WHEN THE ELECTRONFALLS APART

In condensed matter physics, some particles behave like fragments ofan electron.

Philip W. Anderson

It is ironic that, in the year when we celebrate thecentenary of the discovery of the electron, the most

exciting developments in the theory of electrons in solidshave to do with the "fractionalization" of the electron—thediscovery of particles that behave as though the electronhad broken apart into three or five or more pieces eachcontaining one-third or one-fifth of its charge, or intoseparate particles, one containing its charge and one itsspin. (See figure 1.) No longer is the quantum theory ofsolids confined to the boring old electron; we now have aremarkable variety of fractional parts of electrons: com-posite fermions, composite bosons, spinons and holons, inaddition to the heavy electrons, quasiparticles and smallpolarons of older stages of condensed matter theory.

These developments of course have nothing to do withthe internal structure of the electron as an elementaryparticle; we are told that the electron is one of the fewparticles that is so far seen as truly elementary in thatsense. But they do mean that in describing the low-energyexcitations of solids, condensed matter theorists have hadthe opportunity to play with many of the interesting newconcepts that modern field theory has developed to dealwith strings, non-Abelian gauge fields and the like, aswell as with some interesting new ideas of our own. Ashas often been the case in the past, the mathematicalstructure of modern condensed matter theory is exhibitingparallel development with elementary particle theory.

In fact, right from the start of the quantum theoryof solids in the 1920s, it was found useful to invent newparticles to describe the collective behavior of the largenumber of electrons that occupy the energy bands of solids.The most useful and ubiquitous of these is the "hole,"invented by Heisenberg already in 1926 to describe nearlyfull atomic shells, which exhibit a reversed sign of spin-orbit coupling. Rudolph Peierls's arguably greatest con-tribution to physics1 was to extend this concept to theenergy bands in solids, and to realize that the missingparticles in a nearly full band would behave dynamicallylike positively charged holes. His application of this wasto the "anomalous" Hall effect in metals. The Hall effectis the transverse voltage that develops from sideways

PHILIP ANDERSON is Joseph Henry Professor of Physics atPrinceton University in Princeton, New Jersey.

acceleration of a moving electron by a magnetic field. Theforce on the moving particle is

and clearly which way the voltage deviates from thecurrent is a measure of the sign of e. It had long beenpuzzling that in metals on the right side of the periodictable, e often came out positive. This idea of the hole wassoon taken up by semiconductor physicists, with enormousconsequences for technology, as we all know. (And, ofcourse, it was taken up by Dirac, in his subsequentrelativistic theory of the electron, in which he originallypostulated the positron as a hole in a filled band ofnegative-energy solutions of his equation.)

The hole is not fractionalized, unless one views -1 asa fraction, but it has in common with the soliton-likeexcitations discussed below that collective behavior of thewhole band is responsible. It is the Pauli exclusion prin-ciple (together with basic principles of symmetry) thatforces a filled shell or band to be electronically inert andto have zero net velocity, so that the shell or band withone missing particle responds in many ways like thenegative of a real particle.

BCS quasiparticlesThe concept of the hole plays an important role in thenext example of a bizarre particle, the quasiparticle of theBardeen-Cooper-Schrieffer (BCS) theory of superconduc-tivity. One may think of the ground state of the BCStheory as a coherent linear combination of states withdifferent numbers of electron pairs, in which the membersof the pairs have equal but opposite spin and momentum,and hence zero total spin and momentum, but charge 2e.Thus, when one adds an electron of momentum k andspin 11), one arrives at a state equivalent to one that couldbe reached by subtracting an electron of momentum -kand spin |1) from such a pair, leaving behind its partner.Thus, in this case, particle states—the exact eigenstatesof the system—are linear combinations of electrons andholes, with opposite charge but the same momentum andspin.2

Again, it is the background of the entire Fermi seaof electrons that absorbs the missing quantum number,so there is nothing really there but physical electrons with

