cambridg~, MA 02 J38, USA Receive~

29 May 1989

We investigate the statistical mechanics of a gas of fractional statistics particles in 2 + I dimensions. In the case of statistics very close to Ferm~ statistics (statistical parameter (J =-= n(1 - I/n), for lalge n). the effect of the statistics is a weak attractiol1. Building upon earlier RPA calculation of Fetter, Hanna, and Laughlin for the case n - 2, we argue that for large n perturbation theory is reliable and exhibits superfluidity (or superconductivity after coupling to electromagnetism). We attempt to describe the order puameter for this superconducting phase in terms of "spontaneous breaking ofcommutat\vity of translations" as opposed to the usual pairing order parameters. The vortices of the superconducting anyon eas are charged, and superconducting order parameters of the usual type vanish. We investigate the characteristic P dnd T violating phenomenology.

1. Introduction

Since the early days of quantum mechanics it has been appreciated that the behavior of assemblies of identical particles is influenced not only by conventional "forces" but also by the particle statistics. Indeed, the ideal Bose and Fermi gases are the points of departure for most studies of condensed matter at low temperature. It has been extremely useful to have these simple paradigms; for example such ubiquitous concepts as the Fermi sutface and Bose condensation were abstracted from their study. While Bose and Fermi statistics are the only logical possibilities in three spatial dimensions (and the whole notion of quantum statistics degenerates in one • Research supported in part by DOE contract DE-AC02-76ER02220. Research supported in part by NSF Grant 86-20266 and NSF Waterman Grant 88-17521. Research supported in part by NSF Grant DMR 88-17291.

t

*

1001

34J 1002

Y.-H. Chen el al.

spatial dimension), in two dimensions the situation is more interesting. In two spatial dimensions, the possibilities for quantum statistics are not limited to bosons and fermions, but rather allow continuous interpolation between these extremes. The quantum statistics is defined by the phase of the amplitude associated with slow motion of distance particles around one another. If the phase e'o on interchanging the particles is + 1 the particles are bosons, if it is - 1 the particles are fermions; but other values of the phase are allowed, and give us generically anyons. It is a very attractive probh~m; to figure out the behavior of these new quantum ideal gases. The high temperature, low density behavior was addressed several years ago in a paper by Arovas, Schridfer, Wilczek, anc Zee.: They calculated, in particular. the value of the second virial coefficip.nt. A simple answer was found, that interpolates continuously between bosons and ferI'!1ions. While this result was significant as a check of the consistency of the whole circle of ideas, and as an exercise for sharpening technique, it hardly addressed the central questions regarding the new quantum idc=al gases. The most important effects of quantum statistics, of course, occur only at low temperatures or high density. The existence of a cusp in the vi rial coefficient at Bose statistics was one of several indications that the behavior of anyon gases at low temperatures would be interesting and probably far from smooth. However, it has proved quite difficult to extend the calculations :itarti~~ from the high-temperature end, and since the problem :;eemed both esoteric and inaccessible it was largely abandoned. Recently, however, there has been a sharp increase in the interest in this problem - for reasons we shall review shortly - and important progress, especially through the work of Laughlin,2,4 Kalmeyer and Laughlin, 3 and Fetter, Hanna, and LaughlinS on high-temperature superconductivity. In this paper we report further progress in understanding the behavior of the anyon gases with statistics parametrized by 8

=

1£(1 - lin),

(1.1 )

at zeru temperature. Here n is a positive integer. n = 1 corresponds to bosons, while we approach fermions at large n. In accord with Laughlin and coworkers, we shall argue that these gases generically form superfiuids, and become superconductors if the anyons are electrically charged. The mechanism ofsuperfiuidity seems rathe: different from conventional pairing, and seems to lie outside the usual Nambu-Goldstone-Higgs framework. Our conclusions are based both on detailed calculations in a controlled approximation, and on qualitative symmetry arguments we expect to be quite robust. We will also derive an effective Lagrangian, that summarizes the electromagnetic response of the charged anyon gas. This Lagrangian, which to a first approximation is of the usual London or Landau-Ginzburg form, also contains small but characteristic interactions violating the discrete symmetries P and T. These terms lead to novel effects,

344 On Anyon Superconductivity

1003

whose occurrence (or not) should enable us to determine whether the anyon gas is realized in concrete physical systems. At this point it would be disingenuous not to remark that much of the stimulus for the recent upsurge in interest in the anyon gas are some theoretical speculations that quasiparticles in euo plane~, which presumably are the key actors in high t~mperature superconductivity, are in fact anyons. These speculations were motivated by analysis of c.xcitations arollnd certain types of ordered stat~s (chiral spin liquids) that haye been proposed for the electronic ground state in the planes. Needless to say, the fact that superconductivity is an automatic by-product nlakes these ideas considerably more compelling. For sim plicity) most of the discuss!O~ of this paper win be given for the case in which there is a single type of anyon. The d!scussi~n can be readily generalized to a set of t\\'O or more types of anyon:;~ possessing identical charge and Inass, but distinguished hi" an isospin index t. Although some of the quantitative formulas will be modified, the qualitative results will be generally similar. For reasons which will be discussed elsewhere we believe that in models of relevan~e to hightemperature sup~rconductivity there will always be an even number of anyon species. Bf'fore we embark on the analysis, it seems appropriate to establish the context with a brief quasi-historical account of the develooment of the circle of ideas we are dealing with. Many of the basic principles involved in fractional quantum statistics were clearly stated and illustrated in a remarkable paper by Leinaas and Myrheim. 6 Unfortunately this paper received little notice at the time, and did not enter the general consciousness, presumably because it was felt to be a purely academic exercise without a broader context. The continuou~ modem development of the ideas began as part of the recent interest in peculiar, and in particular fractional, quantum numbers more generally. In fact, it was argued !ong ago in prescient "Nork by Skyrme 7 that in 3 nonlinear sigma model of pions, particles with the quantum numbers of nucleons can emerge in the form of solitons. What was surprising about this is that spin and isospin one-half ~n emerge in a theory in which the elementary fields have integer spin and isospin. Later, Finkelstein and Rubinstein8 clarified the topological considerations responsible for Skyrme's phenomenon, and showed by a topological argument that solitons of half integral spin in fact obey Fermi statistics, as one would expect on general grounds. (This work probably also represented the first study of what would now be called a 8 angle in quantum field theory.) In a somewhat analogous fashion, magnetic monopoles in 3+ I dimensions can be fermions even in a theory in which the elementary fields are all bosons, 9 and can carry fractional 'o and even irrational I I electric charge. Also, Skyrme's spontaneolis generation of half integral spin turns out to have an analog 12 for the case of more than two "flavors" of strong interactions, provided one takes account of the global effects of the Wess-Zumino coupling. I] Closely

