Possible SU(3) Chiral Spin Liquid on the Kagome Lattice Ying-Hai Wu∗ and Hong-Hao Tu† Max-Planck-Institut fu¨r Quantenoptik, Hans-Kopfermann-Straße 1, 85748 Garching, Germany (Dated: November 22, 2016) WeproposeanSU(3)symmetricHamiltonianwithshort-rangeinteractionsontheKagomelattice and show that it hosts an Abelian chiral spin liquid (CSL) state. We provide numerical evidence basedonexactdiagonalizationtoshowthatthisCSLstateisstabilizedinanextendedregionofthe parameterspaceandcanbeviewedasalatticeversionoftheHalperin221fractionalquantumHall (FQH)stateoftwo-componentbosons. WealsoconstructapartonwavefunctionforthisCSLstate and demonstrate that its variational energies are in good agreement with exact diagonalization results. The parton description further supports that the CSL is characterized by a chiral edge 6 conformal field theory (CFT) of the SU(3) Wess-Zumino-Witten type. 1 1 0 2 Introduction – Topological aspects of condensed mat- arenotrelevantinexperimentsbecausethespinsinsolid v ter have been actively studied since the discovery of the state systems are almost all due to electrons so belong o quantum Hall effect. An important development in this to the SU(2) group. However, it was proposed [37, 38] N area is the concept of topological order [1], which de- that the SU(4) Heisenberg model might describe cer- 0 scribes phases that can not be distinguished using the tainmaterialsinwhichSU(4)symmetryarisesfromcou- 2 conventional symmetry breaking paradigm but exhibit pled spin and orbital degrees of freedom [39]. There has exotic topological properties such as fractional charge, also been substantial progress in experiments using cold ] l anyonic braiding statistics, and ground state degener- atoms with several internal states, which brings SU(N) e - acy on high genus manifold. Besides the FQH states, spin systems even closer to experimental reality [40–43]. r it has long been speculated that there could be topolog- The atomic species, lattice configurations, and the forms t s ically ordered states in antiferromagnetic spin systems of interactions in cold atom experiments can be tuned in . t (called spin liquid) which do not break the crystalline a wide range [44, 45], which would enable us to explore a m symmetries or spin rotation symmetries. To suppress the rich physics associated with SU(N) spins. the tendency of magnetic ordering, one should consider The connection between FQH and CSL states was re- - d frustrated lattices in which no simple alignment of spins vealedinaseminalworkbyKalmeyerandLaughlin[46], n can achieve the lowest energy under antiferromagnetic which demonstrated that bosonic FQH states can be o exchange. The Kagome lattice has been very promising mapped to spin states. In this example, one maps the c [ in this regard, and recent numerical and experimental spin-1/2 degree of freedom on a lattice site to a boson studiesindeedpointtotheexistenceofspinliquidstates andimposethehard-coreconstraintsuchthatthereisat 3 in certain systems [2–9]. mostonebosonpersite. Thiscanbegeneralizedtocases v 4 The theoretical study of strongly correlated spin sys- where the lattice sites have higher spins of the SU(2) 9 tems is generally very difficult. Being motivated by the group [47]. We will explain below how to map SU(3) 5 large N expansion in gauge field theory, it has been pro- spins to bosons and establish a correspondence between 2 posed that one may investigate SU(N) spin systems us- SU(3) CSL and FQH states of two-component bosons. 0 . ingsimilarperturbativemethods(organizedinpowersof Being equipped with the mapping between FQH and 1 1/N) to obtain some hints about the physics of SU(2) CSL states, we can use some techniques developed for 0 spins [10–13]. One may worry that the perturbative FQH states to understand CSL. One fruitful way in the 6 1 results obtained in the large N limit would not be ap- FQH context is to express FQH wave functions as chiral : plicable when N is small, so other theoretical methods correlators of CFT [48, 49]. The advantage of this ap- v and numerical calculations are also essential in under- proach is that, by using CFT null field technique [50], it i X standing the physics. Exactly solvable models, such as can give a parent Hamiltonian with the CSL as its exact r the Uimin-Lai-Sutherland model [14–16], the Haldane- groundstate[49,51–53](seeRefs.