42 OCTOBER 1997 PHYSICS TODAY '31-9228-9710-04M

0%

FIGURE 1. ELECTRONFRAGMENTATION can occur ina number of ways. Left: In aBCS superconductor, theparticle states are linearcombinations of electrons(green) and holes (red). Thus,a lone electron is itself acombination of these states.Middle: In systems that exhibitthe fractional quantum Halleffect, the behavior is explainedby states filled with particlesthat have fractions of anelectron's charge (odd fractionsin the simplest cases).Right: In several systems,including the attractiveone-dimensional Hubbardmodel, trans-polyacetylene andperhaps the normal metal stateof high-7c cuprates, the spin(blue) and charge (green) ofelectrons are carried bydistinct particles.

charge -e and spin V%. Nonetheless, quasiparticles can doodd things: One may demonstrate, as Walter J. Tomaschdid,3 quantum interference fringes between electrons andholes from the same region of the Fermi surface (WilliamMacMillan and I supplied the theory). At an interfacebetween a normal metal and a superconductor, electronsin the normal metal can be reflected by the interface asholes, leaving behind the net charge 2e, which is carriedoff by the superconducting condensate. This Andreevscattering is the main current-carrying process at such aninterface, and hence is responsible for our ability to insertcurrent into a superconductor.4

One-dimensional Hubbard modelThe superconducting case, bizarre as it is, still gives usone new quasiparticle state corresponding to one singleold free electron state, so while the electron's propertiesare profoundly modified, we have not yet managed to takeit apart into separate pieces. As far as I can tell, thiswas first envisaged by Elliott Lieb and Fred Wu5 in aspectacular theoretical paper published in 1968 under themisleading title "Absence of Mott Transition in [the One-Dimensional Hubbard Model]." The Mott transition is atheory of the metal-insulator transition for certain typesof materials. John Hubbard had proposed his model of alocal repulsive interaction in a tight-binding band as anappropriate system for modeling magnetism and metal-insulator transitions (it was very similar to a model I hadput forward in 1959 to describe the Mott insulating anti-ferromagnet), and the subject of Mott metal-insulatortransitions had (and has) continued to be rife with confu-sion and controversy. Lieb and Wu's tour de force was anexact formal solution of the one-dimensional version ofthis rather realistic model of interacting, spin-degenerateelectrons, for all parameter sets and sizes (including theN —> oo thermodynamic limit, which is relevant for describ-ing real solids).

Whether they made the point given in their title ismoot—what they showed was that the case with exactlyone electron per site is always an insulator, and all otheroccupancies are metals, and whether that is a "Motttransition" or not is a question that is semantic, not

substantive—but the real kicker came when they de-scribed the excitation spectrum. They wrote that "we findthree types of excitations," which can be interpreted astwo types of spinless charge excitations (holes and parti-cles) and one type of neutral spin excitations. Please notethat none of these are at all like electrons! Lieb and Wureferred back to similar phenomena in a 1963 paper byLieb and Werner Liniger, and in a 1967 paper by C. N.Yang, but none of those are as conclusive or deal withphysical models, though both papers indeed pioneered themathematical techniques used.

It is an understatement to suggest that the signifi-cance of this discovery was not appreciated for many(perhaps twenty) years. For the first time, it was shownthat in a particular interacting many-electron system, thesingle added electron or quasiparticle could be simply nota stable component of the system: that it could and woulddecay into two or a number of constituent parts, and thatthe system's true, exact excitation spectrum is not elec-tron-like at all.

There was a second feature, again foreshadowed inmany papers going back even to early work by Sin-itiroTomonaga and Joaquin Luttinger in the early 1950s: Inspite of breaking apart, the electron still has a Fermisurface (in this one-dimensional case, two "Fermi points"),which are the only points in electron momentum spacewhere excitation energies vanish—and these Fermi pointsremain at the same locations in momentum space. Thislast property is known as Luttinger's theorem and it issurprising that it occurs for this bizarre system: The proofof Luttinger's theorem uses perturbation theory methodsthat should not be valid in this case.

