DynamicalCorrelationsoftheKagomeS = 1/2HeisenbergQuantumAntiferromagnet Andreas M. La¨uchli1 and Claire Lhuillier2 1 MaxPlanckInstitutfu¨rPhysikkomplexerSysteme, No¨thnitzerstr.38, D-01187Dresden, Germany. 2Laboratoire de Physique The´orique de la Matie`re Condense´e, UMR 7600 CNRS, Universite´ Pierre-et-Marie-Curie, Paris 6, 75252 Paris cedex 05, France. (Dated:January8,2009) We determine dynamical response functions of the S = 1/2 Heisenberg quantum antiferromagnet on the kagomelatticebasedonlarge-scaleexactdiagonalizationscombinedwithacontinuedfractiontechnique. The dynamicalspinstructurefactorhasimportantspectralweightpredominantlyalongtheboundaryoftheextended Brillouin zone and energy scans reveal broad response extending over a range of 2 ∼ 3J concomitant with pronounced intensity at lowest available energies. Dispersive features are largely absent. Dynamical singlet correlations–whicharerelevantforinelasticlightprobes–revealasimilarbroadresponse,withahighintensity 9 atlowfrequenciesω/J <∼0.2J. Theselowenergysingletexcitationsdohowevernotseemtofavoraspecific 0 valencebondcrystal,butinsteadspreadovermanysymmetryallowedeigenstates. 0 2 PACSnumbers: 75.10.Jm,75.40.Mg,75.40.Gb n a J Introduction The S = 1/2 Heisenberg quantum antifer- A lot of our present understanding of S = 1/2 systems is 8 romagnet(AFM)onthekagomelatticeisakeymodelforour basedonaseriesofexactdiagonalizationstudies[10],which theoretical understanding of highly frustrated quantum mag- convincingly showed the absence of magnetic order, and re- ] netsintwospatialdimensions. Incontrasttoothermodelsys- vealed a puzzlingly high density of low-energy singlet and l e tems, such as the checkerboard magnet, it continues to hide tripletexcitations,asofaruniquephenomenon.Inthefollow- - r the true nature of its ground state despite a longstanding ef- ingweexplorehowthehugedensityoflow-lyingexcitations t s fort. On the experimental side many kagome like materials affectsexperimentallyrelevantresponsefunctions. . t were discovered and characterized over the years, and some Static spin response The static spin structure factor is a ofthemseemtogetclosetothegoalofanexperimentalreal- givenbythefollowingexpression: m izationofaperfectHeisenbergS =1/2kagomesystem. Re- - 1 (cid:88) d centlythesynthesisofpowdersamplesoftheHerbertsmithite Sz(Q) ≡ √ e−iQ·rj Sz , j n ZnCu (OH) Cl [1]andthesubsequentexperimentalinvesti- N 3 6 2 j o gationssparkedanewwaveoftheoreticalinterestinthislong S(Q) = (cid:104)Sz(−Q)Sz(Q)(cid:105), (2) c standingproblem[2]. [ So far most of the theoretical studies focused on ground where the wave vector Q is not restricted to lie in the first 1 state properties, discussing various possible phases, such as Brillouinzone(BZ). v valence bond solids [3], gapped spin liquids of different Thenumericallydeterminedstaticspinstructurefactorfor 5 kinds [4, 5, 6], and also stable critical phases [7, 8]. Much the 36 sites sample is shown in Fig. 1, as an intensity plot 6 0 less attention however has been paid to the precise nature covering multiple BZs [right hand side, plot (1)], and along 1 and form of the low-energy excitations visible in frequency a path in the extended BZ [plot (2)]. The response is strong 1. resolved probes such as inelastic neutron scattering or light and broad along the zone boundary of the extended BZ, re- 0 scattering techniques. In the present Letter we fill this void vealing the short ranged nature of the antiferromagnetic cor- 9 and present a detailed numerical study of the dynamical re- relations[10]. IntheN =36groundstatethereareadditional 0 sponse of S = 1/2 kagome systems in the spin triplet and smallpeaksatthepoint(g),afeaturewhichisalsoreportedin : v singlet channelsand formulatepredictions tobe tested inin- arecentDMRGstudy[6].ComparisonwithaN =24sample i elasticscatteringexperiments. (notshown)confirmsthatthebroadresponsealongthebound- X WestudytheS = 1/2Heisenbergquantumantiferromag- aryoftheexendedBZisageneric,size-independentfeature, r a netonthekagomelattice,governedbytheHamiltonian: characterizingastatewithspincorrelationswhicharedecay- ingrapidlybeyondthenearestneighborsites. Tocontrastthis (cid:88) H =J Si·Sj (1) resultwithamagneticallyordereds√tatew√eshowthestaticre- (cid:104)i,j(cid:105) sponseintheq=0[Fig.1(3)]and 3× 3[Fig.1(4)]states, whichhavebeenobtainedusingastrongJ couplingwiththe where J > 0 is the antiferromagnetic nearest neighbor ex- 2 appropriatesign. change coupling. Our results are based on large-scale exact Dynamical spin structure factor The energy and momen- diagonalizations of up to N = 36 spins, supplemented by tumdependenceofthedynamicalstructurefactor: thecontinuedfractionmethod[9]fordynamicalcorrelations functions. For the dynamical quantities we performed 500 1 1 S(Q,ω)=− Im(cid:104)Sz(−Q) Sz(Q)(cid:105), (3000) iterations in Hilbert spaces of dimensions 3.8× 108 π ω−(H −E )+iη GS (4.5×109)forthespinandsingletdynamicsrespectively. (3) 2 30 g 30 f 30 e (1) 20 20 20 10 10 10 0 0 0 0 1 2 3 4 0 1 2 3 4 0 1 2 3 4 ω/J ω/J ω/J 0.6 (2) 30 h 30 d 20 20 Q)0.4 z( S 10 10 0.2 0 0 0 0 1 2 3 4 0 1 2 3 4 Q ω/J ω/J (3) i c 4 4 Γ X K g f e 2 2 hX dM 0 0 iM cK 0 1 2 3 4 0 1 2 3 4 X ω/J ω/J b (4) 3 Γ b 2 1 0 0 1 2 3 4 ω/J FIG.1: (Color)DynamicalspinstructurefactoroftheN=36sample. TheeightpanelsdisplayfrequencyscansS(Q,ω)(η = 0.02J)at labeledwavevectorsQintheextendedBrillouinzoneshowninthelowerrightcenter.Notethattheintensityscalesdifferamongthedifferent panels.TheΓpointhasnoweightandisnotshown.Theblueverticallinesshowthepolelocationandintensityofthecontinuedfraction.The verticaldottedmagentalinedenotesthefinitesizespingapinthecorrespondingmomentumsector. Thedashedredlinemarkstheposition (cid:82) ofthefirstfrequencymomentω¯ = dωωS(Q,ω)/S(Q). IntherightmostcolumnthestaticspinstructurefactorofthepureHeisenberg modelonthek√agome√latticeisshown,asanintensityplot(1)andalongthepathΓ−(e)−(g)−Γ(2). Thestaticstructurefactorforthe q=0(3)and 3× 3(4)Ne´elorderstatesinducedbyappropriatesecondneighborcouplingsarealsodisplayed. is directly relevant for inelastic neutron scattering (INS) ex- of 2 ∼ 3J, starting immediately above the (finite-size) gap. perimentsandthereforeaquantityofcentralinterest. Inmag- Furthermore there seems to be a pronounced enhancement netically ordered systems we expect to see dispersive, long- of the intensity at small ω. The different spectral functions lived spin waves [11], while one-dimensional systems in ap- look rather similar, suggesting an approximate factorization propriate regimes can display spinon continua with a rich S(Q,ω)∼S(Q)×f(ω),atleastatintermediateandhighω. structure[12]. Theoverallpictureisdefinitelyquitedifferentfromthespec- Our numerical results for the N = 36 kagome lattice are trumofaNe´elorderedsystemonthesamesystemsize,where presented in the left part of Fig. 1. The shaded panels dis- anoverwhelmingpartofthespectralweightiscarriedbyvery playanenergyscanatthewavevectorindicatedbythepanel fewpolesineachQ-sectorassociatedtotheBraggpeakand position and its label, referring to specific points in the ex- theone-magnonmodesrespectively[13]. Stillatsomewave tended BZ. Each panel