# Zhihao Hao and Oleg Tchernyshyov- Fermionic Spin Excitations in Two- and Three-Dimensional Antiferromagnets

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Fermionic Spin Excitations in Two- and Three-Dimensional AntiferromagnetsZhihao Hao and Oleg TchernyshyovDepartmentofPhysicsandAstronomy,JohnsHopkinsUniversity,Baltimore,Maryland,USA(Received28April2009;revisedmanuscriptreceived14October2009;published28October2009)Spin excitations in an ordered Heisenberg magnet are magnonsbosons with spin 1. That may changewhen frustration and quantum uctuations suppress the order and restore the spin-rotation symmetry. Weshow that spin excitations in the S 1,2 Heisenberg antiferromagnet on a kagome lattice are spinonsfermionswithspin1,2.Inthegroundstatethesystemcanbedescribedasacollectionofsmall,heavypairsofspinonswithspin0.Amagneticexcitationoflowestenergyamountstobreakingupapairintotwospinonsatacostof0.06J.DOI: 10.1103/PhysRevLett.103.187203 PACSnumbers: 75.10.Jm, 71.10.Pm, 75.50.EeTheHeisenbergantiferromagnet, describedbytheex-change HamiltonianH 1Xh|]iS| S], (1)where1 > 0istheexchangecouplingand h|]idenotesapairofneighboringsites,isasimplemodelwithrealisticprototypes. Abipartite antiferromagnet inthree dimen-sions exhibits long-range spin order that breaks the globalspinSU(2) symmetry. Low-energyexcitationsof suchamagnet arespinwaves, whosequantizationyieldsmag-nonsbosons with spin 1 [1]. In one dimension, spin orderisdisruptedbyquantumuctuationsandtheSU(2)sym-metry is restored. For spins of length S 1,2, the excita-tions are spinonsquasiparticles with spin 1,2 [2]. Along-standing question is the existence of similarly unusualexcitations in higher-dimensional magnets where the sym-metryofthegroundstateisrestoredbyacombinationofstrong quantumuctuations (small S) and geometricalfrustration (a nonbipartite lattice). In 1973, Anderson pro-posed a resonating valence-bond state for spins 1,2 on thetriangular lattice [3]. In this state, spins form singlet bonds,or quantum dimers [4], with their neighbors. Although theground state on the triangular lattice turned out to beordered,kagomeandhyperkagomelatticeshaveemergedaslikely candidates for exotic quantum physics [5].The exchange energy (1) on a kagome lattice [Fig. 1(a)]isminimizedwhenthetotalspinofeverytriangleattainsits lowest allowedvalue [6]. For S 1,2, that canbeachievedbyputtingtwospins of atriangleinasingletstate; the third spin is free to form a singlet on the adjacenttriangle.Ifadimercouldbeplacedoneverytriangle,wewould construct a ground state with frozen singlets, avalence-bondsolid. That doesnot work: oneinfourtri-angles on a kagome lattice lack a dimer. Quantum uctua-tionsinducedbydefecttrianglesmakethegroundstateanontrivialsuperpositionofvalence-bondstates. Somere-cent theoretical works [7,8] lend support to a valence-bondcrystal proposed by Marston and Zeng [9], while others areconsistent withavalence-bondliquid[10]. Exact diago-nalizationstudies[11]reveal alargenumberoflow-lyingsingletstates,presumablyassociatedwithdifferentdimercongurations. Spin-1 excitations appear to have an energygap of 0.05 to 0.10 J [1012].Weproposeasimplephysical pictureof theS 1,2kagome antiferromagnet, in which the system is viewed asan ensemble of spinons, fermions with spin 1,2. Theantisymmetry of the many-body wave function can betracedtoquantuminterference. As twospinons areex-changed, they drag around pairs of spins entangled in S 0 states.Intheend, thebackground singletpairsreturntothe same positions. However, an odd number of themreverse their orientation, thus altering the sign of themany-body wave function.The defect triangles turn out to be small, heavy pairs ofspinons bound by exchange-mediated attraction. The bind-ingenergyisEl 0.06J. Spin-0excitationsareassoci-ated with the motion of pairs; their heavy mass is reectedin the large density of singlet states