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Spin-Peierls and spin-Hquid phases of Kagomb quantum antiferromagnetsJ. B. Marston and C. ZengLaboratory f Atomic and Solid State Physics, CorneII Vniversi?v, Ithaca, New York 14853 -2501The ground state of the spin-i nearest-neighborHeisenbergquantum antiferromagnet on theKagomC attice probably lacks &in order; therefore, conventional spin-wave analysisbreaks down. To ascertain the ground state, we instead use a systematic l/n expansionwith

.SU(n) fermions. Two distinct states occur in the large-n limit, dependingon the sizeof the biquadratic interaction z When ?=O, there are an infinite number of degenera te.ground states consisting of disconnecteddimers. At finite n, however, this degeneracy sbroken by local resonance. n qontrast, a globally resonating chiral spin-liquid phasewith nospin-Peierlsmodulation-is. he likely large-n ground state at sufficiently large f Forintermediate values of J, a phase ransition from the dimer state to the chiral phaseoccurs asthe temperature &-eases. At a higher temperature, here is a second ransition to aparamagneticstate. We comment on the possibility that these phasesare experimentallyrealized by the nuclear magneti.q.motientsof a second ayer of 3He atoms lying on a graphitesurface.

Novel ground states of low-spin quantum antiferro-magnetsmay arise on l ow-dimensional, highly frustratedlattices. If spin order is absent, he usual spin-waveexpan-sion about the ordered state-breaks own and other meansmust be employed o find the ground &ate. (By spin orderwe mean he existenceof a local spin moment. N6el orderon the square attice is the simplest example of such or-der.) The spin-4 nearest-neighborHeisenberg antiferro-magnet on the two-dimensional KagomC lattice exhibitsthese eatures:The small value of the spin, the low coor-dination number (z&4), and the frustrating interactionsall work against he formation of spin order. Indeed, exactdiagonalization studies of clusters of up to 21 sites suggestthat the local spin moment vanishes.*However, it is dif-ficult to extract the nature of the order (or lack thereof)that occurs n the thermodynamic limit from thesestudies.Here, we report on a different approach that relies on asystematic expansion n powers of l/n, where n labels ageneralizationof the physical SU(2) spin to an antisym-metric, self-conjugate, epresentationof the group SU( n).[Note that n > 2 doesnot correspond o a higher-spin rep-resentation of SU(2).] This technique was previously ap-plied to the square attice Heisenbergantiferromagnet,butit is particularly well suited for the KagomC problem be-cause spin order never occurs in the exactly solvablelarge-n imit. Though the expansionparameter s of order1 for the real problem, our approach makes definite pre-dictions about possible ordering and the phase diagramthat can then be tested by other means.

ten as a four-fermion interaction involving the SU(n> de-struction and creation operatorsc,, and cp, where x labelsthe lattice site and a: = 1,2,..., is the flavor index. It is thendecomposedvia a Hubbard-Stratonovich transformationby i ntroducing complex fields xXY hat live on the links ofthe lattice. The phasesof the xXY ields transform as spatialgauge ields under local U( 1) gauge ransformations of theferinions; and the time component of the gauge ield ap-pears n the guise of a Lagrangemultiplier field (I&) thatenforces he local particle-number constraint (n/2 fermi-ons livl on each site, where ~1~s an even integer). Thelarge-n solution then corresponds o the problem of findingthe saddle point of the following functional integral repre-sentation for the partition function:Z=&I [4lev - n&&541.Here, the effective action is given by an integral over theGrassman ields:

ew - ~S&741 = s [dc*l Idclew - Nx,~,c*,cl,where the imaginary time action is

+ 2 (2) xxy12Cwv*ncxaH.c.)WY) \J 1 - IMany different SU(n) generalizations of the spin-i The implicit sum over the repeated lavor index makesSU( 2) antiferromagnetare possibleon unfrustrated, bipar- each term in the action of order n. By choice of gauge,tite lattices; however, those approaches based on (pX an be set to zero. The integration over the fermions canSchwinger bosons require different representations of in principle be done exactly (since they appear only atSU( n) on the two sublattices and thereforedo not work on quadratic order) and the problem of finding the correctthe Kagomk lattice. In contrast, the fermionic path- saddle point reduces o one of determining the static con-integral approach works on all lattices and we employ it figuration of the xXxy ields which minimizes the effectivehere. Since this method was discussed n some detail in action. Clearly, xXY unctions as the order parameter. SinceRef. 2, we limit our discussion o key points. The nearest- it is invariant under global SU(n) transformations, spinneighborHeisenbergspin-spin coupling JSx-Sy is first writ- order is impossible n the large-n imit.5962 J. Appt. Phys. 69 (8), 15 April 1991 0021-8979/91/085962-03$03.00 0 1991 American Institute of Physics 5962

