Variational study of the ground state and spin dynamics of the spin- 1 2 Kagome antiferromagnetic Heisenberg model and its implication on Hebertsmithite ZnCu 3 (OH) 6 Cl 2
VVariational study of the ground state and spin dynamics of the spin- Kagomeantiferromagnetic Heisenberg model and its implication on HebertsmithiteZnCu (OH) Cl Chun Zhang and Tao Li
Department of Physics, Renmin University of China, Beijing 100872, P.R.China (Dated: September 9, 2020)We find that the best RVB state of the spin- Kagome antiferromagnetic Heisenberg model(spin- KAFH) is described by a Z gapped mean field ansatz, which hosts a mean field spinon dispersionvery different from that of the widely studied U (1) Dirac spin liquid state. However, we find thatthe physical spin fluctuation spectrum calculated from the Gutzwiller projected RPA(GRPA) theoryabove such an RVB state is actually gapless and is almost identical to that above the U (1) Diracspin liquid state. We find that such a peculiar behavior can be attributed to the unique flat bandphysics on the Kagome lattice, which makes the mapping between the mean field ansatz and theRVB state non-injective. We find that the spin fluctuation spectrum of the spin- KAFH is notat all featureless, but is characterized by a prominent spectral peak at about 0 . J around the M point, which is immersed in a broad continuum extending to 2 . J . Based on these results, weargue that the spectral peak below 2 meV in the inelastic neutron scattering(INS) spectrum ofHebertsmithite ZnCu (OH) Cl , which has been attributed to the contribution of Cu impurityspins occupying the Zn site, should rather be understood as the intrinsic contribution from theKagome layer. We propose to verify such a picture by measuring the Knight shift on the Cu site,rather than the O site, which is almost blind to the spin fluctuation at the M point as a result ofthe strong antiferromagnetic correlation between nearest neighboring spins on the Kagome lattice. PACS numbers:
I. INTRODUCTION
The spin- Kagome antiferromagnetic Heisenbergmodel with nearest-neighboring exchange (spin- KAFH) is extensively studied in the last three decadesin quest of quantum spin liquid . While it is generallybelieved that the ground state of the spin- KAFH is aquantum spin liquid, it is strongly debated what is theexact nature of such a novel quantum state of matter.Variational studies based on the resonating valencebond(RVB) theory have accumulated extensive evidencefor a gapless U (1) Dirac spin liquid scenario ,which is also implied by some recent studies usingother numerical approaches . On the other hand, agapped Z spin liquid ground state has been claimedby many other studies . Clearly, a study of thespin fluctuation spectrum of the system is the moststraightforward way to resolve the controversy amongthese ground-state-oriented studies.On the experimental side, HebertsmithiteZnCu (OH) Cl is argued to be an ideal realiza-tion of the spin- KAFH, with possible Cu impurityspins occupying the Zn sites between the Kagomelayer . Inelastic neutron scattering(INS) measure-ment on Hebertsmithite ZnCu (OH) Cl finds that thespin fluctuation spectrum of the system is characterizedby a broad and featureless continuum above 2 meV.Below 2 meV, a broad peak emerges around the Mpoint of the Brillouin zone . This peak is believed tobe contributed by the Cu impurity spins between theKagome layers. Such a scenario is supported by a laterNMR measurement on the system , which implies a finite gap of ∆ = 0 . J ∼ . J in the intrinsic spinfluctuation spectrum of the Kagome layer. A spin gapof similar size is also reported in the NMR study of a re-lated Kagome material ZnCu (OH) FBr . However, itis puzzling why the low energy spectral peak can exhibitsuch a strong momentum dependence as observed in INSmeasurement, if it is indeed contributed by impurityspins. More recently, a refined NMR measurement onHebertsmithite ZnCu (OH) Cl finds that the intrinsicspin fluctuation spectrum of the Kagome layer is actuallygapless .Within the RVB theory framework, attempts havebeen made to reconcile the discrepancy between the dif-ferent theories regarding the spin excitation gap of thespin- KAFH. Using projective symmetry group(PSG)analysis, people find that a gapped Z spin liquid statecan indeed be realized in the vicinity of the U (1) Diracspin liquid state if one introduce longer-ranged RVBparameters . However, VMC calculations along thisline generate controversial results . In a recentwork , we find that the mapping between the mean fieldansatz and the RVB state becomes non-injective aroundthe U (1) Dirac spin liquid state as a result of an uniqueflat band physics on the Kagome lattice. More specifi-cally, we