at low energies. Spin-1excitationscorrespondtobreakingupaboundpair into(nearly)freespinonswithparallel spins. Becausespinon1+2+4(b) (a)FIG. 1(coloronline). (a)Kagomelatticeandthemajorplay-ers:quantumdimers(thickbonds),adefecttrianglecontainingnodimer (red), andanantikinkspinon(bluearrow). (b) Themapping to a compact U(1) gauge theory on a honeycomblattice. Electricuxonlinkshas strength 1andisdepictedasarrowsinthemanner of Elser andZeng[15,25]. TheinsetshowstheU(1)chargesofthelatticesitesandquasiparticles.PRL 103, 187203 (2009)P HYS I CAL RE VI E W L E T T E RSweekending30OCTOBER20090031-9007,09,103(18),187203(4) 187203-1 2009 The American Physical Societypairs retain their individual character on kagome, the spingapisdeterminedprimarilybytheirbindingenergy, %El 0.06J. This number is in line with the previousestimates ofthe spin gap [1012].Thispictureisbasedonastudyoftwotoymodelsonlattices that share with kagome the triangular motif: the chain[13,14]andtheHusimicactus,atreelikevariantofkagome [15,16]. Both systems have dimerized groundstates with exactly one singlet bond on every triangle.Thechainprovidesinformationaboutthepropertiesofindividual spinons, while the Husimi cactus sheds light ontheirquantum statistics and interactions.The chain has two ground states, one with singlets onthe left bonds of the triangles (L), the other with singlets ontheright bonds (R), Figs. 2(a) and2(b). Alocal spin-1excitationdecays intotwospinonsquasiparticles withspin 1,2, Figs. 2(c)2(e). They serve as domain wallsandcomeintwoavors[13]: kinksinterpolatebetweenstate L on the left and state R on the right, antikinks do theopposite. A kink is fully localized and has zero energy cost:every triangle in its vicinity has a total spin 1,2 and is thusin a state of lowest energy, which is of course a stationarystate. Onemight thinkthat zeroexcitationenergywouldleadtoaproliferationofkinks,butthatdoesnothappen:kinkscanonlybecreatedinpairswithantikinks, whoseexcitation energy is positive. That hints at the existence ofaconserved charge, which will be dened later.Antikinksaremobile; their dynamicsisdescribedap-proximately as hopping to the nearest triangle [13]:Hjni 514 jni 12jn 1i 12jn 1i . . . (2)wherenisthetrianglecontainingtheantikink.AFouriertransform yields the quasiparticle dispersion ek 51,4 1 cosk % 1,4 k2,2m in the limit of low latticemomenta; the antikink mass m 1,1. Alocal spin-1excitation results in the creation of a kink and an antikinkwithaminimumenergyof1,4.TermsomittedinEq.(2)create additional excitations by promoting the two singletsadjacenttotheantikinktotriplets;alternatively,theycanbeviewedasthecreationof anextrakink-antikinkpairnext to the antikink. These virtual excitations renormalizethe antikink mass to m % 1.16,1and lower the bottom oftheantikinkbandto 0.219J[17]inagreementwithexact diagonalization [18]. Such perturbations appear to beharmless and will be ignored.Wenotethat thesignofthehoppingtermsinEq. (2)dependsonthesignconventionforquantumdimers[19]:theS 0stateof twospinsisantisymmetricunder ex-change: j|, ]ij "|#]i j #|"]i,2p j], |i. Whennecessary,we willdepictadimerstate j|, ]i asanarrowpointing from | to ]. The reversal of the arrow amounts tomultiplyingthe statewave functionby 1. Anegativehopping amplitude isenforced by thepivoting rule: whenanantikinkhopsfromonetriangletoanother,aquantumdimermovesintheoppositedirectionbypivotingonthesite shared by the triangles, Figs. 2(d) and 2(e).Agroundstateofthecactushasaquantumdimer oneverytriangle[16]. Likeonthechain, alocal spin-1excitationdecaysintoalocalizedkinkandamobileanti-kinkpropagatingalongaone-dimensional path(Fig. 3).The kink has energy 0, while the antikink propagates alongthe allowed path with a minimumenergy of 1,4 and a mass(d)(c)(e)(a)(b)FIG. 2(coloronline). GroundstatesL(a)andR(b)ofthechain.Alocaltripletexcitation(c)decaysintoalocalizedkinkand a mobile antikink (d)(e). Arrows on the dimers illustrate thepivotingrule(seetext).