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EUROPHYSICS LETTERS

Europhys. Lett. , 6 (4), p. 353-358 (1988)

15 June 1988

2e or not 2e: Flux Quantization in the ResonatingValence Bond State.

S. A. KIVELSON(*),D. S. RoKHsAR(**)(*)and J. P. SETHNA

(*) Department of Physics, S U N Y at Stony Brook, Stony Brook, N Y 11794, U S A(**) Laboratory of Atomic and Solid State Physics, Clark HallCornell Univ ersity, Ithaca, N Y 14853-2501, U S A

(received 29 December 1987; accepted in finalform 5 April 1988)

PACS. 74.00 - Superconductivity.PACS. 74.20 - Theory.PACS. 75.105 - Heisenberg and other quantized localized spin models.

Abstract. - The * resonating valence bond. (RVB) st at e has been proposed as th e basis for anexplanation of high-temperature superconductivity. Recently, we have described the chargeand spin excitations about this sta te, and have show n tha t th ey a re solitons, precisely analogousto tho se found in polyacetylene. Since th e charged solitons are+e bosons, it is natural to askwh eth er flux quantization will occur in unitso f hcl2e, as in traditional BC S supercondu ctivity, orwill come only in larg er units of hcle. We show he re th at flux quantization in units ofhcl2e willoccur unless a condensation of cooperative ring exchanges occurs analogous to th at found in th efractional quantized Hall effect.

Resonance, the descriptionof the quantum ground sta teof a system (say benzene)as asuperposition of several bond configurations, has been the chemist's wayof incorporatingsome of the delocalization energy whichis naturally described using electronic energybands. L. Pauling originally introduced the .resonating valence bond. (RV B) st at e in th ehope of describing simple metals. T his st at e, a quantu m liquidof valence bonds, is kept fromcrystallizing into a Peierls s ta te by it s <<zero oint. or resonance energy. Anderson [l l hasrecently proposed that it may be realized in the recent high-temperature ceramicsuperconductors.

Here we consider a tight-binding model on a square lattice, withM electrons occupyingN > M sites. Since we are interested in topological properties (ra th er than energetics), w eneed not examine the true ground state -a ny sta te th at is adiabatically connected to th eground statewill do. Th e sta te s we will examine willbe coherent superpositionsof nearest-neighbor single t bond configurations, w here each electron participates in a singlet bond withone neighboring electron. We define RVB statesas those which can be constructedperturbatively from one of these superpositions(I).

(*) IBM T. J. Watson Research Center, P.O. Box 218, Yorktown Heights, NY 10598.( l ) We should caution th at this definition probably excludes the s ta te s discussed by Anderson and

collaborators, which have no gap to spin excitations. Such sta tes will have a la rge densityof neutralsolitons, while the unperturbed states we discuss have none.

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s. A. KIVELSON et al.: 2e OR NO T 2e: FLUX QUANTIZATION ETC. 355

internal degrees of freedom,so long as their extent is small compared to the hole in theannulus (so that their angular center of charge6, is well defined).

What goes wrong when we apply this argument to valence bond configurations? We canimagine writing a many-body wave functionY$ for the bonds as a superposition of bond

configurations, and label each configuration in term s of th e ce nte r6, and orientation of eachof the sin glet bonds. Can we use eq.(1) o cr eate a new wave function for flux@ + hclze? Thelack of orthogonality betw een th e bond configuration .position eigenstates. pre ve nts usfrom doing so. Chan ging the re lative phase of two ove rlapping pieces of th e wave functionchanges the normalization and th e potential en ergy a s well as th e kinetic energy. Two bondconfigurations a re orthogonalif th e solitons are in different places, bu t th e cen ter of chargecan move (and th e relative phase can change) without moving the solitons: as shown in fig.1,rearrangements of bonds with enclose the hole in the annulus can change the center ofcharge Ce, by Z.

Thus exchange rings which encircle the hole(fig. 1) change t h e m icroscopic Aharonov-Bohm flux periodicity of th e ene rgy fromhc12e to hcle. Does this exten d to flux quantization?For large ann uli, th e rearra ng em ent s which span th e hole have exponentially small overlapswith the original configuration ( - 2 - L , where L is the circumference); each of theircontributions to the flux-dependent energy is also exponentially small(5). There are,how ever, many distinct rea rran ge me nts which encircle th e hole-if the y are imp ortant toth e w ave function, then th e periodicity of th e macroscopic energ y could also behcle. This isprecisely analogous to t he physics of the fractional qu antum Hall effect, where exponentiallysmall contributions from an exponentially larg e num ber of exch ange loops add co herently tothe energy.