find that the U (1) Dirac spin liquid state withonly nearest-neighboring RVB parameter can be gen-erated from a continuous family of gauge inequivalentRVB mean field ansatz. The mean field spinon disper-sion within this family changes violently as we tune theRVB parameter on the second and the third neighbor-ing bonds. The implication of such a peculiar behavioris twofold. First, it implies that the optimization of the a r X i v : . [ c ond - m a t . s t r- e l ] S e p RVB parameters around the U (1) Dirac spin liquid stateis a rather subtle practice. In particular, we should in-clude simultaneously the second and the third neighborRVB parameters in the variational optimization. Second,it implies that the spin fluctuation spectrum calculatedat the mean field level is unphysical. We must go beyondthe mean field theory to resolve the ambiguity in the spinfluctuation spectrum.With these considerations in mind, we have performeda systematic variational Monte Carlo(VMC) study on theground state and spin fluctuation spectrum of the spin- KAFH within the RVB theory framework. More specif-ically, we have performed a large scale variational opti-mization of RVB state for the spin- KAFH with boththe second and the third neighboring RVB parameters.We then calculated the spin fluctuation spectrum of thesystem with the Gutzwiller projected RPA(GRPA) the-ory, which has been proved to be rather successful in thestudy of dynamical property of strongly correlated elec-tron systems .We find that the best RVB state of the spin- KAFH isdescribed by a Z gapped mean field ansatz, which hostsa mean field spinon dispersion very different from thatof the U (1) Dirac spin liquid state. However, when wego beyond the mean field description, we find that spinfluctuation spectrum above the optimized RVB state isactually gapless and is almost identical to that abovethe U (1) Dirac spin liquid state. Unlike the mean fieldprediction, we find that the spin fluctuation spectrum ofthe spin- KAFH is not at all featureless, but is char-acterized by a prominent spectral peak at about 0 . J around the M point of the Brillouin zone. We find thatsuch an in tense spectral peak is immersed in a gaplessspin fluctuation continuum, which extends to an energyas high as 2 . J . We find that the spectral characteris-tic predicted by the GRPA theory agrees well with theprediction of recent dynamical DMRG simulation on thespin- KAFH .We have compared our variational spin fluctua-tion spectrum with the INS result on HebertsmithiteZnCu (OH) Cl . Since the intense spectral peak aroundthe M point is such a robust feature in the spin fluc-tuation spectrum of the spin- KAFH, we argue thatthe spectral peak below 2 meV in the INS spectrum ofHebertsmithite ZnCu (OH) Cl should be attributed tothe intrinsic spin fluctuation of the Kagome layer, ratherthan the contribution from the Cu impurity spins be-tween the Kagome layers. This assignment resolve imme-diately the puzzle concerning the strong momentum de-pendence of this low energy spectral peak. It also impliesthat the ground state of Hebertsmithite ZnCu (OH) Cl is much closer to magnetic ordering instability towardthe q = 0 order than we thought before. We propose toverify such a picture by measuring Knight shift on theCu site, rather than the O site, which is blind to the spinfluctuation at the M point as a result of the strong an-tiferromagnetic correlation between nearest neighboringspins on the Kagome lattice. The paper is organized as follows. The theoretical for-malism of this work is presented in the Sec.II. In thissection, we will introduce the RVB theory for the spin- KAFH in both its U (1) and Z form and discuss the sub-tleties of applying the RVB theory on the Kaogme latticecaused by its unique flat band physics. We then presenta Gutzwiller projected RPA(GRPA) theory for the spinfluctuation spectrum on the RVB ground state. The nu-merical result generated from the above RVB theory ispresented in Sec.III. In this section, we will show that theoptimized RVB state for the spin- KAFH is described bya gapped Z mean field ansatz. We then show that con-trary to the prediction of the mean field theory, the spinfluctuation spectrum above the optimized RVB state isactually gapless and is almost identical to that above the U (1) Dirac spin liquid state. We then compare the the-oretical prediction and the INS result on HebertsmithiteZnCu (OH) Cl . In particular, we will show that thelow energy peak around the M point observed in theINS spectra of Hebertsmithite ZnCu (OH) Cl shouldbe attributed to intrinsic spin fluctuation of the Kagomelayers. The last section of the paper is devoted to a con-clusion of the results and a discussion of some remianingproblems. II. AN RVB THEORY OF THE SPIN- KAFHA. The RVB ground state of spin- KAFH