(d) (c)(a) (b)FIG. 3 (color online). A dimerized state on the Husimi cactus.Alocaltripletexcitation(a)decaysintoalocalizedkinkandamobileantikink(b)(d). Displacedvalencebondsareshowninbluecolor.Shadedtrianglesin(d)marktheallowedpathoftheantikink.PRL 103, 187203 (2009)P HYS I CAL RE VI E W L E T T E RSweekending30OCTOBER2009187203-2m 1,1. (Again, we neglect virtual excitations in thevicinity ofthe antikink.)Next we examine a single defect triangle on the cactus,Fig. 4(a). TheHamiltonianconnectsthisstatetoastatewithasingletbondofalongerrange[15],Fig.4(b).Thelonger-rangebondcanbeviewedas twoantikinks withtotal spin0. Byjumpingtoa neighboringtriangle, theantikinks areabletopropagatealongthreebranches ofthe cactus meeting at the original defect triangle, Fig. 4(d).Thus a defect triangle is made fromtwo antikinkswithtotal spin0. Astateof twoantikinks withatotalspin 1 is also of interest to us, so we will consider a genericcase of two antikinks with some specied spin projections,Fig. 4(c).The Husimi cactus is a toy model in which spinons havejust enoughfreedomtoexchangetheirlocationsbyhop-ping along the three branches. That allows us to determinetheirquantumstatisticsbyexaminingtheresultingBerryphase acquired by the spinons. The two-spinon wave func-tion turns out to be antisymmetric under exchange, soantikinks are fermions.Werst sketchaninformal argument alongthelinesof Arovaset al. [20]. StartingwiththestatedepictedinFig. 4(c), weperformanadiabaticexchangeof thetwoantikinks usingthepivotingrule. Wearriveat thestateshowninFig. 4(d)withdimersinthesamepositionsbutwith one dimer reversed. If we use the same dimer basis todescribe both states, the exchanged state acquires an extrafactor of 1. The extra factor appears for any initialpositions of theantikinks.Theinformalargument issupportedbyanexplicitde-terminationof thegroundstateof twoantikinks onthecactus[17]. Wendanondegenerate, symmetricspatialwavefunctionfortotalspinS 0andadoublydegener-ate, antisymmetric spatial wave function for S 1, as oneexpectsfortwofermions. ThegroundstateintheS 1sector has energy 1,2, twice the minimum energy of a freeantikink. In contrast, two antikinks withS 0formaboundstate whoseenergylies0.06Jbelowthebottomofthetwo-spinoncontinuum. Theboundstatehasasmallsize ( % 2.8 latticeunits of the chain.Curiously, antikinks in the S 0 ground state are boundnot only to each other, but also to the original defecttriangle: a pair is localized. The localization is topologicalinorigin:likeonthe chain,twoantikinkscannotmovethroughthesamebranchof thecactus: anantikinkmayonly be followed by a kink.Let us discuss implications for the Heisenberg model onkagome. The cactus can be viewed as a tree made oftriangles. Extrapolatingresultsfromatreetoaperiodiclattice(evenwiththesamecoordinationnumber)isonlywarrantedforthosephysicalquantitiesthatarenotsensi-tive to the long-distance properties of the lattice. Forexample, the bandstructure of a tree is quite differentfromthatofaperiodiclattice.Fortunately,thesmallsizeofaspinonpairmeansthatitsinternalstructureisdeter-mined by the local geometry of the lattice, so that the pairbindingenergyisexpectedtoberoughlythesameonthecactus, kagome, and hyperkagome.On both the kagome and hyperkagome lattice one in fourtriangles carries a spinon pair. We are now facing a many-body problem of fermionic spinons with a concentration of1,3per site. Theexistenceof spinonswithFermi-Diracstatistics in this and other frustrated two-dimensional mag-nets was conjectured previously [9,21,22] on the basis of alarge-Ngeneralizationof theHeisenbergmodel [23]. Inthesetheories, spinons interact withanemergent latticecompact U(1) gauge eld[24]. We exploitedanarrowrepresentationof dimer coveringsonkagome[15,25] todeneactitious U(1) gaugeeldlivingonlinks of