displays the broadened (η=0.02J) vectors the lowest pole carries significant weight [especially spectral function (black line), the locations and weights of at(g)]. Theoriginofthisfeatureremainstobeelucidatedbut the poles of the continued fraction expansion (blue vertical could potentially come from an algebraic divergence in one lines), the finite size spin gap in the corresponding momen- scenario [8] or from triplon excitations on top of (remnants tum sector (dotted vertical line), and the first frequency mo- of)avalencebondcrystalinadifferentscenario[14]. (cid:82) mentω¯(Q)= dωωS(Q,ω)/S(Q)(dashedverticalline). Inordertoaddressfinite-sizeeffectswepresenttwospec- Consistent with the static structure factor [by virtue of the tral functions at the wave vectors (g) and (i) for N = 24 (cid:82) sum rule S(Q) = dω S(Q,ω)], the dynamical spin re- and36spinsinFig.2(a)and(b). Thecharacteristicwidthin sponse function concentrates essentially along the boundary energy as well as the prominent response at low ω for wave of the extended BZ. The main feature of this system is the vector (g) are clearly stable with respect to finite size ef- stretchingofthemagneticresponseineachQ-sectoronavery fects. Inpanels(c)thelocaldynamicalspincorrelationfunc- (cid:82) large number of excited states spanning a large bandwidth tion S (ω) ∝ dQ S(Q,ω) is shown for different system loc 3 N=36, Kagome 1 0.15 0.8 (a) g point N=24 (b) i point, N=24 g point N=36 i point, N=36 0.8 N=32, Checkerboard 0.8 ) ω ωQ,)0.6 0.1 ωQ,) mics(0.6 0.4 D Woamllasin Multi-Triplon Continuum S(0.4 S( na 0 y 0 1 2 3 4 0.05 r D0.4 ω/J 0.2 me N=26, Square 0.8 Di 00 1 2 3 4 0 1 2 3 40 cal 0.2 0.4 ω/J ω/J Lo 0 0 2 4 6 ω/J 0.3 (c) zzωS()Loc00..0012..5521 NNN===233406 0 0 0.5 1 1.5 ω/2J 2.5 3 3.5 4 0.05 FIG. 3: (Color) Dynamical singlet fluctuations [Eq. (4)] for three %]1000 differentsystems.Mainplot:kagomelattice.Upperinset:Checker- AccumulatedSpectral Weight [ 24680000 (d) bLpreolosoawptrtoedenrdlsaiqentutsoiaecfnte:tthiwUteyintthrhferrapuerespetlrssaayeqtnseuttdesemtwsteqse-.ullalikrteehevlaaqtluteiacnleciteaetxbihvoienbdfietcianrtygusrNtease´loeglfroothurednedRr.asmtTatahene. 0 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 ω /J thusrevealsthequalitativefeaturesoftheinelasticlightscat- FIG. 2: (Color online) Upper panel: Finite size behavior of the teringresponse. Andsecond,weexpectaspontaneoustrans- spectralfunctionsattwodifferentpointsintheBrillouinzone: g(a) lationalsymmetrybreakingduetodimerizationtoshowupas andi(b)intheconventionofFig.1. Lowerpanel: (c)Localspin animportantω →0contribution. (cid:82) autocorrelationfunctionS (ω) ∝ dQS(Q,ω)forN =24,30 loc The fluctuation spectrum for the N = 36 kagome system and 36 sites. (d) The cumulative spectral weight as a function of is shown in Fig. 3, where a broad response from the lowest ω/J.Allspectralfunctionshavebeenbroadenedusingη=0.05J. singlet up to energies ∼ 4J is seen together with a strong increase of the response towards the lowest energies. The sizes, again highlighting the stability of the overall shape of kagomeresultcanbecomparedtotheresponseoftheHeisen- the spectral function. Finally panel (d) presents the cumula- berg model on a checkerboard lattice (upper inset in Fig. 3) tivespectralweightasafunctionofω/J, revealinge.g. that and the square lattice (lower inset in Fig. 3), where in both ω ∼0.95Jisthemedianfrequencyforthetotalspinresponse. casesthephysicaloriginoftheresponseisessentiallyunder- Effectofimpurities Wehavestudiedtheinfluenceofalow stood. OntheNe´elorderedsquarelatticethedimer-dimerre- concentration