The spin- KAFH studied in this work has the Hamil-tonian H J = J (cid:88) S i · S j . (1)The sum is over nearest-neighboring bonds of theKagome lattice. To describe the spin liquid ground stateof the system in the RVB scheme, we introduce Fermionicslave particle f α and represent the spin operator as S = (cid:80) α,β f † α σ α,β f β . Such a representation is exact whenthe slave Fermion satisfy the constraint (cid:80) α f † α f α = 1.The RVB state is generated from Gutzwiller projectionof BCS-type mean field ground state | RVB (cid:105) = P G | BCS (cid:105) . (2)Here P G denotes the Gutzwiller projection into the singlyoccupied subspace. | BCS (cid:105) is the ground state of the fol-lowing BCS-type Hamiltonian H MF = (cid:88) i,j ψ † i U i,j ψ j . (3)Here ψ i = (cid:18) f i, ↑ f † i, ↓ (cid:19) , U i,j = (cid:18) χ i,j ∆ ∗ i,j ∆ i,j − χ ∗ i,j (cid:19) . χ i,j and ∆ i,j denote the RVB parameters in the hopping and pairingchannel. The structure information of the RVB state isencoded in the mean field Hamiltonian H MF , which isusually called a mean field ansatz of the RVB state.We note that the RVB state so constructed is invari-ant when we perform a SU (2) gauge transformation ofthe form U i,j → G † i U i,j G j on the RVB parameter U i,j ,in which G i is a site-dependent SU (2) matrix . Thus,to generate a symmetric spin liquid state, the RVB or-der parameter U i,j should be invariant under the symme-try operations only up to a SU (2) gauge transformation.The structure of the RVB state can thus be classifiedby the gauge inequivalent way to choose such a gaugetransformation . For example, in a Z spin liquid state,the translational symmetry can be realized either by as-suming a translational invariant RVB mean filed ansatz,or an RVB mean field ansatz that differs by a Z gaugetransformation from the translated ansatz. Here we onlyconsider Z spin liquid state of the second type, whichcan have a smooth connection with the U (1) Dirac spinliquid state studied in Ref.[25]. At the same time, we willkeep RVB order parameters U i,j up to the third neigh-boring bonds.The mean field ansatz of the U (1) and the Z spin liq-uid state studied in this work is illustrated in Fig. 1. Theyellow parallelogram denotes the unit cell of the Kagomelattice, with a and a as its two basis vectors. Theblue, yellow and pink lines denote the first, second andthe third neighboring bonds on the Kagome lattice. Forthe spin liquid state studied here, U i,j is translational in-variant along the a direction, but will change sign whentranslated in the a direction by one lattice constant, ifthe cell index of site i and j in the a direction differ byan odd number.For the U (1) spin liquid state, the RVB parameterstake the form U i,j = − s i,j τ first neighbor − s i,j ρτ second neighbor − s i,j ητ third neighbor (4)Here a chemical potential term is implicitly assumed toenforce the half-filling condition on the Fermion number. τ = (cid:18) − (cid:19) is a Pauli matrix. ρ and η are two realvariational parameters. s i,j = ± U i,j in the a direction. They equal to 1 on the blue, yellow and pinkbonds shown in Fig. 1. When ρ = η = 0, U i,j reduces tothe mean field ansatz of the U (1) Dirac spin liquid statefirst studied in Ref.[25]. We will refer to such a state as U (1)-NN state for brevity below.In a previous work , we have shown that the map-ping between the mean field ansatz and the RVB statebecomes non-injective when ρ = η . More specifically, for − . ≤ ρ = η ≤ .
27, the mean field ground state corre-sponding to the ansazt Eq.(4) is independent of the valueof ρ . Such a peculiar behavior can be understood if werewrite H MF as H MF = H + H ρ + H η , (5) FIG. 1: (Upper panel) Illustration of the mean field ansatzof the U (1) and the Z spin liquid state studied in this pa-per. The yellow parallelogram denotes the unit cell of theKagome lattice, with a and a as its two basis vectors. Thespinon unit cell is doubled in the a direction and containssix sites(site µ = 1 , ...., U i,j is translationalinvariant along the a direction, but will change sign whentranslated in the a direction by one lattice constant, if thecell indices in the a direction of site i and j differ by anodd number. (Lower panel)Illustration of the path in themomentum space(Γ − M (cid:48) − M − K − M (cid:48)(cid:48) − Γ) along whichthe spin fluctuation spectrum is calculated. The elementaryhexagons plotted in dashed line denote the Brillouin zones ofthe Kagome lattice. in which H , H ρ and H η denote the part of H MF con-tributed by spinon hopping between the first, second andthe third neighboring sites. It is then easy to check that[ H , H