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In a remarkable paper,3 Rokhsar proved that the bestsaddle-point configuration on almost all lattices consists ofa dimer covering of the lattice. That is, lxXYl is nonzero(and equal to 3= J/2) on one and only one of the linksattached to each site. Rokhsars proof applies to theKagomC lattice, and since the coordination number z=4,only 4 of the links are nonzero in this phase. The dimer-ization corresponds to spin-Peierls modulation since onlyspins connected by a dimer are correlated via a singletbond. There are, however, an infinite number of ways tolay down the dimers and all of these states are degeneratein the large-n limit.To ascertain how l/n corrections break this degener-acy, we follow Read and Sachdev4and expand the effectiveaction for the xXY fields in fluctuations QXY about anygiven dimer configuration J&. The order n(6x)2 contri-bution was calculated in Ref. .4 for the square lattice, butdoes not lift the dimer degeneracy on the Kagome lattice.It merely corrects the free energy (which is order n) by aconstant of order 1 since ( (6~)~) = O( l/n). The firstterm that removes some degeneracy occurs at ordern (6~) 3 n the effective action expansion and involves hexa-gons that have dimers on three of the links and fluctuationson the remaining three links. Clearly, this term contributesto the total free energy only at second order [that is,n2( (a~)~)] because the quadratic part of the effective ac-tion is invariant under SxXY ) - Z&. Since it is secondorder, it always lowers the free energy (by an amount oforder l/n). Indeed, the hexagon behaves ike the benzenemolecule: Both lower. their energy through resonance.Other corrections to the free energy appear at order l/n,but are independent of the dimer configuration.Thus, the I/n term picks patterns which maximize thenumber of hexagons with three dimers. A simple countingargument now shows that the maximum fraction of suchhexagons (we call them perfect hexagons) is 4. Let N3,N3d, NC, and N6p be, respectively, the number of triangles,triangles without a dimer which we call defect triangles,hexagons, and perfect hexagons. It was shown in Ref. 5that Nsd = iN3 for any dimer. covering. According to theobservation that (1) exactly three defect triangles are at-tached to every perfect hexagon, and (2) none of six near-est hexagons around a perfect one can be perfect (and,consequently, no two perfect hexagons will share a com-mon defect triangle), we have 3Nep< N3& Combining theseobservations and using N3 .= 2N6, we obtain Nep < 2 NeFurther constraints on the 12 second-nearesthexagonsaround a central perfect one reveal how to construct pat-terns with the maximum fraction. First of all, only six ofthese 12 hexagons.are potential sites for another perfecthexagon. The six hexagons that are separated from thecentral perfect one by two triangles (bow ties) cannotsupport a perfect hexagon because he site in the center ofthe bow tie will not be attached to a dimer. Second, if welink the center of the central perfect hexagon to the centersof the two next closest possible perfect ones by two straightlines, then the angle between lines has to be either 120 or180.These two local rules generate various patterns which.we classify by the size of the respective unit cells. Note that

HG. 1. Unit cells with the maximumfraction of perfect hexagons wherethe thick l ines denote dimers. (a) 1%site unit cell with perfect hexagonsthat form a oblique lattice. (b) 36siteunit cell with perfect hexagons thatform a honeycomb lattice.