Let us not suppose the Hubbard model is an isolatedinstance of charge-spin separation. In 1974, Alan Lutherand his collaborators6 (following ideas of Dieter and UrsulaSchotte) developed a much simpler and more generallyapplicable, but less rigorous, technique called bosonization,which allows many variants of the original model to betreated, and the above phenomenon of "charge-spin sepa-ration" seems to be almost ubiquitous for physical one-di-mensional systems. (Bosonization is essentially the de-scription of a fermionic entity as a function of bosonic

OCTOBER 1997 PHYSICS TODAY 43

FIGURE 2. TRANS-POLYACETYLENEhas alternating double and singlebonds (green) between its carbon

atoms (black). Although the electronsare not as localized as this diagram

implies, the C = C bonds are shorterthan the C-C bonds and the electrondensities form a Peierls charge-density

wave along the chain.

operators—the requisite anticommutators typically arisefrom an interplay between normal ordering and thebosonic commutators.) Groups in the USSR also devel-oped techniques relying on resummation of perturbationtheory, by use of which Igor Dzialoshinsky and AnatolyLarkin in 1974 even gave a formal description of exactlyhow the electron breaks up—that is, they gave a formalGreen's function, or propagator, for the electron, valid inthe asymptotic long-time limit.

The remarkable thing to me has not been the bur-geoning of beautiful theoretical work on these one-dimen-sional systems, which continues to this day, but the failureof most of the theoretical work to connect with experimentson real one-dimensional or quasi one-dimensional systems.As far as I know, only in the past year has a single instanceof charge-spin separation been experimentally indicated,and that was on a very simplified example (angle-resolvedphotoemission spectra of SrCuO2).

7 Most of the experi-mental work on these substances is still interpreted ac-cording to the free-electron picture. Theorists have putgreat effort into calculating asymptotic singular behaviorsof perfect systems at low temperatures, but what areavailable experimentally are transport properties of some-what impure systems at finite temperatures. Most sur-prising, the interesting and essential question of how onedeals with the transport of electrons that decay almostimmediately into separate, distinct excitations of chargeand spin (with, presumably, very different scattering prob-abilities and transport properties), has been even statedonly by our group, starting with my work with Yong Renin 1991 and continuing to the present.8 For variousreasons the one-dimensional case is a difficult one, and Icover these ideas under the two-dimensional case below.

Trans-polyacetyleneLieb and Wu and their successors dealtprimarily with the case of repulsive,Coulomb interactions among electrons,because of their interest in the Motttransition. In the Hubbard model therepulsive interactions are representedby a repulsive potential U. Attractiveinteractions between electrons are in-variably the consequence of couplingto lattice vibrations, and the frequen-cies of these vibrations, or phonons,introduce a second, lower-frequencydynamical scale. The "attractive U"Hubbard model is an oxymoron, andone must always include the dynamicsof the phonons or possibly other at-tractive mechanisms to arrive at real-istic results. The second—and first

truly experimental—case of fractionalization of electronsresulted from just such a system with attractive interac-tions between electrons mediated by phonons: the case ofpolyacetylene, culminating in the classic paper of Wu-PeiSu, Robert Schrieffer and Alan Heeger in 1979.9

The polymer chain based on acetylene takes twoforms. The interesting one is called trans-polyacetyleneand, in the ground state, is an insulator in spite of havingjust one odd TT electron in its valence band per molecule,because the electrons couple strongly to the moleculardisplacements. They form stronger bonds—that is, theydisplace the atoms toward each other—between successivepairs of atoms along the chain, giving us what is calleda charge-density wave (CDW), which doubles the peri-odicity over which the chain repeats. This may be drawnschematically as shown in figure 2. The pair bonds inthe figure each imply two electrons but in fact the electronsare by no means totally localized in the regions betweenthe closer pairs of carbons. (See the article on CDWs byRobert E. Thorne in PHYSICS TODAY, May 1996, page 42.)By doubling the periodicity, the tight-binding energy bandopens up an energy gap at the momentum vector IT I acorresponding to the double periodicity. (See figure 3.)