ρ + H η ] = 0 , (6)when ρ = η . The eigenstate of the mean field Hamilto-nian thus does not depend on the value of ρ when ρ = η .The peculiarity of H MF discussed above is deeply re-lated to the unique flat band physics on the Kagome lat-tice. It is well known that there is a flat band (with aneigenvalue of 2) in the mean field spinon dispersion ofthe U (1)-NN state(i.e., when ρ = η = 0). The originof such a flat band can be traced back to the destruc-tive interference between the hopping amplitudes out ofa localized Wannier orbital of the form shown in Fig.2.Interestingly, one find that such a destructive interference FIG. 2: Illustration of the localized Wannier orbital of theflat band of the mean field ansatz Eq.(4). The wave functionamplitude on the red, blue and white sites are +1, -1 and 0.The hopping integral on the dashed bonds has an additionalminus sign as a result of the factor s i,j in Eq.(4). The hoppingamplitudes from the red and the blue sites to any white siteadd to zero when ρ = η . remains effective in the presence of the second and thethird neighboring hopping terms, provided that ρ = η .A nonzero ρ = η thus only shift the eigenvalue of theflat band, but does not change its wave function. In thewhole range of ρ (= η ) ∈ [ − . , . ρ = η . Thusthe U (1)-NN state originally studied in Ref.[25] can ac-tually be generated from a continuously family of gaugeinequivalent RVB mean field ansatz. Such a peculiaritynot only complicates the optimization of the RVB param-eters around the U (1)-NN state, but also implies that thespin fluctuation spectrum calculated at the mean fieldlevel is unphysical, in we insist on relating the excitationcharacteristic of a quantum system to its ground statestructure.For the Z spin liquid state, the RVB parameters takethe form U i,j = − µ(cid:126)n φ · (cid:126)τ on − site − s i,j τ first neighbor − s i,j ρ(cid:126)n φ · (cid:126)τ second neighbor − s i,j η(cid:126)n φ · (cid:126)τ third neighbor (7)Here (cid:126)τ = ( τ , τ , τ ) are the Pauli matrices, (cid:126)n φ =(sin φ, , cos φ ) is a unit vector in the τ − τ plane. µ, ρ, η and φ , , are six real variational parameters of the Z spin liquid state. It can be easily checked that spin liquidstate generated from the U (1) and the Z ansatz respectall physical symmetry of the spin- KAFH. At the sametime, the Z spin liquid state reduces to the U (1) spinliquid state when φ , , = N π , in which N is an arbitraryinteger. The mapping between the RVB parameters andthe spin liquid state becomes non-injective when ρ = η and φ = φ = φ = π . B. The spin fluctuation spectrum of the spin- KAFH above the RVB ground state
The spin fluctuation spectrum of the system can beextracted from the dynamical spin susceptibility definedbelow χ i,j ( q , τ ) = −(cid:104) T τ S i ( q , τ ) S j ( − q , (cid:105) . (8)Here S i ( q ) = 12 (cid:88) k ,α,β,µ e i q · δ µ f † k + q ,µ,α σ iα,β f k ,µ,β , (9)is the i -th component of spin density operator at mo-mentum q . µ = 1 , ...., δ µ is the displacementof the µ -th sublattice with respect to the origin of thespinon unit cell. Since the ground state of the systemis spin rotational symmetric, we will concentrate on thefluctuation of the z-component of the spin density op-erator, which can be expressed in terms of the Nambuspinor as followsS z ( q ) = 12 (cid:88) k ,µ e i q · δ µ ψ † k + q ,µ ψ k ,µ , (10)in which ψ k ,µ = (cid:18) f k ,µ, ↑ f †− k ,µ, ↓ (cid:19) is the Nambu spinor. Atthe mean field level, the dynamical spin susceptibility isgiven by χ z,z ( q , τ ) = 14 (cid:88) k ,µ,ν e i q · ( δ µ − δ ν ) Tr [ G νµ ( k + q , − τ ) G µ,ν ( k , τ )] , (11)in which G µ,ν ( k , τ ) = −(cid:104) T τ ψ k ,µ ( τ ) ψ † k ,ν (0) (cid:105) is the spinonGreen’s function calculated at the mean field level.Anticipating the inadequacy of the mean field theory,in particular the subtleties of the RVB mean field the-ory on the Kagome lattice related to its unique flat bandphysics, we present in the following a Gutzwiller pro-jected RPA(GRPA) theory of the spin fluctuation spec-trum for the spin- KAFH. The essence of such a theoryis to diagonalize the Heisenberg Hamiltonian in a sub-space spanned by the Gutzwiller projected mean fieldexcited state . These Gutzwiller projected mean fieldexcited states are connected to the RVB ground statethrough the operation of the spin density operator S z ( q ).Since [S z ( q ) , P G ] = 0, we haveS z ( q ) | RVB (cid:105) = S z ( q )P G | BCS (cid:105) = P G S z ( q ) | BCS (cid:105) = (cid:88) k ,m,n P G φ k ,m,n ( q ) γ † k + q ,m, ↑ γ †− k ,n, ↓ | BCS (cid:105) = (cid:88) k ,m,n φ k ,m,n ( q )P G γ † k + q ,m, ↑ γ †− k ,n, ↓ | BCS (cid:105) = (cid:88) k ,m,n φ k ,m,n ( q ) | k , q , m, n (cid:105) , (12)in