To make this analogy more precise, we turn to the soliton description of the resonatingvalence bond state . First, let us establish some conventions.It is useful to distinguish a re dand a black sublattice with a checkerboard convention: our singlet bonds always connect ared site to a neighboring black site. V acant site s can lie on either sublattice,so there are redand black charged solitons. We will be interested in ordered pairs{ A , B } of singlet bondconfigurations; the first element can be considered as a bra and the second as a ket in amatrix element (AId IB) .It is also useful to give a direction(6) to each bond: the bonds inth e A configuration a re directed from red to black, in theB configuration from black to red.

Suppose first t ha t th e em pty sites (charged solitons)are in the same places in the twoconfigurations. Draw both configurations on the sam e lattice. Since any occupied site sh are sexactly one A bond and one B bond, the drawing will decompose into nonintersecting

loops [61. (If a particular bond is part of both configurations, it will form a trivial loop of noenclosed are a.) Th e directions of th e bonds give orientations t o each of th e loops, and allowone to reconstruct the original pair of configurations from the oriented loops.

These loops represent the electronic degrees of freedom left after the positions of thecharge solitons are set. We would like to define a quasi-particle wave functionP(R1 , .., RN-M) or the solitons, where the electronic degrees of freedom are considered

3

( 5 ) In the large-U limit, it is easy to see that it goes like tL /UL- l .(6) Many of the arguments presented here, distinguishing red and black sublattices and directions

of bonds, work only on bipartite lattices (e.g. square and cubic). The conclusions presented here arealso valid for triangular lattices; however, the arguments are a bit differen-ince the loops are nolonger oriented, the winding number is only defined modulo two. Also, we should note that an annularregion with an edge dislocation through the hole also has winding number defined modulo two (if thereare an odd number of sites on a loop, it can change a red soliton to a black one). Again, simplemodifications of the arguments give the same answer.

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356 EUROPHYSICS LETTERS

(+

1It

4I

II

( A s o l i t o n )

0 Q--

tII4I

t-c-

t

t

Fig. 1. Fig. 2 .

Fig. 1. - Loop around the hole. Two valence bond configurations are shown, encircling a hole ofcircumferenceL = 8. If we lump the c harges into bonds of charge- e, the A configuration has bon dscentered at I9 = ((112) 2x/L), (512) 2x/L),..., ((Ul ) / 2 ) 2 x / L ) ) ) ,while the B configuration has bondscentered at e = ((312) 2x/L), (712)( 2 x / L ) , . , ((Ul )/2) 2x1~5)). he charges actually sit in the sameplaces in th e tw o configurations; none theless, th e ce nte r of bond ch arges e changes by x as one shiftsfrom A to B. 1

Fig. 2 . - Solitons ac t as sources and sinks for <<bon d-string>> .ne vacant site is different between t heAand B configurations. T he black soliton in th eA sublattice act s as a source, and th e black soliton in theB sublattice a cts as a source. As th eB soliton encircles th e hole in the annulus, i t leaves behind a loopas shown in fig. 1.

slave s t o t h e soliton positions Cjustas hey ar e slaves to the nuclear positions in helium). Theproblem (7) in this approach is the assumption that th ereis a unique .best>> lectronic statefo r each soliton configuration in a multiply-connected geometry.

(7) Th ere is an othe r less subtle bu t more serious problem of which the re ade r should be aw are. Inhelium, th e e lectrons can be removed using th e Born-Oppenheimer approximation: the electrons relaxfas t compared to th e nuclear motions. A t least for the simplest models for highT,, uite the reve rse istrue. The solitons move with hopping matrix elementt and the bonds rearrange via exchange

interactions &U << t: the solitons move fast, and t he bonds adjus t slowly. This may not b e the finalansw er (the quasi-particles m ay behave somewhat differently from < <b are> >acant sites) , but it seemsfor now t ha t a soliton quasi-particle description is not th e obvious one. On th e ot he r hand , we a reinte res ted in topological questions which should not change a s we make t he solitons heavy. E speciallysince this pap er an sw ers why the flux quantum given by the soliton charge is wrong, it seems naturalto examine the limit in which the solitons are the only important degrees of freedom.

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S. A. KIVELSON et al.: 2e OR NOT 2e: FLUX QUANTIZATION ETC. 357

To see this, consider moving one black soliton in theB configuration, keeping theAconfiguration fixed(fig. 2). Th e original blacksite for th e soliton has one bond p ointing out ofit ( the B bond) and noA bond pointing intoit (since th eA configuration has a soliton there ).Th e final position for th e soliton has one bond pointing in, and none o u t i tis a sink fo r.bond-string*, w he re th e original siteis a source. As we c a n y the soliton around the hole inth e annulus and back toits original position, we leave behind a loop encircling the hole(fig.1). Clearly, this nonlocal effect is missed in the quasi-particle description. In the bonddescription, th e solitons have fractional charge ; nonlocal effects of th is kind ar e typical offractionally charged quasi-particles[7].