Now, we ask, what happens if, starting from one endof the chain, we pair atoms 0 and 1, 2 and 3 and so on,and from the other end we pair atoms In + 1 and In andso on, so that they don't match in the middle? The oddatom—say number 4—may keep its electron and be neu-tral, but have a spin; if an electron is added or subtracted,however, the atom is spinless but charged! (See figure 4.)

Schrieffer and his coworkers showed that these twotypes of solitons existed and were locally stable, and thateach was quite extended and mobile, so that dopingpolyacetylene could lead to a sample with charged mobilecarriers but no mobile spins, for instance. These carriers

i

GoAw

^f

E 0 ^L MOMENTUMa

o i

O

^ /

a.

<~\J Gap

— • "

MOMENTUM

FIGURE 3. ENERGY BANDS IN POLYACETYLENE develop gaps because of thecharge-density wave, a: In the absence of a CDW, there is a single tight-bindingenergy band, corresponding to the periodicity a (the distance between successivecarbon atoms), b: The CDW doubles the period of the chain to la and an energygap opens up at the momentum -rr/a that corresponds to this period.

44 OCTOBER 1997 PHYSICS TODAY

FIGURE 4. DEFECT IN POLYACETYLENE CHARGE-DENSITY WAVE leads to an odd spin at the defect (blue,as shown) if the electron numbers are balanced, or to a spinless charge at the defect if they aren't balanced.In either case, the chargeless spin or the spinless charge behaves like a particle.

are quantum particles, in a real sense. Although thisproposal initially met with considerable resistance, theexperimental existence of free solitons with quantum num-bers not corresponding to those of an electron—that is,fractionalized electrons—has been checked out experimen-tally for this system. As Schrieffer also pointed out, ifthe CDW does not have double periodicity but triple ormore, the charged version of the soliton—a so-called dis-commensuration—could carry fractional charge, but thishas not been confirmed in any experimental one-dimen-sional case that I know of.

Fractional quantum Hall effectIn 1982 the most exciting and spectacular example ofelectron fractionalization appeared in physics through anentirely experimental breakthrough: the discovery of thefractional quantum Hall effect by Daniel C. Tsui, HorstStormer and Arthur C. Gossard at Bell Laboratories.10

A few years earlier, Klaus von Klitzing, GerhardDorda and Michael Pepper had discovered the (integer)quantum Hall effect through measuring the Hall effect inhigh magnetic fields on the two-dimensional electron gasin a high-mobility surface inversion layer on silicon.Sharp plateaus of the Hall voltage as a function of fieldoccur at quantized values of the Hall constant,

Ik (1)

where j is an integer.This remarkable observation (and the accompanying

near zeros in the conventional conductivity, which hadbeen observed earlier by Tsui and coworkers) was soonexplained ingeniously by Robert B. Laughlin,11 using theidea that almost all of the electron states are localizedexcept near the centers of the quantized Landau levels.

What Tsui and coworkers saw in 1982 was madepossible by Ga-GaAs heterostructures in which the two-dimensional electron gases had vastly improved mobility.In these the plateaus occurred not only at integer valuesof j (in equation 1) but at rational fractional values withvarious odd denominators: RH = h/e2v, with v= V3, %, 4/3;H, %; V7 and so on, as well as at 1, 2, 3 and so on. (Seefigure 5.)

In each case the center of the plateau occurs at aratio of magnetic field B to two-dimensional density ofelectrons n such that n/B = v/<$>0, with 4>0 being thequantum unit of magnetic flux,

he

That is, for every unit of flux there are v electrons; in thecase of v = V3, for example, there are three flux quantaper electron.