which φ k ,m,n ( q ) = 12 (cid:88) µ e i q · δ µ ( v ∗ k + q ,µ,m u k ,µ,n − u ∗ k + q ,µ,m v k ,µ,n ) . (13) | k , q , m, n (cid:105) = P G γ † k + q ,m, ↑ γ †− k ,n, ↓ | BCS (cid:105) is a Gutzwillerprojected mean field excited state generate by S z ( q ). m and n are the indices of the quasiparticle eigenstate. Thequasiparticle operator γ k ,m,α is related to the bare spinonoperator f k ,µ,α through the Bogoliubov transformation (cid:18) f k ,µ, ↑ f †− k ,µ, ↓ (cid:19) = (cid:18) u k − v k v k u k (cid:19) (cid:18) γ k ,m, ↑ γ †− k ,m, ↓ (cid:19) , (14)in which u k and v k are 6 × u k ,µ,m and v k ,µ,m as their matrix element. It can be shown that ifboth χ i,j and ∆ i,j are real and symmetric, then u − k = u ∗ k , v − k = v ∗ k .We thus choose the subspace spanned by | k , q , m, n (cid:105) =P G γ † k + q ,m, ↑ γ †− k ,n, ↓ | BCS (cid:105) as our working subspace and di-agonalize H J in this subspace to construct a variationalapproximation of the spin fluctuation spectrum for thespin- KAFH. The Gutzwiller projected mean field ex-cited state | k , q , m, n (cid:105) is in general not orthonormal, wethus have to solve a generalized eigenvalue problem ofthe form H ϕ i = λ i O ϕ i . (15)Here the element of the Hamiltonian matrix H and theoverlap matrix O are given by H k (cid:48) ,m (cid:48) ,n (cid:48) ; k ,m,n = (cid:104) k (cid:48) , q , m (cid:48) , n (cid:48) | H J | k , q , m, n (cid:105) O k (cid:48) ,m (cid:48) ,n (cid:48) ; k ,m,n = (cid:104) k (cid:48) , q , m (cid:48) , n (cid:48) | k , q , m, n (cid:105) , (16)These matrix elements can be evaluated with a highly ef-ficient re-weighting technique in VMC simulation . ϕ i denotes the generalized eigenvector of Eq.(15) with eigen-value λ i . It is normalized as follows ϕ † i O ϕ j = δ i,j . (17)The variational spin fluctuation spectrum calculated inthe above subspace is given by S ( q , ω ) = 1 N (cid:88) i | φ † O ϕ i | δ ( ω − ( λ i − E g )) , (18)in which φ is a vector with φ k ,m,n as its components, E g is the variational ground state energy of the RVB state, N denotes the number of lattice site in the system.In recent years, such a variational theory has been ap-plied successfully in the study of dynamical propertiesof several strongly correlated electron systems . Theinternal consistency of the theory can be seen from thefact that the spin fluctuation spectrum so constructedsatisfies the momentum-resolved sum rule of the form (cid:90) ∞ dωS ( q , ω ) = S ( q ) = 1 N (cid:104) S z ( q )S z ( − q ) (cid:105) . (19)Here S ( q ) denotes the static spin structure factor calcu-lated on the RVB ground state. r , h L r h FIG. 3: The optimized value of the RVB parameters in the U (1) spin liquid state. The optimized RVB parameters arefound to be very close the non-injective line ρ = η , where themapping between the mean field ansatz and the RVB statebecomes singular. m , r , h L m r h ( a ) f L f f f ( b ) FIG. 4: The optimized value of the RVB parameters in the Z spin liquid state, (a)the amplitudes µ , ρ and η , (b)the gaugeangles φ , φ and φ . The optimized RVB parameters arefound to be very close to the non-injective line ρ = η and φ = φ = φ = π , where the mapping between the meanfield ansatz and the RVB state becomes singular. III. RESULTS AND DISCUSSIONSA. The variational ground state of the spin- KAFH
The RVB parameters in the U (1) and the Z RVB stateare optimized through variational Monte Carlo simula-tion. The calculation is done on a L × L × E L - 4 U ( 1 ) Z - 5 - 4 - 0 . 4 2 8 7 7 5- 0 . 4 2 8 7 5 0- 0 . 4 2 8 7 2 5- 0 . 4 2 8 7 0 0 L - 4 FIG. 5: The variational energy of the U (1) and the Z spinliquid state as a function of L . The details at large L isshown in the inset. The error bar of the data is smaller thanthe symbol size. The blue solid circles in the inset is the resultof Ref.[44] for the U (1) spin liquid state. studies , it is claimed that the best RVB stateof the spin- KAFH is described by a U (1) gapless meanfield ansatz. We find that this is not true. In Fig.2 andFig.3, we present the optimized RVB parameters for the U (1) and the Z spin liquid state. The largest clustersize that we have achieved good convergence in the RVBparameters is L = 18, beyond which the optimizationprocedure becomes numerically too expensive. The en-ergy of the optimized U (1) and Z spin liquid state isplotted in Fig.4 as a function of L . The Z RVB statehas clearly a lower energy than the U (1) state in thethermodynamic limit.We note that the energy difference between the U (1)and the Z spin liquid state is extremely small. To findout how close the two states are in the Hilbert space, wehave calculated their overlap on finite clusters. We findthat the overlap is still as large as 0 .