[54,55]foralternative a Shastry type models [17–20], the Affleck-Kennedy-Lieb- ways of deriving parent Hamiltonians). However, these Tasakitypemodels[21–26],havebeendesignedandthey parent Hamiltonians usually contain long-range interac- provide useful insight into SU(N) spin systems. Another tions. To be more realistic, it is of great importance to widelyusedmethodistodecomposethespinsasbosonic test whether the CSL states constructed from CFT can or fermionic partons and build exotic spin states using be stabilized using Hamiltonians involving only simple mean field parton states supplemented with Gutzwiller short-range interactions [56, 57]. projection. Based on different approaches, a rich variety Mapping SU(3) Spins to Bosons — To make connec- of physical phenomena has been revealed in SU(N) spin tionsbetweenSU(3)spinmodelsandFQHstatesoftwo- systems [27–36]. component bosons, we briefly review their properties. It might appear at first sight that SU(N) spin systems The generators of the SU(3) group are usually chosen to 2 betheeight3×3Gell-Mannmatricesλ (i=1,2,··· ,8). i Foralatticeinwhicheachsiteisdescribedbythefunda- mental representation and the whole system is described by an SU(3) invariant Hamiltonian, the local Hilbert space dimension is three and there are two U(1) symme- tries. Toformulateabosondescription,wemayinterpret the lattice as being occupied by two-component bosons (thetwointernalstatesarelabeledas↑and↓). Imposing the hard-core constraint that allows for at most one bo- sononeachsiteresultsinalocalHilbertspacedimension three (i.e. empty, one ↑ boson, and one ↓ boson). The FIG. 1. The Kagome lattice with 18 sites. The red circles two U(1) symmetries correspond to the particle number illustrate the three types of terms P , Q , and Q−1 in the st rst rst conservations of these two types of bosons. SU(3)Hamiltonian(4). Thegreenarrowsonthesmalltrian- glesandthenumbersintheirvicinitygivethehoppingphases The simplest FQH state of two-component bosons is in the parton mean field Hamiltonian (5). the Halperin 221 state at filling factor 2/3 [58] M M (a) (b) (c) (d) (cid:89) (cid:89) Ψ = (z↑−z↑)2(z↓−z↓)2 (z↑−z↓), (1) 221 s>t=1 s t s t s,t=1 s t −12.85 −13.85 −12.10 −12.85 −12.95 −13.95 −12.15 −12.90 wherez =x+iy isthecomplexcoordinateintwodimen- −12.20 −13.05 −14.05 −12.95 sions and the superscripts indicate the internal states. −12.25 The low-energy properties of this state is encoded in the −13.15 Egap −14.15 −12.30 −13.00 Chern-Simons theory with the Lagrangian density E−13.25 −14.25 −12.35 −13.05 1 −13.35 −14.35 −12.40 −13.10 L= 4πKIJ(cid:15)µνρaIµ∂νaJρ, (2) −13.45 −14.45 −12.45 −13.15 −12.50 where K is the 2×2 matrix −13.55 Esplit −14.55 −12.55 −13.20 IJ −13.65 −14.65 (cid:18) (cid:19) −12.60 −13.25 2 1 (3) 1 2 FIG.2. (Coloronline)EnergyspectraontheKagomelattice One characteristic signature of topologically ordered with 18 sites. (a) K1 = 0.6 and K2 = 0.4 with PBC; (b) K =0.6andK =0.5withPBC;(c)K =0.6andK =0.4 states, the ground state degeneracy on torus, can be de- 1 2 1 2 with OBC; (d) K =0.6 and K =0.5 with OBC. The value duced from this Chern-Simons theory as |detK| = 3. 1 2 of J is fixed at 1 in all calculations. The Chern-Simons action also provides useful informa- tion about its edge physics: the K matrix has two pos- itive eigenvalues, so there are two copropagating edge constrains on the Hamiltonians and it is usually more modes described by U(1)×U(1) bosons. For bosons in convenient to express them in terms of swapping opera- the lowest Landau level, this state is the exact zero en- tors. For our purpose, we need to define two-body and ergy ground state if there are only contact interactions three-body swapping operators P and Q . When P st rst st (cid:80) δ(rσ − rτ) between the bosons regardless of their isappliedonastate, thespinstatesonthelatticesitess στ spins. The contact interaction forbids two bosons to ap- andtareexchanged. WhenQ isappliedonastate,the rst pear at the same position, which is somewhat equivalent spin states on lattice r, s and t are cyclically permuted to the constraint of having at most one boson per site in thebosonicdescriptionofSU(3)spinmodels. Ingeneral, (a) (b) thespinmodelsdefinedonalatticeappeartobeverydif- ferent from the simple continuum model, but their low- energy effective theories have the same action. This can be seen from the parton construction of the SU(3) CSL K2 K2 state (see below). Exact Diagonalization — The CFT construction pro- vides us parent Hamiltonians for which the SU(3) CSL states are exact ground states [52, 53]. These Hamilto- nians inevitably contain long-range terms but they pro- K1 K1 vide useful hints about what kind of short-range Hamil- FIG.3. (Coloronline)E andE ontheKagomelattice gap split tonians might have ground states in the same phase. A with 18 sites. (a) E at K = 0.0,0.1,··· ,1.0 and K = gap 1 2 general Hamiltonian can be written in terms of the Gell- 0.0,0.1,··· ,1.0;(b)Esplit atK1 =0.0,0.1,··· ,1.0andK2 = Mann matrices, but SU(3) invariance imposes stringent 0.0,0.1,··· ,1.0. ThevalueofJ isfixedat1inallcalculations. 