345 1004

Y.-H. Chen et ale

related phenomena occur in condensed matter systems l4 and in a wide variety of quantum field theories. IS · 16 Of course in three spatial dimensjons the non-trivial commutation relations of the angular momentum algebra ensure that the spin of any particle, regardless of microscopic origins, must be an integer or half-integer. Thus, the above cited results generating half-integer spin from integer spin in 3 + I dimensions are in a sense the best possible. In two dimensions the situation is different. The rotation group has a single generator which in principle can have any real eigenvalue. For instance, particles orbiting arwnd gauge theory strings, or even around ordinary magnetic flux tunes, can readily be seen to carry fractional angular momentum. 17 Once this is realized, it is then natural to asi<. (as Finkelstein and Rubinstein had done in connection with Skyrme"s work in 3 + 1 dirnensions) what happens to the spin-s1 atistics connection in these circumstances. This VIas in\'estigated in a series of papers· lI ,19,1."'O at first largely rediscovering (in ignorance) the results of Leinaas and Myrheim, but soon going beyond them in various ways, particularly in suggesting how objects of fractional statistics could actually be realized in the physical world. (For ~n account of early controversies surrounding these ideas, and their resolution, see Goldhaber and MacKenzie. 21 ) One early application of the idea of fractional statistics '9 was to the 2 + I-dimensional S2 u model, used to model the low-energy excitations of planar ferromagnets and antiferromagnets. It was shown that the c1assiCRI u model does not determine a unique quantum theory. The quantum theory allows inclusion of a new interaction, represented by the so-called Hopf term, which is invi3ible classically. The coefficient of the Hopfterm is an angle 8, closely related to the (J introduced in connection with fractional statistics. Indeed, in the (J model the coefficient of the Hopf term determines the spin and statistics of certain collective excitations, the baby Skyrmions. Roughly speaking, the Hopf term plays a role for these excitations somewhat similar to the role played by the WessZumino interaction in connection with 3 + 1 dimensional Skyrmions. Soon afterward the most important realization of fractional statistics so far established arose from a most unexpected quarter, in studies of the behavior of semiconductor heterojunctions held at millikelvin temperature in a strong external magnetic field. The fractional quantized Hall effect (FQHE), discovered in this context, established the existence of a rich new state, or actually series of states, of matter. The theory of these states was developed mainly by Laughlin,22 with important contributions from Haldane23 and from Halperin. 24 At the foundation of the theory is the irlea that the new states are best described as incompressible quantum liquids, around which the low-energy excitations are localized quasiparticles with unusual quantum numbers, including notably fractional statistics. lJsing this idea, Halperin was able to predict the values of the allowed fractions in the FQHE hierar~hy in a simple and convincing, as well as observationally successful, way. Arovas, Schrieffer, and Wilczek, using !he Berry phase technique, showed directly25 that the quasiparticles had the properties

346

On Anyon Superconductivity

1005

assumed by Halperin. (For an account 0f early objections to these ideas, and their resolution, see Laughlin.26) They also suggested that since the statistical Hinteraction", together of course with ordinary electromagnetism, is the dominant interaction of the quasi particles at long distances, it should be possible to write an effective Lagrangian for the long-wavelength behavior of the quasiparticle gas, usingjust these interactions. The formal implementation of this idea was carried tllrough in the abovementioned paper by Arovas, Schricffer, \\'ilczek, and Zee.' ·\n important denlent of that paper, which has played a key role in the further d,~veioprnent of the subject, is the intrQducti"'n of a local implementation of fractional Quantum statistics, through the Chern-Simons interaction. It is alsc quite likely that fracti()nal statistics eAci!ation~ exist for liquid 3He RIms in the A pha.se. 21 The application of this circie of ideas to superconductivity is by no rneans as certain or well-developed even as it is in the contexts mentioned above. It is surely premature to be writing even tile most informal of histories here. Still, it may be useful to orient ourselves with respect to some of the relevant recent literature on high-temperature superconductivity. Immediately upon ~he experimental discovery of the new superconductors, Anderson 28 stressed their essentially two-dimensional character, the importahce of strong magnetic ordering; and the possible existence of excitations with exotic quantum numbers. A relatively ccncrete propo~al e;nbodying one form of Anderson's vision \Vas put forward by Kivelson, Rokshar, and SetJ..-na. 29 fhey showed that division of valence bonds on a square lattice occupied by approximately one valence electron per sit~ into localized dimers, as suggested by the phase "resonating valence ~ond", could plausibly support excitations specifically, defe-:::ts in the pair-bonding of electrons, tra;>ping a single unpaired site - which are charged, spinless bosons. The initial thought was that Bose condensation of such charged excitations was the mechanism of superconductivity. A closely related proposal was made by Dzyaloshinskii, Polyaicov, and Wiegmann. 3o Their starting point was a a-mudel description of the spira ordering in the euo layers. They proposed that one employ the Hopf tenT., as we mentioned above, with 8 = 1C. (The paper contains the remark, without elaboration, that only 8 = 0 or 8 = 1C are consistent with unitarity. This is mistaken.) The effect of this tenn is to make the baby Skynnions of the pure spin model obey Fenni statistics. The idea then is that the charge carriers plausibly induce or bind to these baby Skynnions, making the composite a boson. Although the microscopic basis of this picture was never clear, and in fact the whole scr.nario now appears rather dubious, this paper caught the imagination of many physicists. Altogether, these early papers focused considerable attention on the possibility of exotic quantum numbers and statistical transmutation in two dimensions. Unfortunately, the most immediate natural consequence of all these suggestions is that, since one has direct Bose condensation instead of pairing, the flux

347 1006

Y.-H. Chen et al.

quantum should be h/ e. Experimentally, it appears to be h/2e, at least in the regimes where it has been studied so far. Various modifications of the ideas have been proposed,29 but it is difficult to know what conclusions to trust when such a seemingly straightforward one must be abandoned. Also, with the loss of the cO'1lpellingly simple concept of Bose condensation as a mechanism of superconductivity, the motivation for the suggestion of exotic quantum numbers becomes much less clear. An essentially new set of ideas was added by Laughlin and collaborators, in Refs. 2, 3 5. Kalmeyer and Laughlin made an approximate mapping of certain frustrated spin models onto Bose ga.ies with short range repulsive interactions and subje~t to a strong external ma~rJetic field. The latter situation is completely analogous to that in the quantiL:c~ Hall effect, and one can therefore take battletested knowledge oftht~ ground state and low-lying ~xcitatio!'\s in the H.an system over into the spin models. Given the previous discussion of the FQHE~ it should not seem shocking that the quasi particles are then found to obey fractional statistics. Wen, Wilczek, and Zee 31 have given a more abstract treatment of the problem, not relying on the details of a specific wave function, indicating what sort of spin ordering is essential to obtain fractionaJ statistics quasiparticles. We follow them in referring to this class of ordered systems as chiral spin liquids. Once one has a chiral spin liquid, it is plausible that charged particles doped into .. he system induce or bind to the fractional statistics quasiparticlt!s, thus themselves acquiring fractional statistics. In several papers, Laughlin and his collaborators have argued that fractional statistics in and of itself leads to superconductivity. The present paper sharpens and extends these arguments. An important feature of most models incorporating anyons is that they violate the discrete symmetries P and T. Thi3 is quite natural for the FQHE, which takes place in an external magnetic field. It occurs spontaneously in 3He-A. It would also have to occur spontaneously in high-temperature superconducton:, if anyon models are to describe them. It is, of course, characteristic of chiral spin liquids. That such symmetry breaking could occur, and can have important experimental consequences, was first emphasized by March-Russell and Wilczek,33 and considerably elaborated recently by these two together with Halperin. 34 Some of the issues have also been discussed re~ntly by Wen. and Zee3S and by Anderson.36 The considerations of this paper suggest some additional possibilities, and allow us to begin to discuss them quantitatively. Calculations of the energy of the undoped spin systems using variational wave functions of the Kalymeyer-Laughlin type have not yielded particularly good energies for simple model Hamiltonians, such as Heisenberg antiferromagnets with any combination of couplings to a few near neighboUrs. Moreover, for the undoped parent compounds of the actual copper-oxide superconductors (e.g., La2CU04) there is compelling evidence that the planes of copper spins are well described by a nearest-neighbour Heisenberg model on a square lattice, with a ground state that has conventional antiferr9magnetic order. 31- 39 It is known,