Laughlin is again responsible for the correct explana-tion of these fractional values.12 He pointed out that hisexplanation for the integer quantum Hall effect could berecast by describing a filled quantized Landau level as anincompressible electron fluid because once a Landau levelis filled with exactly one electron per flux quantum, thenext electron must go into the next higher energy level,which is ho)c = heB/mc higher in energy. He then ingen-iously showed that he could also make a plausible many-body wavefunction that is an incompressible electronquantum liquid at V3, V5, . . . l/(2re + 1) of a filled Landaulevel.

To get an idea how he could do this, notice that thetwo-dimensional coordinates, xj and y,-, of an electron canbe combined into the single complex coordinate

A wavefunction in the lowest Landau level can be de-scribed (aside from a fixed single-particle factor) as anypolynomial function of Z, that is not a function containingany factors of Z*, the complex conjugate of Z. In anymany-body wavefunction that satisfies the exclusion prin-ciple, no two particles can be at the same point, so thefunction must necessarily have a factor (Zj-Zt) for anypair of particles j and I, and a filled Landau level actuallyhas one factor (Z} - Zt) for every pair.

Now, one could satisfy the exclusion principle alsowith factors (Zj-Zi)3 (the exponent must be odd forfermions). If one does just that for every pair, it turnsout that the magnetic field must be three times as big forthe same number of particles. But such a wavefunctionwill be very much favored if the particles repel each otherat short range, as of course they do. With ingenious andextensive numerical calculations, Laughlin showed thestability of the fractional states—that is, the states with(Zj - Zi)n factors for odd n and the usual quantized valuesof the flux.

These states behave as though they are made up offractional electrons. But can we construct a fractionalelectron state? Laughlin also solved this problem in hisinitial paper by using the following ingenious construction.

He began with a wavefunction where all the electronpairs have (Zj - Zt)

3 factors. If an infinitesimal solenoid

OCTOBER 1997 PHYSICS TODAY 45

5/3 7/5 4/5 5/7 3/5

50 100

B (kilogauss)

150

FIGURE 5. QUANTUM HALL EFFECT (upper curve) andmagnetoresistance (lower curve) in a GaAs-Ga(Al)Asheterojunction. Plateaus occur in the Hall resistivity p^ atvalues h/ve2, where v is an integer or an odd-denominatorfraction. Zeroes or smaller dips in the longitudinal resistivitypxx coincide with these plateaus. The scale along the top ofthe figure indicates the filling factor (v) at the correspondingmagnetic field. The effects at fractional filling factors areexplained by Landau levels being filled by states that are likefractional electrons. (In two dimensions, longitudinalresistivity is "per square" because a square's resistance isindependent of its size.)

with one quantum of magnetic flux were to be added atsome point x0 + iy0 = Zo, every electron would have toprecess around it so the wavefunction has to have a newfactor (Zj - Zo) for each j . The electrons are all pushedout a little bit by the new flux, and it turns out that thehole created by their outward motion contains just one-third of an electron, so one has a soliton with charge+V3. By removing a flux quantum (or introducing a nega-tive one) one may also add one-third of an electron. Fora long time, it was not clear whether one had establisheda physical meaning for these fractional charges, but ex-tensive and patient effort by experimenters such as Tsuiand his coworkers has left us quite sure they exist.

In addition to these fractional states, a zoo of moreor less complicated nonclassical excitations of these ex-traordinarily perfect two-dimensional electron gases hasbeen postulated and in most cases demonstrated in thecourse of the intervening years. An ingenious schemepioneered by Steven Girvin and Allan MacDonald13 is toassociate a fictitious flux of the sort described above witha particle. For each quantum of fictitious flux, onechanges the sign of the statistics, so that one or threequanta plus an electron act like a boson, and the plateaus

(at which pxx = 0) act like Bose condensates of the fictitiousparticle, when one compensates the external field with anodd number of fictitious quanta. But at v = V2, the com-pensation occurs for two quanta, and the particle is afermion again. (See PHYSICS TODAY, July 1993, page 17.)Present theoretical ideas (due to Nicholas Read, BertrandHalperin, Patrick Lee and others) propose that thesefermions have an actual Fermi surface, which may havebeen observed.