93 on a L = 18 clus-ter. Both states are also very close to the U (1)-NN statestudied in Ref.[25], since the optimized RVB parametersof the two states are both very close to the non-injectiveline. The closeness of the U (1), Z and the U (1)-NN statecan be made more transparent by comparing their staticspin structure factor. As is shown in Fig.6, the static spinstructure factor of these states are almost identical bothbefore and after the Gutzwiller projection. We note thatthe maximum of the static spin structure factor movesfrom the K point to the M point after the Gutzwillerprojection. The strength of the spin fluctuation is alsoenhanced by the Gutzwiller projection.We now show that the Z spin liquid state has indeeda finite spinon gap at the mean field level. Here we willuse the RVB parameter on the nearest-neighboring bondas the unit of energy. The mean field spinon gap on finiteclusters for both the U (1) and the Z spin liquid state areplotted in Fig.7. We find that the mean field spinon gapin the U (1) spin liquid state extrapolates to zero in thethermodynamic limit as ∆ ≈ αL − , a scaling behavior S (q) U ( 1 ) Z U ( 1 ) - N N G M ' M K G ( a ) U ( 1 ) Z U ( 1 ) - N N S (q) G M ' M K G ( b ) FIG. 6: The static spin structure factor of the U (1), Z andthe U (1)-NN state along Γ − M (cid:48) − M − K − M (cid:48)(cid:48) − Γ. (a)Resultsin the mean field RVB state, (b)Results in Gutzwiller pro-jected RVB state. D L - 1 U ( 1 ) g a p( a ) D L - 2 Z g a p( b ) FIG. 7: The scaling behavior of the mean field spinon gap.The red dashed lines denote the linear fitting of the data.Note that the mean field spinon gap in the U (1) and the Z spin liquid state exhibit different scaling behavior at large L . naturally expected for a Dirac spin liquid. On the otherhand, the mean field spinon gap of the Z spin liquid stateis found to approach a finite value in the thermodynamiclimit as ∆ ≈ ∆ + βL − , with ∆ (cid:39) .
1. Such a scalingbehavior is just what one expect for a gapped system .However, we note that the mean field spinon disper-sion is subjected to the ambiguity related to the peculiarflat band physics of the Kagome lattice and is thus un-physical. In the next subsection, we will present the spinfluctuation spectrum calculated from the GRPA theoryon the RVB state. To one’s surprise, one find that thespin fluctuation spectrum above the Z RVB state is ac-tually gapless and is almost identical to that above the U (1) and the U (1)-NN spin liquid state. B. The variational spin fluctuation spectrum of thespin- KAFH
While the U (1), Z and the U (1)-NN RVB stateare very close to each other in the Hilbert space, theyhost very different mean field spinon dispersion. InFig.8, we plot the spin fluctuation spectrum of thesestates calculated at the mean field level along the pathΓ − M (cid:48) − M − K − M (cid:48)(cid:48) − Γ. The distribution of spectralweight in energy is found to be very different in thesestates, although the integration of spectral weight overenergy, namely, the static spin structure factor S ( q ), arealmost identical in these states. The RVB mean fieldtheory thus fails to provide a reliable prediction on thespin fluctuation spectrum of the spin- KAFH, if we in-sist on relating the excitation characteristic of a quantumsystem to its ground state structure.We now go beyond the mean field theory. The spinfluctuation spectrum calculated from the GRPA theoryis plotted in Fig.9 for the U (1), Z and the U (1)-NNRVB state. The calculation is done on a L = 12 clusterwith a broadening of δ = 0 . J in energy. To one’s sur-prise, the spin fluctuation spectrum above all these threestates are found to be almost identical with each other.This is, however, just what one should expect if we notethat these three states are very close to each other in theHilbert space. Equally surprising is the huge differencebetween the spin fluctuation spectrum calculated fromthe GRPA theory and that from the RVB mean fieldtheory. More specifically, the spin fluctuation spectrumcalculated from the GRPA theory is characterized by aprominent spectral peak around the M point, with anenergy of about 0 . J for L = 12. The remaining spinfluctuation spectral weight is distributed in a broad andalmost featureless continuum extending to an energy ashigh as 2 . J .We note that the intense spectral peak around the Mpoint lies within the spin excitation continuum, whoselower boundary is marked by the white lines in Fig.9.The lower boundary of the continuum reaches its mini-mum at the Γ , M , M (cid:48) and the M (cid:48)(cid:48) point(as is predictedcorrectly by the RVB mean field theory) and defines the FIG. 8: The mean field spin fluctuation spectrum of the U (1)(a,b), Z (c,d) and the U (1)-NN(e,f) RVB state alongΓ − M (cid:48) − M − K − M (cid:48)(cid:48) − Γ, plotted in linear(left column) andlogarithmic(right column) scale. The RVB parameter on thenearest neighboring bond is used as the unit of energy. Themean field spin fluctuation spectrum of the three states arefound to be very different, although they have almost identicalstatic spin structure factor(see Fig.6a). spin gap of the system. We find that the spin gap sodefined decreases with L and extrapolates to zero in thethermodynamical limit for all these three RVB states, ascan be seen in Fig.10. The Z RVB state we found isthus actually a gapless spin liquid state, although it isdescribed by a gapped mean field ansatz.To see more clearly the spectral weight distribution inenergy, we plot in Fig.11 the spin fluctuation spectrumat the M point of the L = 12 and L = 24 cluster. Herewe only present the result of the U (1)-NN state for clar-ity, since the spectrum of the other two states are almostidentical. The most prominent feature of the spectrumat the M point is the intense spectral peak inside thebroad continuum, which contains about half of the totalspin fluctuation spectral weight. We note that the samespectral characteristic is found also in dynamical DMRGsimulation of the spin- KAFH , the result of which isshown in the inset of Fig.11(a) for comparison. The ex-istence of such an intense spectral peak around M pointis thus a robust feature in the spin fluctuation spectrumof the spin- KAFH. We find that the energy of thispeak, here denoted as ω , decreases with the system sizeand extrapolates to a value of no less than 0 . J in thethermodynamic limit(see Fig.11(b)). S (q=M, w ) w U ( 1 ) Z U ( 1 ) - N N( g )