348 On Anyon Superconductivity 1007

however, that the addition of a relatively small concentration of holes is sufficient to destroy the anti ferromagnetic order. It is certainly possible that the holes also induce an effective multispin-interactian which favors a chiral spin state for the remaining copper spins. If this is the case, then it is reasonable to approach the superconducting state by starting with a model Hamiltonian where the spins form a chiral spin liquid even in the absence of free charges. Laughlin has shown that there exists in fact a model Hamiltonian (with long-range foul-spin interactions, and with ex.plicitly broker. time-reversal and chlral symmetries) for which the quantu111-Hall--effect wave function is the exact ground state.)2 There is little re~son to doubt that there exists also a class of Hamiltonians which only have finite range interactions, and are invariant under P and T, fer which the ground state is a chira! spin liquid. Shraiman and Siggia4U have argued that a very dilute concentration o~· holes in a copper-oxygen plane lnay lead to a ground state with a spiral spin structure, assuming that one can ignore the effe~ts of the compensating charged impurities, which must be present and would tend to localize the holes in an actual system at low concentrations. A spiral spin structure, in general, would have a chiral character, as well as a broken translational invariance. It is then plausible that above a certain critical concentration of holes, the broken translational symmetry will be destroyed by fluctuations, but the chiral character will persist. Finally let us note that while the work reported here was proceed~ng, I-!osotani carried out some calculations of the properties of the anyon gas using a somewhat different approximation scheme. Where they overlap, our conclusions agree. Also, Wen and Zee41 have attempted to study some questions related to those studied in this paper, by perturbing from bosons. Also, interesting numerical itudies of small systems ofanyon~ subject to an exte~nal magnetic field have been reported recently.42 2. The Hamiltonian In this section we derive a non-local Hamiltonian formulation of the anyon interaction, starting from a formulation in terms of a Chern-Simons Lagrangian. The Chern-Simons formulation is local, but contains redundant variables. The point of the exercise is that each description has its virtues. The Chern-Simons form clearly exhibits the full symmetry and global nature of the interaction. The Hamiltonian form, on the other hand, has the great advantage that its variables represent true physical degrees of freedom. It is therefore better suited to approximations and explicit calculations. The Lagrangian for an ideal gas of fractional statistics particles is

349 1008

Y.-H. Chen et al.

Here the Xa are particle coordinates and a is a vector field. The coupling of the particles to the gauge field is standard, but the gauge field action is unusual. Instead of a conventional kinetic energy for the gauge field, one has the final term in ( 1). This term, th~ so-called Chern-Simons term, is special to 2 + 1 dimensions. The action is gauge invariant, despite the explicit appearance of undifferentiated vector potentials. This is because these vector potential always appears contracted with conserved currents - either the conventional particle current, or the unusual Hcurrent'" efX1T JUT which is automatically conserved because of the Bianchi identity. Varying with respect to a, we find the field equations eifl J

= i1-eptlT r 2· Jar'

(2.2)

where j is the standard i>oint-particle current and f the standard field strength. These equations indicate that the gauge invariant content of the vector field a is entirely determined by the particle current. In otherwords~ a has no independent dynamics. To avoid confusion with the true electromagnetic potentials and fields, it is convenient to refer to these a fields, whose only purpose in life is to be integrated out and implement fractional statistics, as "fictitious" fields. It f"lIoms from the. field equation that the field strength f is confined to the particle worIdlines, and determi~ed localiy by the-current of these lines. Thus there are no classical Lorentz' forces among the particles. Integrating the 0 component of the fi~ld equation, we find the fundamental relation

eN =

11~,

(2.3)

where N is the particle number and the fictitious flux. This indicates that the effect of the Chern-Simons term is to associate with ~ach particle fictitious flux e/p. Of course, the particles also carry fictitious charge e. Thus as they wind around one another, they acquire phase through the Aharonov-Bohm effect. The consequence of all this is that the sole result of adding the fictitious fields is to alter quantum-mechanical amplitude for trajectories where the particles wind around one another, or are interchanged, by a phase proportional to t!le amount of winding. In other words, the quantum statistics has been altered. A simple calculation shows this alteration of statistics is parametrized by (2.4)

in terms of the angle 8 mentioned before.

JSO On Anyon Superconductivity 1009

Turning to the Hamiltonian formulation, we find again that the system has a uniqu~ underlying simplicity. Writing out the Lagrangian in more extended form:

we see that apart from the first tenn, the rest are either linear in ao or linear in time derivatives. Since the time derivative of ao n~ver appears, varying with respect to it simply yields the constraint (2.6)

Also) when we pass from the Lagranllian to the Hal!1iltonian terms linear in time derivatives cancel. Thus the Hamiltonian is numerically equal to the free-particle Hamiltonian - the net effect of all the extra ttnns is to enforce an unusual relationship between the canonical momentum and the velocity. The classical equations of motion are just those of non-int~racting free particles; the nontrivial dynamics arises entirely from the altered quantum commutation relations. Since a is a redundant variable we can eliminate it. To do this conve~ier:~!y, we impose the gauge condition (2.7)

Then we can solve the constraint e

a;(x) = -2 1tjJ.

~6)

to find

J.

e;j(x- J')j e " d"y 1 12 p(y) = -2-- L x- y IljJ. CI

(x-xar)j t;j

IX -

XCI

12 .

(2.8)

The final result is that t"e Hamiltonian is simply

H

1

= -2 m

L (Par - eo(x,,»)2 ,

(2.9)

CI

with a given as a function of x according to (8). The Hamilti>nian (9) forms the starting point for most of the further considerations in this paper. It was also the starting point adopted by Fetter, Hanna, and Laughlin. S As far as we know it has not previously been explicitly derived in full generality from the Chern-Simons Lagrangian, though the result was stated in Ref. I and a proof has been sketched before. 54 To conclude this section we add a few remarks that are not strictly essential to the logical development, but address some points that might be puzzling.