ahoneycomblattice, Fig. 1(b). Thearrowsdepict aquan-tized electric eld of unit strength. Medial sites carrybackgroundchargeQ 1, whichisneutralizedonav-erage by the charges of antikinks, Q 2, and their pairs,Q 4; kinks have charge Q 2. It is noteworthy thatkinksandantikinksareindeedexpectedtocarryasmallreal electric charge[26].The gauge-theory connection sheds light on the origin ofone-dimensional trajectories of spinons onkagome. AnobjectofchargeQpassingthroughalinkincrementstheelectricuxinitby Q,soantikinkscanonlymoveinthedirection of electric ux, Fig. 1(b). As a result, spinons areconstrained to move along one-dimensional paths that maybe innite (as on the Husimi cactus) or may terminate in aloop. A nite length of an antikink trail R would raise the) b ( ) a ((d) (c)FIG. 4(coloronline). Astatewithasingledefecttriangle(a)evolves into a state with one longer-range bond (b), which can beviewed as two antikinks, shown as open and closed blue dots (c),withatotal spin0. State(d) isobtainedfromstate(c) byanadiabatic exchange of the antikinks. Shaded triangles markallowedpathsoftheantikinks.PRL 103, 187203 (2009)P HYS I CAL RE VI E W L E T T E RSweekending30OCTOBER2009187203-3tripletexcitationenergyofapairbyEc$r21,2R2[17].SinceRdependsonthearrangement ofdimersandmayeven vary from one spinon pair to the next, we only give anupper bound for the connement energy. On kagome, R !10sothat Ec 0.05J. Thusthespingap ElEcmayvaryfrom0.06Jtoasmuchas0.11J,dependingonthe valence-bond pattern in the ground state.Localization of antikink pairs can be traced to theirinteraction with the gauge eld, whose electric ux isconstrained to take on values 1. Two antikinks travelingalong the same line would leave behind links with anunphysicalelectricux 3.Therefore,inordertomove,thefermionsmakingapair must followdifferent paths.The tunneling amplitude is suppressed by a factor of ordereL,(, where ( % 2.8 is the pair size and L is the length ofthe shortest loop on the medial lattice: L 6 for kagome,10forhyperkagome, and 1fortheHusimi cactus. (ThesmallnessofeL,(alsomakesit possibletotreat heavyspinon pairs as stationary objects on time scales relevant tothe motion of light spinons.) Singh and Huse [8] found thatthe presence of other pairs increases the amplitude of pairtunneling. Thuspairsformsmall localizedclustersreso-nating around hexagons and preserving valence-bond orderelsewhere.While our work agrees with the large-Napproaches[9,2123] onthebigpictureof fermionicspinonsinter-actingwithaU(1)gaugeeld,thereareimportantdiffer-ences. First, thespinonsarenotfree:exchange-mediatedattraction binds them into small bosonic pairs. Second, theU(1) gaugeeldmanifests itself as aquantizedelectriceld, rather thanabackgroundmagneticux, andfrus-tratesthemotionofspinonpairs.Themotionofspinons,whichtendstoscramblevalence-bondorder, isstronglysuppressedasaresult. Theseobservationslendsupport tothe picture of a valence-bond crystal proposed by Marstonand Zeng [79].Asidefrominelasticneutronscattering,anotherwaytobreak spinon pairs is to apply a magnetic eld strongenough to fully polarize antikink spins, so that the magne-tization reaches 1,3 of the maximum value. The liberatedfermions will move at high speeds, inducing strong uctu-ationsof theelectriceldandrestoringtranslational in-variance. Amean-eldtheoryof spinonsmovinginthebackground of the emergent magnetic ux may be a goodstartingpointforthispartiallypolarizedspinliquid.Thatwould be interesting in light of recent numerical evidenceforanincompressiblequantumspinliquidat 1,3offullmagnetization [27,28].Weacknowledgehelpful discussions withL. Balents,C. D. Batista, C. L. Henley, andV. Cvetkovic. Researchwassupportedinpart bytheU.S. DOEGrant No. DE-FG02-08ER46544.[1] T. Giamarchi, C. Ruegg, andO. Tchernyshyov, NaturePhys.4,198(2008).[2] L. D.FaddeevandL. 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