configurations that differ by a rotation of the three dimerson a perfect hexagon by 60 are energetically equivalent atorder l/n. We draw one of the several patterns for thesmallest possible unit cell (18 sites) and one pattern fittingin the next smallest unit cell (36 sites) in Fig. 1.Although we have not examined higher-order (beyondl/n) corrections, we fully expect these terms will break theremaining degeneracy and select out a phase with finitedegeneracy. These terms involve increasingly large reso-nance patterns that eventually link the unit cells together.Thus, the dimers should solidify at zero temperature into acrystalline solid built out of unit cells constructed accord-ing to the above rules. Clearly, this long-range dimer orderwill melt at a nonzero temperature since the dimers do notbreak a continuous symmetry, but rather the discrete two-dimensional lattice translational symmetry.As we mentioned earlier, there are actually many dif-ferent SU(n> generalizations of our SU(2) Hamiltonian.Even in the context of our fermionic path-integral ap-proach, we can add a nearest-neighbor biquadratic term tothenHamiltonian, without changing the physical SU(2)limit because 2 ( S,S,J2 = - ( 3 /2)S,S, + const forspin f. Clearly, adding the biquadratic interaction 2 justamounts to a trivial renormalization of the usual bilinearspin-exchange constant J (at least if 7~ W; otherwise, thesystem becomes a ferromagnet) . But for n > 2 this interac-tion represents new processes hat simultaneously inter-change two flavors of fermions on one site with two onanother neighboring site. Thus, the inclusion of the biqua-dratic term furnishes us with a one-parameter family ofSU(n) theories with the same SU(2) limit. For J>O the,bi-quadratic term frustrates spin-Peierls order because tfavors states that spread out singlet correlations uniformlyon all the links. Indeed, upon decomposing the biquadraticterm via a two-stage Hubbard-Stratonovich transforma-tion* (by introducing new real-valued link fields QXY), wediscover that flux-type states with uniform lxXYl are vi-able. (Bicubic, biquartic, and higher terms can also be in-cluded, but do not lead to any other new ground states. Ifsuch a state persists down to n=2, then it realizes Ander-sons resonating valence bonds (RVB) in the sense hat

5963 J. Appl. Phys., Vol. 69, No. 8,15 April 1991 J. B. Marston and C. Zen9 5963Downloaded 06 Dec 2002 to 128.148.60.17. Redistribution subject to AIP license or copyright, see http://ojps.aip.org/japo/japcr.jsp

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each spin on a site participates equally in singlet correla-tions on all four links emanating from the site.We study the crossover from the spin-Peierls phase tothe globally resonating RVB states by integrating out thefermions in a number of different trial configurations of thexXYand Qp,, ields. Since we are also interested in the effectof nonzero temperatures on the crossover, we compute thesaddle-point free energy by using a Fermi-Dirac distribu-tion for the fermion occupancy. The free energy per site (attemperature T and J= 1) isF(x,@) = =2an( 1x1 - (P2/T ) - (n/2)b[~,,;T].Here, a denotes he fraction of links with nonzero 1xX,); nthis case,a = i for the spin-Peierls phase and a= 1 for theflux phases.Also, RX,= ( 1 + 2QX,) xx, and b[&,;T] isthe free energy at half-filling of the fermions that movefrom site to site along the nonzero fi, links. We are as-suming here that IxXYl and QXY ake on only two possiblevalues on any given link: zero or a constant. We thenmust minimize F with respect to the ,yXY ields, but maxi-mize it with respect to the Cp, fields (see Ref. 2).

The best RVB state (a= 1) we found has a flux of+ ?r/2 passing through the each triangle on the KagomClattice and zero flux in the hexagons. At zero temperature,this uniform flux phase drops below the spin-Peierls statein energy when 3>.0.92J. It is clearly twofold degeneratebecause t breaks time-reversal symmetry (the state withflux - r/2 has the same energy). This statedoes not breaktranslational or rotational symmetry, and it has no spin orspin-Peierls order. Rather, it is a chiral spin liquid: ForSU( 2) the expectation value (Sl&2~S3) is nonzero andconstant on every elementary triangle (with vertices atsites 1, 2, and 3). The fermion. spectrum is completelygapped, and so we expect the spin-spin correlation functionto exhibit exponential decay in the chiral phase.The state with flux r through the hexagons and flux7r/2 through the triangles and the state with no flux at allboth have higher energy, regardless of the temperature.This result is in accord with heuristic rules found by Rokh-sar.6He approaches he problem of finding the optimal fluxstates by first breaking up any lattice into its smallestclosed polygons. He then notices that disconnected poly-gons with an odd number of sides (like the aforementionedtriangles) prefer to have flux ?r/2 to minimize the fermionenergy. Polygons. with 4m sides, where m is an integer,want flux r, and finally polygons with 4m + 2 sides (thehexagons) prefer zero flux. Indeed, we may view the zeroflux phase as an excited state of the chiral flux state withdefect antiflux tubes passing hrough the triangles that can-cel out the ~-/2 flux. Similarly, the phase with uniform 7~/2Aux in each triangle and rr in the hexagons can be seen asan array of flux defects passing through the hexagons.Rokhsars rules do not distinguish between the statewith uniform ?r/2 flux per triangle and the state with astaggered lux of *r/2. In fact, the uniform phase is en-ergetically preferred. We gain some understanding of thisfact by studying the Fermi surface of the staggered statewith flux f -s-/2 in the upward pointing triangles and- r/2 in the downward ones. In the gauge n which only