351 1010

Y.-H: Chen el at.

If one were given the Hamiltonian (9) without any explanation of its origins, it might be hard to believe that this Hamiltonian does not lead to classical forces among the particl~s. Indeed, H looks like the Hamiltonian for a charged particle interacting with an electromagnetic field, in a gauge where ao = O. Since the vector potential depends on the particle position4\, it varies in time, and one might therefore expect there to be electric fields depending on the relative positions of the particles, and therefore forces among them. Of couse we know from the preceding discussion that it is not so: what gives? Another puzzle is this: how does our H, lacking as it does the standard scalar potential piece, manage to give gaugeinvariant results? The resolution of these puzzles is really quite silnplc. The resemblance between our H and the standard Hamiltonian for an assembly of particles interacting \lfith an external gauge field is In one crucial respect misleading. That is, our a is g\ven as an explicit lion-local function of the particle positions. This means, in particular, a(xa ) depends not only on the position of particle lX, but on the position of all the other particles as well. Thus when we derive the Hamiltonian equations of mc,tion, there will be additional terms that do not appear in the usual equatiuns for particles interacting with an external gauge field. Keeping this in mind, a straightforward analysis of the equations of motion derived from the Hamiltonian H resolves both our puzzles at the same time. It is found that the additional terms serle exactly to reconstitute the full fictitious electric field, including specifically the gradient of the scalar potential ao, as determined from (2) in the gauge (7), in the Lorentz force equation. And the full fictitious electric field, as we discussed before, does not depend on the positions or velocities of distant particles, and does not generate classical interparticle forces. At the risk of being pedantic, we wish to emphasize explicitly one implicatioil of the preceding discussion. No approximation has been made in deriving H. Especially - despite apparent instantaneous interaction terms - retardation effects have not been neglected. 3.

Approach to the Problem

The statistical mechanics of an ideal gas of anyons has a very different flavor from that of the more familiar quantum ideal gases of bosons and fermions. In the case ofbosons or fermions, one can construct the eigenstates of the many particle Hamiltonian directly from the ~igenstates of the single-particle Hamiltonian, simply by taking tensor products. The sole effect of the statistics, in these two cases, is that one restricts to the subspace of many-body wave functions either symmetric or antisymmetric under permutations, respectively. The reason why this familiar, simple procedure fully incorporates the quantum statistics, is ultimately that the rule for assigning amplitudes to trajectories beginning at x I, X2, •.• and ending at xp" x P2 ' ••• depends only on the sign of the permutation P- Thus symmetry or antisymmetry in these coordinates is a condition stable in

352 On Anyon Superconductivity 1011

time. Also, we can obtain all trajectories with the proper weighting from trajectories along which the particles do not change their identity, if we allow all permutations of identity, with the appropriate sign factors, in the initial state. (Indeed, we have just the same trajectories, but with p-I acting on the initial config~ration instead of P on the final one.) For generic anyons, the situation is different. Tt,e amplitude assigned to a trajectory depends not only upon the permutation suffered by the particles as thl.~y follow the trajecto£ 'i, but aiso on other aspects of the traj~ctories by which they wind around one another. Mathematically, while the Hilbert sp3.ce of a system of identical bosuns and fermions gives a representation of the permuta~~'Jn &fOHP, the Hilbert spase of a 5~rstem of identical anyons gives a representaf;')i1 of the 4;br3id group", in which one distinguishes topologicaliy inequivalent trajectories leading to the same perrl1utatiuns of the particles. IncidentaHy. in 2 + 1 dimensional Inany-uody physics it is possible: in principle to have a system even more exotic than Hordinary" fractional statistics, in which trajectories that involve braidings of identical particles are represented by non-commuting matrices, not just by abelian phases. (It is far from straightforwarci to construct representations of the N particle braid group (lJN that are compatible with all the physical requirements of locality and cluster qecomposition, but the Jones repre~entations of the braid group43 satisfy all of the physical conditions, and in fact have a realization in lo~al quantum field theory via a non-abelian ChernSimons theory.44) Leaving aside these more exotic possibilities, which mayor may not eventually playa role in condensed matter physics, our interest here is with the anyon gas in which particle trajectories are represented by phases. In fact, the phase associated with a given trajectory is the product of the stati~tical parameter and the linking number of tJ1e trajectory.ls.1 ()nce the permutation group ic; replaced by the braid group, the simple construction passing from the solution of one-particle problems to the sulution of many-particl~ problems, familiar for free bosons and free fermions, does not work any more. It seems most unlikely that there is any comparably simple substitute. For this reason. even an ideal gas of anyons must be regarded as an interacting system. Since an exact solution seems out of reach, it seems a good strategy to attempt to begin to understand anyon gases by perturbing around the familiar cases of free bosons or fermions, taking advantage of the tools developed over many years for the study of inter~cting systems of identical particles. There is an extremely naive argument, which' suggests that in general excluding fermions - an anyon gas will be superfluid (or. for electromagnetically charged anyons, superconducting) at zero temperature. It goes as follows. Fermions with arbitrarily weak attractive forces are known to form superfluids at zero temperature. But there is a real sense in which anyons in general can be considered as fermions with an additional attractive interaction. Indeed, the most important effect of quantum statistics at short distances is that it determines the

353 1012

Y.-H. Chen et al.

allowed values of kinetic angular momentum, and thus the strength of the centrifugal barrier. For bosons the allowed values are even integers; for fermions they are odd integers, and for general 8 they are 19/a+ even integer. Thus the minimum allowed absolute value is generically smaller than it is for fermions; and so generic anyons can be regarded as fermions with an additional attractive interaction. Although it will become evident in the following that this argument is really much too naive, clearly it points us in the direction of suspecting superfluidity in the anyon gas at zero temperature. With this suspicion, it might seem logical to try to perturb around Bose statistics. After all, the ideal Bose gas exhibits the phenomenon we are after superfluidity - already in the zeroth approximation. (It is sometimes said that the ideal Bose gas requires a repulsive interaction to become superfluid. We think it is more accurate to say that the ideal Bose gas is a superfluid with zero critical velocity, and poised on the brink of instability - a weak attraction will cause it to cease to have a sensible thermodynamic limit.) On further reflection, however, several difficulties with this approach become apparent. The most important one is the following. Consider the gas with statistical parameter

*=n.n

(3.1)