Another type of fractionalized particle occurs in theso-called Hall edge states, states that travel around theboundary of a quantum Hall fluid.14 They behave like aspecial kind of one-dimensional system that, in the frac-tional v case, has fractional solitons as its basic excitations.(See PHYSICS TODAY, June 1994, page 21, and September1996, page 19.)

High-To cupratesThe glorious successes of the theory for the fractionalquantum Hall effect contrast with the bitter controversyand lack of consensus associated with the high-T^ cuprates.Nonetheless, a substantial number of theorists and ex-perimenters in this field agree that it is likely that in thecuprate normal metal (above Tc), the transport propertiescannot be understood with a conventional quasiparticletheory. The most striking of many unusual observationsis the observation of two transport relaxation times T inyttrium barium copper oxide (as well as other systems inwhich good single crystal measurements are available).The evidence is shown in figure 6.

It is a well-known feature of optimally doped cupratesthat the resistivity p in the plane of the cuprate (whichin simple kinetic theory is p = m / ne2T, where n is the totalnumber of electrons of effective mass m, and T is the meanfree time) is roughly proportional to the absolute tempera-ture, with mysteriously little or no intercept or "residualresistance." The coefficient m can be determined frominfrared measurements, so we actually find that

h

to within 50% or so, where rp is the relaxation time asdetermined by resistivity. This rate, l/rp, is a very largerelaxation rate. (From infrared measurements, one findsthat, at high frequencies, hlT^hw, also.)

There is a second simple way of determining T, fromthe Hall effect. The angle through which an electron'smomentum precesses in a relaxation time TH is the Hallangle between current and voltage in a magnetic field,and is

a eB

flH-*VH-—TH,

where wc is the cyclotron frequency, the frequency of theprecession. But when we measure dH in a wide varietyof samples, we find, quite reliably,

So there are two separate relaxation times in the cuprateplanes, one of about h/kT for accelerations parallel to theelectronic momentum (and hence perpendicular to theFermi surface), and one, somewhat longer and propor-

46 OCTOBER 1997 PHYSICS TODAY

G

3.

WPi

180

160

140

120

100

80

60

40

20

0

-

/

/

y

11 I 1 1 1 1 1

50 100 150 200 250 300

TEMPERATURE T(kelvin)350

1-1

I

o

tional to T 2, for accelerations perpendicular to the mo-mentum and parallel to energy surfaces. I believe thatthis is a consequence of decay of the electrons into separatespin and charge excitations. The time rp is the relaxationtime for the electron to fall apart (and is proportional tothe electron's energy), while TH is the relaxation time ofthe separate excitations (or proportional to it).

The reason15 that the two relaxation rates differ isthat the magnetic field does not affect the relative popu-lations of charge (holon) and spin (spinon) excitations, sothe magnetic field doesn't change the equilibrium amongelectrons, spinons and holons. The electric field, on theother hand, accelerates the electrons to higher energy andthe accelerated electrons decay with their faster breakuprate. This decay affects the resistivity only because thecharge excitations (holons) are rapidly scattered and rap-idly lose their momentum: The electron is acceleratedonly as long as it stays together.