3 in a counterclockwise way. dant states in the fermionic Hilbert space are removed The short-range Hamiltonian we have studied is de- by a Gutzwiller projector P which locally enforces sin- G fined on the Kagome lattice with two-body terms acting gle occupancy on each site, i.e. (cid:80) c† c =1 ∀s. α sα sα on all nearest neighbors and three-body terms acting on We assume that the partons are described by the free all small triangles fermion Hamiltonian (cid:88) (cid:88) (cid:88)(cid:88) H =J P +(K −iK ) Q H = f c† c , (5) st 1 2 rst parton st sα tα (cid:104)st(cid:105) (cid:104)rst(cid:105) α (cid:104)st(cid:105) (cid:88) +(K1+iK2) Q−rs1t, (4) where fst is the hopping parameter of fermionic par- (cid:104)rst(cid:105) tons between nearest neighbors (to be determined be- low). This Hamiltonian can be viewed as a “mean field” where Q−1 means permuting the spin states clockwisely rst theory of the original SU(3) spin problem. However, at (equivalent to two counterclockwise permutations). In the mean field level, the particle number constraint is Fig. 1, we show a Kagome lattice with 18 sites (3 unit only satisfied on average. A trial wave function in the cells along one direction and 2 unit cells along the other physicalspinHilbertspaceshouldsatisfythesingleoccu- direction) and illustrate the terms in the Hamiltonian. pancyconstraintrigorously,whichcanbeobtainedusing Thenumericalresultspresentedbelowareforthislattice Gutzwiller projection as but we have obtained similar results for the Kagome lat- ticewith12sites(2unitcellsalongbothdirections). The |Ψ(cid:105)=P |Ψ (cid:105), (6) G parton Hamiltonian (4) is SU(3) invariant, so its eigenstates be- long to definite representations of the SU(3) group. We where |Ψ (cid:105) is the Fermi sea ground state of (5) at parton choose J = 1 as the energy scale and vary K over a 1/3 filling. 1,2 broad range to search for the optimal values that may Forourpurposeofdescribingthenumericallyobserved stabilize an SU(3) CSL corresponding to the Halperin CSL state, we choose the hopping integral f in (5) st 221 state. to be complex numbers whose phases depend on only The energy spectra for a few systems with periodic one parameter θ as shown in Fig. 1. The value of θ boundary condition (PBC) or open boundary condition determines the fluxes in the triangles and hexagons of (OBC) are shown in Fig. 2. For a system with PBC, the Kagome lattice. With this prescription, the par- theSU(3)CSLthatweseekhasthreedegenerateground ton Hamiltonian (5) with PBC has three energy bands states in the thermodynamic limit but the ground states and,at1/3filling,thelowestbandiscompletelyfilledby generally split in finite size systems. On the contrary, fermionicpartons. ForOBC,thepartonwavefunctionis such a system has only one ground state if it has OBC. similarly constructed by assuming an open boundary for Inbothcases,thegroundstate(s)areseparatedfromthe the parton Hamiltonian (5). For both PBC and OBC, excited states by an energy gap. The numerical results the Gutzwiller wave functions (6) are SU(3) singlets. in Fig. 2 are consistent with these theoretical expecta- To optimize the variational ansatz, we choose many tions. We have also confirmed by explicit calculations different θ values and compute the energy of the wave that the ground states are SU(3) singlets. For the cases function (6) with respect to the Hamiltonian (4) to se- with PBC, the eigenstates also have good momentum lect the one giving the lowest energy. This has been quantum numbers and we found that the three quasi- done in several cases on the lattice with 18 sites but we degenerate ground states all have K = 0 and K = 0. focus here on the following two sets of parameters, i) x y The energy spectra on torus can be characterized quan- K =0.6,K =0.4andii)K =0.6,K =0.5. Thevari- 1 2 1 2 titatively using two variables E and E as shown ationalenergyasafunctionofθisshowninFig.4(a)and gap split in Fig. 2: the former is the difference between the third (b). The best results in the two cases with PBC (OBC), state and the fourth state and the latter is the splitting which both appear at θ ≈0.88π, are −13.12 (−12.30) min of the lowest three states. It is desirable to have a suffi- for K = 0.4 and −14.08 (−12.89) for K = 0.5. For 2 2 ciently large E and a small enough E . These two PBC, they are quite close to the energies of the three gap split variablesareplottedinFig.3forawiderangeofparam- quasi-degenerate ground states [see Fig. 2 (a) and (b)]. etersandonecanseethatsuchrequirementsaresatisfied The variational energies are, however, less satisfactory in a region around K ≈0.6 and K ≈0.45. for OBC [see Fig. 2 (c) and (d)]. 