Now if we imagine that superfluidity is characterized by an effective condensation into bosons - generalizing ordinary Bose condensation or Cooper pairing - then we must ask: how many of these anyons does it take, to form a boson? If we take one m-tuple around another, we find the accumulated phase nm*/n. Thus the condition is $ = 0 (mod 2). Clearly, the minimum required number grows with n, roughly as the square root. It is not easy to see how to obtain this behavior smoothly, starting from condensation of single particles in the Bose gas. Anyons near 19 near zero are similar to a system of bosons with a weak repulsion of statistical origin (representing the centrifugal barrier that is present at 8 # 0) and in a background magnetic field (representing the interaction of one particle with the average statistical background of the others; this interpretation will be clearer in Sec. 4). Now, bosons with a weak repulsion undergo bose condensation and become superfluid. In the presence of a magnetic field, bose condensation still occurs but not in a translationally invariant fashion; one should expect to form some sort of vortex lattice. Our approach instead will be to work near Fermi statistics:

354 On Anyon Superconductivity 1013 ‘g=a

( 1-i 1 n

’

(3.3)

Then as n gets large the expectation that condensation requires more and more particles appears rather as a virtue than as an embarrassment - it allows us to lose superfluidity in the limit of fermions. One reason that we think it is natural to work near 8 = a is the following. In order to establish that the statistical attraction (relative to fermions) of a departure from 8 = A gives rise to superfluidity, it seems to us that the key case is to show that even a weak statistical attraction among a system of otherwise free fermions leads to superfluidity. Once it is established that a weak statistical attraction gives superfluidity, it is natural to expect the same for the strong statistical attraction that arises at the case (0 = n/2) that is believed to be of most interest. Once the effects of a weak statistical attraction are understood qualitatively, it is reasonable to hope that the effects of a strong statistical attraction are similar qualitatively. Our basic strategy is thus to attempt to understand the statistical mechanism for superfluidity starting from the regime of 8 near 7c where this mechanism is operating weakly and can be studied in a controlled way. Both the qualitative arguments of the next section and the detailed calculations which follow are based on an approximation procedure suggested by Arovas et al.’ and employed to great effect by Laughlin* and by Fetter, Hanna, and Laughlin.’ We now describe this procedure, and identify a limit in which it is expected to be valid. Above, we have seen that in a precise sense the statistical interaction can be implemented by attaching fictitious charge and flux to fermions. It is, however, very awkward to deal with the resulting long-range interactions directly. In other problems involving long-range interactions, it is sometimes valid to replace the effect of many distant particles by a mean field or collective variable, with the deviations from the mean represented by residual weak or short-range interactions. Could something like that occur in our problem? We will argue that in fact very plausibly it does. To get started, let us consider the self-consistency of the approach. Suppose, then, that we do replace the total effect of the distant particles by their average. In our context, this means we are replacing the many singular flux tubes by a smooth magnetic field with the same flux density. For 8 = K( 1 - l/n), the resulting magnetic field is related to the average particle density jj by

In such a magnetic field, the particles move along cyclotron orbits with radius

355 1014

Y.-H. Chen et al.

mv

(3.5)

r = eb .

Taking for the velocity the velocity at the nominal Fermi surface, we substitute

~4xp v=-m

(3.6)

and find that a typical cyclotron orbit contains (3.7)

particles on the average. If the number of particles inside the typical significant orbit is much greater than I, we should expect that it is indeed valid as a first approximation to replace the field generated by the particles by its average value. since fluctuations will be small compared to the total. While this argument can and should be sharpened, it seems clear that in the limit of large n it is at least self-consistent &s a first approximation to replace our anyon gas by a gas of fcrmions carrying fictitious charge and propagating in a fictitious magnetic field tied to their density according to (3.4). 4. A Qualitative Picture Several of the most important qualitative features of the anyon gas can be understood readily from the simple starting point defined in the previous section. There, the anyon gas was replaced to a first approximation by fermions propagating in a uniform backgro~nd fictitious magnetic field given by b = 21£P / n. In the fictitious background field b, the energy eigenstates of the fermions .form Landau bands, each with degeneracy

Pt

eb 21£

= -

p

=-

n

(4.1)

per unit area, with energy eigenvalues

(4.2) where I = 0, 1, 2, .... When the statistical parameter is () = x( 1 - 1/ n), the density is just such as to fill n Landau levels exactly. (In the next approximation we will find a massless particle that will give the feunions a logarithmically divergent self-energy, which we ignore for the present.)

356 On Anyon Superconductivity

1015

The fact that the bands are exactly filled suggests that the ground state will have a particularly favorable energy at these values of the statistical parameter. Exactly filling the top band ought to be analogous to completing a shell in atomic or nuclear physics, or filling an ordinary band in a solid. If this is true, the ground state should exhibit a certain rigidity, and exhibit an energy gap. To test and quantify these expectations, let us consider the effect of adding a small real magnetic field B to the fictitious one b. The situation is asymmetric with respect to the sign of the real field relative to the fictitious field, and we must consider the two cases where the fields add or cancel separately. I f the real field is in the same direction as the fictitious one, the density of states per Landau level will be somewhat greater, and we will not quite completely fill n ~C\'el~ a!lymore. Let us der.ote the fractional filling of the highest level by I-x. Then from the conservation of particle number we derive

(b+ B) (n - x) = bn; (b+ B)x = Bn.

(4.3)

For the total energy we have then

E -_

1) ( 1)

e(b+B)e(b+B){~( e L t+- - n-- x }_n --21£

X

m

1-0

2

2

2 2

41£m

1 (1 -;;1) B 2} . {b 2+~bB-

(4.4)

Thus the energy relative to the ground state is positive, and grows linearly with B for small B. If the real field is in the opposite direction from the fictitious one, the density of states per Landau level will be smaller, and we will have to promote some particles to the (n + 1) level. Denoting the fractional filling of this level by x, we have from particle conservation

(b - B) (n + x)

c::

bn; (b - B)x = Bn,

(4.5)

and for the energy

E=

e(b-B) e(b-B) 21£

m

1) ( I)}

{~( L t+- + n+- x 1-0

2

2

2

n e

2

=--

41lm

(4.6)

Thus in this case too the energy relative to the ground state is positive, and grows

357 1016

Y.-H. Chen et al.

linearly with B for small B. Despite the asymmetry of the situation, the coefficients of the terms linear in B are equal in the two cases. The quadratic terms differ. These arguments though simple are quite significant. They suggest that the anyon gas, at the statistics considered, will strive to exclude external magnetic fields. This is the germ of the Meissner eff~ct, a hal!mark of superconductivity. At the same time they suggest the existence of an energy gap in the charged particle spectrum. Indeed, the energy to create a separated particle-hole pair should be just the energy to excite a fermion into the lowest empty Landau band, viz. eb 21CP E· = - = P~1r

m

mn

(4.7)