FIG. 9: The spin fluctuation spectrum calculated from theGRPA theory for the U (1)(a,b), Z (c,d) and the U (1)-NN(e,f)RVB state along Γ − M (cid:48) − M − K − M (cid:48)(cid:48) − Γ, plotted in linear(leftcolumn) and logarithmic(right column) scale. The Heisenbergexchange coupling J is used as the unit of energy. The spinfluctuation spectrum above the three states are found to bealmost identical, as can be seen more clearly in (g), in whichwe compare the spin fluctuation spectrum of the three statesat the M point. The white lines in (a)-(f) mark the lowerboundary of the spin fluctuation continuum. C. Comparison with the experimental results onHebertsmithite ZnCu (OH) Cl Previous INS measurement finds that the spin fluctua-tion spectrum of Hebertsmithite ZnCu (OH) Cl , whichis believed to be an ideal realization of the spin- KAFH,is characterized by a featureless continuum above 2meV . Below 2 meV, the spectral intensity increaseswith decreasing energy and aggregates toward the M point in momentum space. As a result of such a trend, abroad peak emerges around the M point below 2 meV. U ( 1 ) Z U ( 1 ) - N N D FIG. 10: Scaling of the spin gap in the U (1), Z and U (1)-NNRVB state with 1 /L . The Heisenberg exchange coupling J isused as the unit of energy. S (q=M, w ) w L = 2 4 L = 1 2
S(q=M, w ) w ( a ) w ( b ) FIG. 11: (a)The spin fluctuation spectrum at the M pointfor L = 12 and L = 24. The Heisenberg exchange coupling J is used as the unit of energy. Shown in the inset is theDMRG result reproduced from Ref.[32]. (b)The energy ofthe intense spectral peak at the M point, here denoted as ω ,as a function of 1 /L . It is widely believed that such a low energy spectralpeak should be attributed to Cu impurity spins occu-pying the Zn site between the Kagome layers, ratherthan the intrinsic spin fluctuation of the Kagome layer.According to such a picture, the intrinsic spin fluctua-tion spectrum of the Kagome layer is characterized by afeatureless continuum with probably a small gap. Thisis supported by a later NMR study on the system , inwhich the Knight shift on the Oxygen site is measured.It is found that the Knight shift vanishes in the zero tem-perature limit for Oxygen site far away from the Cu spectral weight w S ( q = M , w ) S o x y ( q = M , w ) FIG. 12: Comparison between the fluctuation spectrum ofthe local field on the Oxygen site and the full spin fluctuationspectrum at the M point. The Heisenberg exchange coupling J in the spin- NN-KAFH is used as the unit of energy. TheOxygen site is found to be almost blind to the intense spectralpeak in S ( q = M , ω ). impurity spin. A spin gap of the order of 0 . − . J is claimed by fitting the temperature dependence of theKnight shift data on such Oxygen site.However, there are three reasons to object such a pic-ture. First, if the spectral peak below 2 meV is indeedcontributed by the Cu impurity spins, it should not ex-hibit such a strong momentum dependence as observedin the INS measurement. Second, as the intense spectralpeak around the M point is such a prominent featurein the spin fluctuation spectrum of the spin- KAFH,in particular, as it contains almost half of the total spinfluctuation spectral weight at the M point, it must ap-pear somewhere in the INS spectrum of HebertsmithiteZnCu (OH) Cl , if the latter is indeed an ideal realiza-tion of the spin- KAFH. It is thus very likely that thepeak below 2 meV in the INS spectrum of HebertsmithiteZnCu (OH) Cl corresponds just to such a theoreticallypredicted spectral feature. Third, since the nuclear spinon the Oxygen site is coupled symmetrically to the twoneighboring Cu spins but is almost decoupled from thethird Cu spin in the Kagome unit cell, a very smallKnight shift on the Oxygen site does not necessarily im-ply a very small uniform spin susceptibility of the system.It may simply imply that the two neighboring Cu spinsthat the Oxygen nuclear spin are symmetrically coupledto are antiferromagnetically correlated with each other.We note that the most recent Knight shift result on theOxygen site of Hebertsmithite ZnCu (OH) Cl is consis-tent with a power law(rather than exponential) decayof the uniform susceptibility with temperature, whichimplies that the intrinsic spin fluctuation spectrum ofKagome layer is gapless .To verify such a picture, we have studied the fluctua-tion spectrum of the hyperfine field at the Oxygen site,which is proportional to the imaginary part of the follow-ing local spin susceptibility χ z,z oxy ( q , τ ) = −(cid:104) T τ S z oxy ( q , τ ) S z oxy ( − q , (cid:105) . (20) Here S z oxy ( q ) = 12 (cid:88) k ,µ =1 , ψ † k + q ,µ ψ k ,µ , (21)denotes the sum of the spin density operator on sublat-tice 1 and 2 at momentum q . In Fig.12, we comparethe spectral function of such fluctuation, here denotedas S oxy ( q , ω ), with the full spin fluctuation spectrum S ( q , ω ) at q = M . It is found that S oxy ( q = M , ω ) is al-most blind to the intense spectral peak in S ( q = M , ω ).This indicates that the Knight shift on the Oxygen siteis not a sensitive probe of such a prominent spectral fea-ture of the spin- KAFH. We suggest to perform Knightshift measurement directly on the Cu site to resolve thisissue.