1 2 Parton Wave Functions — To gain further insights For the optimal choice θ , the three parton energy min into the nature of the ground states of the SU(3) Hamil- bandsof(5)haveChernnumbers−1,0,+1,respectively tonian (4), we now resort to a parton wave function de- [see Fig. 4(c) from top to bottom]. This means that the scription of the numerically observed CSL phase. This partontrialwavefunction(6)describesaGutzwillerpro- relies on a fermionic representation of the SU(3) spins, jected Chern insulator with Chern number +1. To de- where the three local states are encoded using singly oc- scribethethreequasi-degenerategroundstatesontorus, cupied fermions, |α(cid:105) = c†|0(cid:105) (α = 1,2,3). The redun- one may construct parton wave functions by adopting α 4 -2 -2 imation of the exact eigenstates. The parton description -4 (a)PBC -4 (b)OBC also helps us to deduce the low-energy effective theory. on -6 on -6 KK11==00..66,,KK22==00..45 It has been shown in Ref. [63] how to derive a Chern- ati -8 ati -8 Simons theory for SU(2) spin systems on arbitrary lat- ari-10 ari Ev-12 Ev-10 tices, so one might expect that the Chern-Simons theory -14 K1=0.6,K2=0.4 -12 for the SU(3) CSL can also be derived without reference -160 Kπ1/=40.6,θK23=π0.5/4θminπ -140 π/4 θ3π/4θminπ toIpnagretonnersa.l,onecanestablishamappingbetweenSU(N) spins and (N-1)-component bosons, which suggests that 4 (c) θ=θmin 100 (d) SU(N) CSL and FQH states of multi-component bosons on 2 ax10−1 are closely related. It would be very interesting if one Epart 0 λ/m10−2 cspaninalssyostiedmenstitfyhasthocratn-rahnogsetinCtSeLracsttiaotness.in oAtnhoetrhSeUr (eNx)- λ -2 10−3 citing direction opened by our current work is to in- -4 10−4 vestigate SU(N) CSL with non-Abelian anyons. The Γ Γ K M 0 5 10 15 CFTconstructionofmulti-componentnon-AbelianFQH index states [64, 65] is a good starting point in this direc- tion. The relation between these non-Abelian spin- FIG. 4. (Color online) (a) PBC and (b) OBC variational singlet (NASS) states and the Halperin 221 state is very energiesofthepartontrialstatesforthe18-siteKagomelat- much the same as that between the Moore-Read state tice as a function of the parameter θ. The lowest variational energies for the Hamiltonian (4) with K = 0.6, K = 0.4 and the Laughlin state. While the Kalmeyer-Laughlin 1 2 (green crosses) and K = 0.6, K = 0.5 (blue open circles) state is designed for spin-1/2 systems, the lattice ver- 1 2 both appear at θ ≈0.88π. For PBC, the gap between the sion of the Moore-Read state may be realized in spin- min lowest and the middle bands of the parton Hamiltonian van- 1 systems. It should be possible to reformulate the ishesatθ=π/4andθ=3π/4(denotedbytwodottedlines). NASS states in SU(N) spin systems where the spins are (c) Band structure of the parton Hamiltonian at the opti- described by higher-dimensional representation of the malvariationalpointθ . Thethreebandsareseparatedby min SU(N) group. It would also require numerics to see if energy gaps and have Chern numbers −1, 0, and +1, respec- tively (from top to bottom). (d) (Normalized) eigenvalues such states can be stabilized by sufficiently simple short- of the overlap matrix of Gutzwiller wave functions with 15 range Hamiltonians. different twisted boundary conditions for partons on the 48- Upon finalizing the manuscript we noticed two recent site Kagome lattice. The existence of three large eigenvalues preprints [66, 67] on closely related topics. suggests that there are three linearly independent states on Acknowledgement — We are grateful to Meng torus. Cheng for helpful discussions. HHT acknowledges A.E.B.NielsenandG.Sierraforanearliercollaboration on the SU(N) CSL. Exact diagonalization calculations twistedboundaryconditionsforthepartons[59,60]. We are performed using the DiagHam package for which we have checked that, by computing the eigenvalues of the thankalltheauthors. ThisworkissupportedbytheEU overlap matrix, 15 different twisted boundaries for par- project SIQS. tons on the 48-site Kagome lattice (4 unit cells in both directions)indeedyieldthreelinearlyindependentstates [see Fig. 4(d)]. Thus, these three parton wave functions provide a complete approximation of the ground state manifold on torus. 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