Considered more closely, these arguments also suggest a close connection between vortices and fermion excitations that seems to be something new in the theory of superconductiyity. This connection is characteristic of anyon superconductivity, and will playa key role below both in its deeper theory and in its phenomenology. The point is this: since the fictitious field is uniquely tied to the iJarticle density, and is appropriate to n Landau levels being exactly filled, to accommodate any additional real magnetic field we will necessarily have to excite particles across the gap. (Or to create holes, a process which we have ~een is also characterized by a gap.) Conversely, if the particles do not fill the Landau levels exactly, there must be a real magnetic field present to account for the mismatch. Anticiilating that the filled Landau level state, and its possible adiabatic modulations, is the superfluid component, we are led to conclude that in anyon superconductivity, charged quasiparticles and vortices do not constitute two separate sorts of elementary excitations - they are one and the same. We can also infer the value of the flux quantum, from this identification. Adding a single fundamental unit 21C/ e of real flux increases the number of available states by one per Landau level. Thus, for n filled Landau levels, the act of piercing the material by a unit flux tube creates n holes. Clearly this is not the most elementary excitation. The most elementary excitation is to produce just one hole. Thus the elementary tluxoid is 1/ n of the fundamental unit, or 21C/ ne. Although these simple arguments have taken us a long way, there remains a central feature of superfluidity that is not at all obvious, or even true, in the simple approximation described thus far. This featare is the existence of a sharp Nambu-Goldstone mode, or concretely an excitation with the dispersion relation 2 (J)2 ex: k at low frequency and small wave vector. It does exist. It was discovered in a remarkable calculation by Fetter, Hanna, and laughlin. S They calculated the effect of adding back the residual interactions, and found that these interactions produced the necessary pole in the current-current correlation function. In physical terms, this means that there are particle-hole bound states at zero energy. In the following two sections we shall review and generalize these calculations.

358 On Anyon Superconductivity 1017

Unfortunately these calculations do not by themselves make it clear why the massless mode exists. Aside from being emotionally disturbing, it is not objectively satisfactory to lack such understanding. Without it, one may be left uncertain whether this central qualitative feature of the anyon gas is robust, or an artifact of the approximations employed in the calculation. Similarly, one may be left uncertain 'Nhether small changes in the model Hamiltonian itself - which after 311, is highly idealized - might change this feature. Fortunately, the exi:;:ence of the massless mode can also be derrlonstrated simply, and it can b~ ul1d~rstood qualitatively using arguments closely related to those in the present "eel Ion. This is the subject of Sec. 7.

To conclude the present section we would like to make some brief remarks the enyon B3S at other v3.1ues of the statistical paralneter, when 6* 1l(1-1/n). I f [he lOp Landau level were not completely filled, !hen the second of our calcu·· lations above (leading to Eq. (4.6» would be valid for either sign of the field. The energy is then analytic in B, and the presence of a iinear term is indicative of the fact that the ground state of the anyo:l system possesses an orbital ferromagnetic moment in this case. (We also find that there is an orbital magnetic moment when 8 = n(1 - lIn) but the analysis is considerably more complicated. 34) For more general rational values of 8/n, it is possible that the anyons in the highest Landau level will form a correlated many-body state, similar to the states of the fractional quantized Hall effect. 45 In this case there is again an energy gap for vortex excitations, and we expect again to find a superfluid ground state. For most of our discussion, up to and including the previous sentence, we have assumed that the ground state is homogeneous. (An exception was when we discussed the expected ground state for fractional statistics near bosons.) This is almost surely true for the values 8 = n( I - lIn) which are our main concern. However, it is almost surely not true in general. For example, let us consider again statistics very close to, but not equal to, one of our favored values, say n = no. Then c1ear!y instead of expanding around n = 00 - fermions - we should expand around n = no. The particles will then have a small residual interaction. More important, the particle density will then not quite fit the density appropriate to the fictitious magnetic field. It seems very likely that the best way to accommodate this situation is to allow an occasional normal partic!e - or equivalently, an occasional vortex - rather than to disrupt the superfluid state globally. Thus, operationally, one would separate the anyons into t~NO c1assesthe first, with fractional density nol n to be treated as an anyon gas with 8 = n( I - 1/ no) and the remainder to be treated as vortices or anti vortices in that background. Readers familiar with the fractional quantized Hall effect45 will recognize a strong resemblance to the situation that occurs there, when the density is close to but not quite equal to one of the favored rational filling fractions. These considerations are by no means rigorous or complete, but they do serve cO~':t'rning

359 1018

Y.-H. Chen et al.

to suggest that the physics of the any on gas at general values of 8 is likely to be quite rich and to depend quite strongly on "number-theoretic" properties of 8. 5. The RPA Calculation In this section we discuss the mechanics of calculations in the random phase approximation. The method follows closely that of Fetter, Hanna and Laughlin; we have merely adcted a few observations and elaborated several points left implicit in their very concise presentation. To begin with, 3S we discussed in Sec. 2, the Hamiltonian of the anyon gas is (changing notation slightly to agree with Ref. 5) (5.1)

where ra is a two-dimensional vector specifying the position of particle a and (5.2)

with raj) = ra - rp. Here the particles are to be regarded (in the absence of interactions) as fermions; the interactio~ then makes them anyons with statistical parameter (J = n(1 - 1/ n). It will be convenient to use second quantized notation, in which (5.3)

Here '¥ is a spinless fermion field, and 1 a(r) = -

n

Jd

2

r'

iX(r-r')

Ir- r'1 2

'I't (r')'I'(r').

(5.4)

.

The Hamiltonian describes a system of spiriless fermions interacting through long range gauge potentials. Actually these expressions are somewhat formal, in that if the density is constant the integral for a will diverge. For this reason, and also to implement the ideas of Sec. 3, it is useful to separate a into a background part and a fluctuating part. This is analogous to the 'familiar use of normal ordering or subtractions in defining the vacuum quantum numbers in quantum field theory. It should be considered as part of defining the theory. We shall have to check whether the theory so defined retains the properties - and in particular, the symmetries -

360 On Anyon Superconductivity

1019

we expected of the naive model. Alternatively, one could in principle fonnulate the theory in a finite geometry, say on a torus. If we ignore fluctuations and substitute the average density p for the density operator in a(r), we expect that the system should reduce to spinless fermions propagating in a constant fictitious magnetic field. Thus we are led to define 1

a(rl = a(r) + -

J

d 2 r'

Z X (r- r')

Ir- "12

n

('JIt'JI(r) -

.0),

(5.5)

where a(r)

I

= - biX r, 2

21lp

b=-.

n

(5.6)

(5.7)

This definition of a 'replaces (5.4). However; thp formula (5.6) for a(r) requires some explanation. The mean vector potential a should naturally be defined by the same integral I

a(r) = -

n

Jd r' iX(r- r') .0 2

Ir- r'1 2

(5.8)

as (5.4), with the true charge density 'l't'JI replaced by the mean density p. The only problem with this is that the integral in (5.8) is not unambigaously convergent if p is strictly constant. To interpret this integral, note that for arbitrary .0 such that the integral in (5.8) is well-defined, that integral computes an abelian gauge field li such that b = 21lp / n, where b = a. a2 - a2a., and moreover such that V . ii = 0, and such that ii vanishes at 00. a is uniquely determined by those conditions, and the integral in (5.8) has exactly the kernel required to produce the field a obeying those conditions. For the limiting case in which the support ofp extends over all of space, the integral in (5.8) is ambiguous (not absolutely convergent), and it is impossible to obey all of the conditions that would hold ifp had compact support (to give the right b, ii cannot vanish at (0). We interpret the integral in (5.8) as giving an average a field that gives the right band obeys the gauge condition and has a behavior at 00 that is as good as possible. The proposed form in (5.6) obeys these desiderata, but is not quite unique since without changing b or viola ring the gauge condition or worsening the oehavior at 00, one could add a constant to ii. This ambiguous integration constant is actually closely related to the pnysics that we will eventually find. Modulo an integration constant, the answer in (5.6) is certainly what one would get by doing the integral