of nonmagnetic impurities on the spin dynam- sponseisassignedtothetwo-magnoncontinuumwithamaxi- icsbydepletingaN = 27samplebyonesite. Theaveraged mumstrengtharound3J.Thisisobviouslyverysimilartothe dynamical spin response closely resembles Fig 2(c) with an Raman response of the square lattice Heisenberg AFM [16]. additional resonance-like feature at ω ∼ J due to the strong On the checkerboard lattice with its plaquette valence bond singletformingonthetwobondsnexttothevacantsite[15]. crystal ground state [17] the dimer-dimer response function Due to its local nature, this feature is expected to be generi- shows different frequency domains: a single low-lying peak callypresentinS(Q,ω)aswell. (shaded in red) originating from the valence-bond symmetry Singlet fluctuations In order to assess the importance of breakingpartnerofthegroundstate,followedbytwodomains theabundantnumberoflowenergysingletexcitationsforop- at non zero-frequency, first a range of singlet excited levels tical probes and to investigate the tendency towards valence (shaded orange) which have been convincingly explained as bond crystal ordering, we study the local dynamical fluctua- valence bond crystal domain-wall excitations [17, 18] and a tionsofanearestneighbordimeroperator: second range ω >∼ J corresponding to a multi-triplon con- tinuum [17] (shaded in brown). It is clear from this com- D = S ·S −(cid:104)S ·S (cid:105) parisonthatthekagomelatticedoesnotshowtypicalvalence i,j i j i j 1 1 bond crystal characteristics exemplified by the checkerboard Di,j(ω) = −πIm(cid:104)Di,jω−(H −E )+iηDi,j(cid:105) (4) magnet. Still the response on several levels up to ω <∼ 0.2J GS seemstobeparticularstrong. Howeverthisresponsespreads Theinterestinthisquantityistwofold. FirstEq.(4)isclosely on many low lying levels in any symmetry sector which can related to the Raman or RIXS reponse of a spin system and be excited by the dimer-dimer operator. We do not see any 4 clearprecursorofaspecificspatialsymmetrybreakingpattern of quasiparticles in a doped kagome system [25]. Building when comparing the excited levels to different valence bond on these results it will be interesting to understand the evo- crystalsymmetrypredictionssummarizedinRef.[19]. Possi- lutionofthedynamicalresponseatfinitetemperatureaswell blereasonsare: i)theabsenceofVBCorder(seeRef.[6]for astheeffectofamagneticfield. Giventhatmanyexperimen- asimilarconclusion)orii)averyweakorderingwithavery talkagomesystemshavesome(small)Dzyaloshinsky-Moriya large unit cell which has not still emerged from competing interactionstheirinfluenceonthedynamicalresponseisalso orders. worthstudying. Comparison to theoretical proposals Both the spin and WethankC.L.Henley,F.Mila,R.Moessner,D.Poilblanc the dimer dynamical fluctuations of a S = 1/2 kagome sys- and K.P. Schmidt for discussions and comments, and M. de temareintrinsicallybroad,andarenoteasytoreconcilewith Vries,G.NilsenandH.M.Rønnowfordiscussionsandaccess the excitation spectrum scenarios for the various proposed totheirunpublishedneutronscatteringdata. Weacknowledge ground states. For example the dynamical spin correlation supportbytheSwissNationalFunds. Thecomputationshave functions lack the characteristic coherent triplon excitations, been performed on the machines of the MPG RZ Garching which are expected on top of a valence bond crystal [14]. andtheCSCSManno. On the other hand the spectrum is much too dense to be ex- plained by a spinon continuum with a energy scale of order J [4]. 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