the horizontal links are imaginary (xXu = ilxl with the linkorientation pointing either all to the left or all to the right),the Fermi surface (which is at zero energy because ofparticle-hole symmetry j has the shape of an isosceles.ri-angle with vertices at momenta (0, f %-/V5) and (~~0). Ap-parently, the complete gap in the fermion spectrum of theuniform chiral phase orces the fermions into lower energystates than in the staggered phase. It also means that theuniform chiral phase is likely to be locally stable againstS,X.,~ erturbations at arbitrary wave vector.If we choose 7 to be slightly smaller than the criticalvalue required to favor the chiral spin-liquid state over thespin-Peierls phase at zero temperature, then we find a first-order transition to the chiral liquid state at a nonzero tem-perature. For example, at f=O.9OJ the transition occurs ata temperature of T~0.05J. The liquid phase then survivesup to T=J/4, where a second-order transition to a para-magnetic (gas) state marked by )xXYl = aXs = 0 occurs.(A small-IX1 expansion of the free energy shows that thisparamagnetic transition always occurs at-T= J/4, regard-less of the saddle point or the value of J, Note that theentropy from different dimer configurations is of order 1,and so its contribution to the free energy can be neglectedin the large-n limit.)The above spin-Peierls solid to chiral liquid and chiralliquid to paramagnetic gas transitions may correspond tothe two peaks in the specific heat seen in an exact diago-nalization of a 1Zsite cluster5 and an approximatedecoupled-cell Monte Carlo calculation, but further nu-merical work is needed to test for the existence of spin-Peierls or chiral order. In nature, the KagomC system maybe realized by a second ayer of He atoms adsorbed onto agraphite substrate at a particular coverage density.5 Thenuclear spins of the 3He atoms interact via simple antifer-romagnetic two-atom-exchange processes and ring ex-changes nvolving more than two atoms, but the nearest-neighbor Heisenberg antiferromagnet may be a goodstarting point for further studies. In fact, experiments showat least one peak (entropy arguments suggest there areactually two peaks) in the specific heat at a temperature ofabout 2.5 mK.8 It may be necessary o include other spin-exchange erms in a realistic model Hamiltonian, and themethods developed here can easily be applied to more com-plicated systems hat possess lobal spin-rotational symme-try. We thank V. Elser, C. Henley, T. Hsu, and D. Rokhsarfor helpful discussions. This work was supported by anIBM postdoctoral fellowship (J.B.M;) and NSF GrantNo. DMR-88-18558 (C.Z.)

C. Zeng and v. Elser, to appear in Phys. Rev. B 42, 8436 (1990).J. B. A4arston and I. Affleck, Phys. Rev. B 39, 11,538 (1989).D. S. Rokhsar, Phys. Rev. B 42, 2526 (1990).4N. Read and S. Sachdev, Nucl . Phys. B 316, 609 (1989).*V. Elser , Phys. Rev. Lett. 62, 2405 (1989).D. S. Rokhsar, Phys. Rev. Lett. 65, I506 (1990).7J. B. Marston and D. S. Rokhsar (unpublished).D. S. Greywall and P. A. Busch, Phys. Rev. L&t. 62, 1868 (1989).M. Roger, Phys. Rev. L&t. 64, 297 (1990).5964 J. Appl. Phys., Vol. 69, No. 8,15 April 1991 J. B. Marston and C. Zeng 5964

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