361 1020 Y.-H. Chen et aJ.

in (5.4) for some almost constant p of compact support, and then taking the limit as the support of p extends over all space. The value that one would get for the integration constant would depend on exactly how one took the limit. For later use, we define current operators j;(r) EE 'I't (r)~(p; + a;(r) ) 'I' (r) m

(5.9)

.. t I j;(r) == \f' (r)-(p; + a;)'I'(r) . m

(5.10)

Since on the one hand it is a soluble problem, and on the other we have argued it contains much of the important physics, we will treat the system of otherwise free fermions propagating in the average field as the reference problem, and regard the re~t of the Hamiltonian as a perturbation. The interaction Hamiltonian for this perturbation scheme is

=

H.

+ H"

(5.11)

(5.12)

H2

=

_1_2 2mn

JJJ

d 2 r d 2 r' d 2 r"'I't(r)'I'(r) {e;j(r- r;j ('I't(r')'I'(r') -

Ir-r'l

P)}

X {eik(r - r")k ('I't (r")'I'(r") _ -)}

Ir-r"12

=

I 2mn

--2

p

JJJ

2

2

d rd r' d 2 r" p(r)

(r- r'). (r- r") 2

Ir- r'l Ir- r"1

2

(p(r') -

p) (p(r") - p). (5.13)

In the same spirit let us reorganize H2 iuto two pieces, usingp(r)

=

p + (p(r) - p).

362 On Anyon Superconductivity

1021

The first half of the resulting expression is expected to dominate for large n, when fluctuations in density are relatively small. Its meaning becomes transparent upon doing the integral:

J

d2 r

(r- r')' (r- r")

Ir -

r' 12 ·1 r - r" 12

= - 21r Inlr' -

r"I.

(5.14)

It represents an effective COulomb interaction! The interaction is repulsive

L'etween like-signed particles, attractive between oppositdy-signed particles. The existence of such a~ interaction is important in two respects. First, it generates.an effective long range repulsion between two particles, or two holes. Given the identification of these excitations with vortices, this is responsible for the anyon superconductor being type II. Second, it generates an effecti ve long range attraction between particles and holes. This is responsible for the formation of the zero-mass bound state. The nature of the interaction can be given an interesting interpretation. Imagine that a massless gauge field has developed dynamically, such that our particles couple to this field. Then there would be a logarithmic interaction of precisely the calculated form. Later we shall see that the premises in this interpretation do actually hold. If we simply drop the other half of H 2, we are left with two-body interactions only, and can make great progress. Note that the discarded term, besides being intrinsically small, is manifestly translation, rotation, and (even if we couple in electromagnetism) gauge invariant. The remai~ing interactions c~n be written

(5.15)

where the spatial part of

i

has been defined before, and

Jo(r)

= p(r) - p .

(5.16)

There is no distinction betweenJo and jOe V takes a simple form in Fourier space. To exhibit this, we take a momentum vector q with component only in the t and x directions, and we order the coordinates as (t, x, y). Then one has

363 1022

Y.-H. Chen et al.

p(21C)2 mnq2 n

0 -i21l

0

i21C q

0

0

0

0

(5.17)

q

The appearance of the Coulomb interaction suggests the importance of summing bubble graphs, as in the standard treatment of the electron gas. Since the interaction Hamiltunian can be written in terms of j, the correlation function of .f obeys a simple geometric equation, in this approximation. Thus defining (5.18) as the matrix of time-ordered expectation values in the true ground state, and D~v as the corresponding object in the non-interacting ground state, we have in this approximation (5.19) The product is to be regarded as convolution in real space or simple multiplication in Fourier space. Solving this equation, we find (5.20) Another perspective on the bubble-graph approximation, thai is actually superior from a logical point of view, is to regard it as simply a perturbative evaluation of the inverse propagator D- 1• The previous equation, in the form (5.21) is then simply lowest-order perturbation theory. Why is it more appropriate to perturb in the inverse propagator than in the propagator itself? That is a standard story that we shall not belabor here; the key point is that the inverse propagator, unlike the propagator itself, is regular at small frequency and wave vector, so whereas for the propagator itself we find immediately that the limits 00, q ~ 0 and n - 00 do not commute, there is every reason to expect the perturbative evaluation of the inverse propagator to become accurate as n ~ 00. The calculation of !}~v is straightforward though rather arduous; it is presented in Appendix A. The result may be parametrized in the form

364

On Anyon Superconductivity 1023

(5.22)

~n

writing this result we have specialized to the case q~, = 0; this involves no real toss of generality. D is not quite the object we want. The electromagnetic response is rather given in terms of the true c~rrent-current corrdation function ~

(1 2) --

.. "p,'I'

__ 0;~ \III Trt J ~I l(I) JAY (2'. ' !1'\, "

(5.23)

where ( I ) denotes the dependence on ," t I, and I) denotes the exact ground state. Fortunately, A and Dare ciosely related. Consider, for exa:nple, the 010 entry. We have

(A - D)IO

-; =" -

m

(IT['I't'l'(I)(a - a)(l),

'I't'l'(2) - p]l)

(5.24)

In the now familiar manner, we sp.parate p into an average and a fluctuating part:

(5.25)

The contribution to A - D involving the average can be simply expressed in terms of D itself; the contribution from fluctuations is small in the n -+ 00 limit and ·Ne drop it. Passing to Fourier space, we arrive at (A-D)IO =

21CP I

+ ;--Doo. mn q

A generalization of this argument leads easily to

(5.26)

365 1024 Y.-H. Chen et al.

0 i) U=m:2 (00 00. 0 0 0

(5.27)

q

Finally, the true electromagnetic response includes not only the current-current correlation (which essentially represents the iteration of the first order term in the true electromagnetic potential A) but also a contact term, from the direct appearance of A 2 in the Lagrangian, which is quadratic in momentum. Thus the final expression for the response function, defined according to (5.28)

is (5.29)

Collecting the various formulae, we find (setting, for reasons discussed in Appendix A, 1:3 = I)

e2n

K=-21ldet

q2 W iqE -1:0 q-l:o We We iw'E. W w2 q-1:o -1:0 We We -iqE -iwE we(E -1: 1+ 1:2 + det)

(5.30)

where (5.31 )

and (5.32)

In arriving at this expression, we have made approximations at three stages: in the perturbative evaluation of the inverse propagator D -I, in formulating the interaction Hamiltonian, and in passing from D to A. We have discussed the first of these above, now let us address the other two. Both these approximations were

366 On Anyon Superconductivity 1025

of the same general form: in an expression involving the correlation of the density at one point with density fluctuations at two other points, we replaced the density with its average. In concluding this section, we wish to remark that this approximation can be justified in the large n limit. Indeed, the triple correlations of density fluctuations satisfy a simple Dyson equation. Although we will not present the details here, a straightforward analysis based on this equation shows that the terrns dropped invoive a highe:- POVllcf of the lr~tera,~tion than the terms kept, and thus a higher power or 1; n. Lieady, these re.rnarks also pOInt the way to a practical method of calculating to higher ord·~r. 6.