For fixed soliton positions, we can classify the bond configurations by a topologicalwinding numb erW. he difference in winding numb er betw een tw o configurationsis the ne tnum ber of loops enclosing th e hole counterclockwise. We also w an t to assign rela tive va luesof W for configurations with differing soliton positions; clearly, configurations in which onlya few solitons have moved sho rt distances a nd only local bond configurations have changed

should have nearly the same winding number. However, because moving a solitoncompletely around th e hole changes the winding number by one, we m ust add to W a factorinvolving th e ce nterof charge to m ake a winding angle0 hich changes continuouslyas thesolitons move around.

Le t us imagine se ttingQ = 0 for some initial configuration of bonds (say, unstaggeredvertica l bonds every wh ere, with all th e solitons symm etrically disposed in one angu lar slicearound 6 = 0). Any st at e can be produced by suitable local rearra nge me nts of the bonds andsolitons. Let OF and 07 denote t he ang ular positions of th e red and black solitons, includingth e ne t num ber of times th ey encircle th e hole from thei r original positions. We can defineth e winding angle0 or a general rearrangem ent to be th e ne t angular motion of solitons

needed to produce the final configuration

If th e solitons in two configurations a re in th e same places, t he winding num berW definedabove equals the difference in0 etween the two configurations, divided by2x.

Th us th e bond configurations .remember, th e ne t num ber of solitons th a t hav e circledthe annulus. The quasi-particle wave function can depend on0 n addition to t he solitonpositions. Now, half the angular center of charge for all the solitons

is a single-valued functionof 0 n d R I , ..., R N - M , o

is a single-valued quasi-particle wave function.Do these two w ave functions have th e same e nergy, for large annuli? Roughly speaking,

we wan t to know the energy costof changing the relative phaseof the even and odd windingnumber sectors: flux will not be quantized in the larger unitshcle unless that energy cost

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358 EUROPHYSICS LETTERS

grows with the system size. We have seen (using both the bond and soliton descriptions)th at if th e energy costis E , the n time evolution for a timehlE will introduce significant bondrearra nge me nts which encircle th e hole in the annulus without a nex tflux of solitons (themixing sta tes w ith different winding numbers).If th e Hamiltonian is local ( i . e .not involvingproducts of creation and annihilation operators linking for distant sites), then a loop ofshifted bond s of lengthL involves powers of the Hamiltonian of orderL. For a loop to formwithout moving a soliton around, th e system has t o go through higher-energy interm ediatest at es around th e perim eter; th e contribution of such a loop to th e time evolution will go asexp [- aL] or some constant a. Th e number of paths aro und th e hole grows with a similarform, and one might imagine a phase transition (as is found in the fractional qua ntized Halleffect [5]) whe n larg e loops begin t o dom inate. Otherw ise, i t will tak e an exponentially largetime to excite large loops, and the hcle periodicity to the energy will therefore beexponentially small.

We have examined this question in two dimensions, using both a mean-field theory andby approximately mapping the 2-d lattice onto a1 + 1 dimensional continuum fieldtheory [81. Both approaches suggest that condensation of ring exchanges does not occur intw o dimensions. We have no constructive argum ent t h at flux quantization in units ofhcle isimpossible in a hypothetical resonating-valence-bond states with 3-d connectivity(*).

* * *These ideas were prompted by discussions withE. ABRAHAMSnd G. BASKARAN.he

connection betw een ring exchange and flux quantization was developed during t he Marchmeeting of th e A PS, in a discussion with AB RAH AMS, ASK ARA N,. W. ANDERSON,nd anumber of other physicists. We thank the Institute for Theoretical Physics at Santa

Ba rbara , t he solid st at e group at the Massachusetts In stitute of Technology, and th e appliedphysics group a t Stanford University for their hospitality during this work. The work w asfunded in pa rt by NSF g ra n ts n umb er DMR-83-18051 (SAK), D MR-85-03544 (DSR& JPS),PHY-82-17853, and DMR-84-18718. Two ofus (SAK and JPS) acknowledge support throughfellowships from the AlfredP. Sloan foundation.

(8) If the experiment is not difficult, we suggest measuring the flux quantum in Ba(Pb/Bi)03.

REFERENCES

[ ll ANDERSON. W., Science, 235 (1987) 1196.[2] KIVELSONS. A., ROKHSAR D. S. and SETHNA . P., Phys . Rev. B , 35 (1987) 8865.[3] YANG C. N., Rev. Mod. Phys . , 34 (1962) 694.[4] GAMMELP. L., BISHOPD. J., DOLANG. L., Kwo J. R., MURRAYC. A., SCHNEEMEYER . F. and

[5] KIVELSONS., KALLINC., AROVAS . P. and SCHRIEFFER . R., Phys . Rev. Lett . ,56 (1986) 873.[6] KASTELEYN . W ., J . Math. Phys . , 4 (1963) 287.[7] KIVELSONS. and R&EK M., Phys . Lett. B , 156 (1985) 85.

[8] KIVELSON . A., work in progress.

WASCZCZAK. V., preprint.