The next centuryAs we remarked earlier, the transport theory of compositeelectrons is unfamiliar, and standard methods (such asfluctuation-dissipation theory) do not apply. These stand-ard methods have served us faithfully over many years,and it is very hard, particularly for young physicists who

FIGURE 6. TRANSPORT RELAXATION RATE in the high-7c

cuprate YBCO in the normal metal state above Tc, shows twodifferent temperature dependences depending on how it ismeasured. Measurements based on resistivity reveal lineardependence on temperature from Tc = 93.5 K to about 300 K(a), but those based on the Hall angle 9H yield a rateproportional to T2 (b). Such behavior may be evidence thatthe electrons decay into separate spin and charge excitations.(Adapted from N. P. Ong, Y. F. Han, J. M. Harris, in High-Tc

Superconductivity and the C60 Family, Sunqui Feng, Hai-CanReng, eds., Gordon and Breach, 1995.)

have never known anything else, to abandon the appar-ently trustworthy certainty of the textbook formulas. Mis-application of a valid theorem proved by Larkin, whichsays that holons and spinons must move together, is oneof several excuses theorists have been using to ignore theproblem. (The theorem, while true, does not constrainthe motion of holon and spin excitations, only of excitationsplus backflow of the Fermi liquid as a whole, which cancarry additional quantum numbers.) Some theorists haveattempted to fit data like those in figure 6 to complicatedsingle-particle theories, but such attempts invariably leadto implausibly complex band structures, not to mentionthat they are usually highly parametrized fits to singlesets of data, while the experimental picture is very broadlyapplicable and includes magnetoresistance and AC Halleffect data as well.

It is very appropriate to end this article on an open-ended note. Fragile electrons in the quantum Hall regimehave been one of the great successes of the last decade ofthe electron's first century: With many imaginative andingenious schemes, theorists seem to have them undercontrol. But the apparently simpler problem of motion ofspin-charge separated electrons in one- and two-dimen-sional systems has only tentatively, and controversially,begun to yield to our understanding, and will surely begood for a decade or so of the electron's second century.

References1. R. Peierls, Phys. Z. 30, 273 (1929).2. A clear exposition of these coherence effects is given in J. R.

Schrieffer, Superconductivity, Benjamin, New York (1964).3. W. J. Tomasch, Phys. Rev. Lett. 15, 672 (1965); 16, 16 (1966).4. Well described in articles in R. D. Parks, Superconductivity,

Dekker, New York (1969).5. E. Lieb, F. Y. Wu, Phys. Rev. Lett. 20, 1445 (1968).6. A. Luther, E. Peschel, Phys. Rev. Lett. 32, 992 (1974).7. C. Kim, A. Y. Matsuura, Z.-X. Shen, N. Motoyama, H. Eisaki,

S. Uchida, T. Tohyama, S. Maekawa, Phys. Rev. Lett. 77, 4054(1996). C. Kim, Z.-X. Shen, N. Motoyama, H. Eisaki, S. Uchida,T. Tohyama, S. Maekawa, to appear in Phys. Rev. B.

8. An attempt to pinpoint some nonclassical behavior in quasione-dimensional systems has been begun; see D. G. Clarke, S.Strong, J. Phys. Cond. Matt. 48, 10089 (1996).

9. W. P. Su, J. R. Schrieffer, A. J. Heeger, Phys. Rev. Lett. 42,1698(1979); Phys. Rev. B 22, 2099 (1980).

10. D. C. Tsui, H. L. Stormer, H. C. Gossard, Phys. Rev. Lett. 48,1559(1982).

11. R. B. Laughlin, Phys. Rev. B 23, 5632 (1981).12. This and related work is described in the book The Quantum

Hall Effect, R. E. Prange, S. M. Girvin, eds., Springer, NewYork (1987).

13. S. M. Girvin, A. MacDonald, Phys. Rev. 58, 1252 (1987). Animportant generalization is due to Jain; see J. K. Jain, Phys.Rev. Lett. 63, 199 (1989); Phys. Rev. B 40, 8079 (1989).

14. X.-G. Wen, Phys. Rev. Lett. 64, 2206 (1990).15. For more discussion, see P. W. Anderson, The Theory of Super-

conductivity in the High Tc Cuprates, Princeton U. P., Prince-ton, N.J.( 1997). •

OCTOBER 1997 PHYSICS TODAY 47