Results.,( the RPA

Cal~lation

\Ve now evaluate the

e:e/~tror.lag\l("'li(' rEspon~e

K;;..Jq, .::0) for snlall q

~nd

w

explicitly. From Appendix A we derive in this limit Io~

-1-(~)\3n(!r we 8 \,t ,

Il~

-I-(:r + 3; (*r.

I2~

-I-(~J +n (1)'.

(6.1)

It is noteworthy that to this order only transitions between the two tcp filled Landau levels and the two bottom empty ones contribute. There is evidently a pole in the response funeticn, at (6.2)

The physical significance of K becomes more transparent if we reformulate it in terms of an effective Lagrangian. We have found that we can reproduce the response function at low frequency and small wave vector using an effective model which contains a massless scalar field interacting with the electromagnetic gouge field, of the form

367 1026

Y.-H. Chen ·et ale

Notice that this Lagrangian is invariant under the gauge transformation:

(6.4) This model exhibits the Higgs mechanism in its pristine form (due to Stuckelberg): tP, which in the absence of electromagnetism represents a scalar degree of freedom - essentially a sound wave, with v 2 equal to the speed of sound squared - loses its independent significance when thus coupled to electromagnetism. I ndeed, it can ~e set to zero by a gauge transformation. The f~rst two terms in L are familiar in the theory of superconductivity. They generate the ordinary London equations. The next two terms are higher order in gradients, and thus formally subdominant. However, we have kept them because they display a qualitatively new feature. Whereas the first two terms are automatically invariant under parity and time reversal, the next two are not. They are of course fully rotationally and gauge invariant, but violate both P and T, in such a way that PT is conserved. In a word, they obey the symmetries of our underlying microscopic model - the anyen gas - and we have every right to expect that they should occur. The fact that these terms are in a real sense small is both entertaining and Sif;a!ficant. It is entertaining, in that it is a rather unexpected analogue of a familiar situation in high-energy physics. There, it is an important result that in QCD, parity violation and time-reversal violation cannot occur through low-dimension (renormalizable) i:lteractions. It is this fact that makes it comprehensible that parity and the time-reversal violation are hard to observe, even though neither is fundamentally a good symmetry. Similarly here, it is "'ery significant that parity and time-reversal symmetry are in some sense automatically hidden in anyoD superconductivity. This inakes the phenomenology more challenging to work out and the experiments t\l meaningfully test the symmetries necessarily subtle. If we put tP = 0 inside the Lagrangian, we see that these new terms are closely related to gauge theory Chern-Simons terms. It is amusing that upon dropping the requirement of relativistic in variance we find there are two possible ChernSimons like terms. To a first approximation the charge density and electric current associated with tP are

p = -C(cP- CAo)

(6.5)

as follows from varying the Lagrangian with respect to Ao, Ai respectively and dropp!ng the terms proportional to a and b. Using these approximate expressions, we can write the new terms in a more transparent form:

368 On Anyon Superconductivity

1027

(6.6)

(6.7)

Thu~ we see that a cor rd~tes electric charge density with magnetic fields, and b correlates current with perpendicular electric field in a nlanner reminiscent of the Hall effect. Also, we see that a change tJp in the density is generally accompanied by a change in the ffi:Ignetic moment density, proportional to a. The num~rica! evaluation is carried out by comparing the photon two-point function calculated froln L with the response function K. A few details of the calculation are presented In Appendix B. OUT results, valid in the limit n - 00, are:

m2

c=

'

e~ 2:1i' (6.8)

a=

ne~ 327tm Ii

,

b= 0 where proper units have been restored. The values of v 2 and C are just such as to reproduce the standard formula for the London penetration depth. The vanishing of b can be understood on physical grounds. We will discuss this, as well as some phenomenological implications of L, in Sec. 8. One can also obtain the coefficient a by another type of analysis, which we believe to be exact, and whose details will be given elsewhere. 34 The correct formula for a differs from that in (6.8) by an additillnal factor (I - n -2). Note that this gives a = 0 for the case of bosons (n = I) as we expect for this situation, where P and T are actually good symmetries. To conclude this section we would like to comment on the relation of the effective Lagrangian discussed above to a more complete effective Lagrangian, and how the latter might be calculated. These comments illustrate certain points but do not incorporate the special featurc3 of the order parameter discussed in Sec. 7; thus the equations that follow should be interpreted metaphorically. In the Landau-Ginzburg generalization of the London framework one con-

369 1028

Y.-H. Chen et al.

siders that the density as well as the phase of the superfluid condensate can vary. In this generalization, we wouid have instead of the Lagrangian considered above, a Lagrangian of the form

(6.9)

Our previous considerations on the anit of quantized flux suggest q = nee This form goes over into the London Lagrangian if we s&>ecialize the complex scalar to the form (6.10)

.J

where V = m 2/2A. is the vacuum expectation value of <1>, and work to lowest order in gradients. Notice that the mass term m and the self-interaction A. lose their significance in this limit. We determined the coefficients of the London Lagrangian by matching to the electromagnetic response at low frequency and small wave vector. One could in principle determine the coefficients of the Landau-Ginzburg Lagrangian, or an appropriate modification of it) within the framework of the calculations reported above, by matching to the response at higher frequency and larger wave vector. It should be remarked, however, that the unique feature of the statistical interaction - its long-range n!lture - does not guarantee, or even make it reasonable to expect, that it is a good guide with respect to short-distance or small-time behavior. Other interactions of a more prosaic sort will surely come into play. Therefore the idealization involved in treating the quasiparticles in any real material as an ideal gas of anyons generally becomes more severe as we move away from the London regime, except for certain qualitative questions of a global character. We might also step back one more step, and try to build into an effective Lagrangian the fact that the P and T violation, which we have been treating as if it were fundamental, must actually have its origins in spontaneous symmetry breaking. A simple possibility is the following. Let " be a real scalar field, meant to parametrize the degree of chiral spin liqui.d order. Then let

370 On Anyon Superconductivity 1029

L =

fL.G.

(6.11 )

+ L"

where I .

LL.G.

2

V

2

•

= :;-1<1> - iqA o 1 - -::;-ld, - iqAj I2 + ia17eijd,A;{t( - iqAo
..

-+- ib1'/Gjj(aaA, - d,A,,,) {~t (dj