IV. CONCLUSIONS AND OUTLOOKS
In this work, we have performed a systematic varia-tional Monte Carlo study on the ground state and spinfluctuation spectrum of the spin- KAFH in the RVBtheory framework. We find that the best RVB statefor the spin- KAFH is described by a Z gapped meanfield ansatz, which hosts a mean field spinon dispersionvery different from that of the U (1) Dirac spin liquidstate(denoted in this work as U (1)-NN state) originallystudied in Ref.[25]. However, we find that the two statesare actually very close to each other in the Hilbert spaceas a result of the non-injective nature of the mapping be-tween the mean field ansatz and the RVB state aroundthe U (1)-NN state. We find that such a singular behavioris deeply related to the unique flat band physics on theKagome lattice and signals the failure of the RVB meanfield theory for the spin- KAFH.Going beyond the RVB mean field theory, we showwith the GRPA theory that the spin fluctuation spec-trum above the Z RVB state is almost identical to thatabove the U (1)-NN state and is actually gapless. We findthat the spin fluctuation spectrum of the spin- KAFHis not at all featureless, but is characterized by a promi-nent spectral peak around the M point at low energy,which contains about half of the total spin fluctuationspectral weight in that momentum region. Interestingly,we find that such an intense spectral peak, which hasan energy ω ≈ . J in the thermodynamic limit, lieswithin a gapless continuum extending to 2 . J . Such aspectral characteristic is found to agree well with theprediction of recent dynamical DMRG simulation on thespin- KAFH. We thus believe that the intense spectralpeak at ω should be a robust feature in the spin fluctu-ation spectrum of the spin- KAFH.We argue that the spectral peak below 2 meV in theINS spectrum of Hebertsmithite ZnCu (OH) Cl shouldbe attributed to the intrinsic spin fluctuation of theKagome layer, or, more specifically, the prominent spec-tral peak we found around the M point, rather than the0fluctuation of Cu impurity spins occupying the Zn site between the Kagome layers. A smaller(or even zero)value of the observed peak energy ω as compared to thetheoretical prediction(according to which ω ≈ . J )implies that Hebertsmithite ZnCu (OH) Cl is muchcloser to the q = 0 magnetic ordering instability thanwe thought before. We show that the Knight shift onthe Oxygen site is blind to such an intense spectral peakaround the M point as a result of strong antiferromag-netic correlation between nearest neighboring spins onthe Kagome lattice. We propose to use the Knight shifton the Cu site as a direct probe of this important spectralfeature of the spin- KAFH.The results presented in this work constitute a con-crete example in which the RVB mean field theory failseven at a qualitative level. At the same time, it demon-strates once more the power of the GRPA theory in de- scribing the dynamical properties of the quantum magnetsystems. However, giving the fact that the Z and the U (1)-NN state exhibit almost identical excitation behav-ior in the spin triplet channel, one can not help askingif there exists any qualitative difference in the excitationbehavior of the two states in the spin singlet channel. Inparticular, does the Z RVB state host a gapped gaugefluctuation spectrum, or, as in the case of the U (1)-NNstate, is gapless in the gauge channel? 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Nat.Phys. , 62(2015). J. W. Mei and X. G. Wen, arXiv:1507.03007. F. Ferrari, A. Parola, S. Sorella, and F. Becca, Phys. Rev.B , 235103 (2018). F. Ferrari and F. Becca, Phys. Rev. B , 100405(R)(2018). F. Ferrari and F. Becca, Phys. Rev. X , 031026 (2019). K. Ido, M. Imada, and T. Misawa, Phys. Rev. B ,075124 (2020). F. Ferrari and F. Becca, Phys. Rev. B , 014417 (2020) We are grateful to S. Sorella for pointing out to us thisfact. We note that the spin fluctuation spectrum of the U (1)-NN state has been calculated earlier with the GRPA the-ory in Ref.[49]. However, the spectral weight distributionthey found is very different from what we presented here.We think that such a difference may originates from themuch larger broadening they have used( δ = 0 . J as com-pared to δ = 0 . J used in our calculation), which mayhave masked the important spectral signature of the spin- KAFH. We note that the cluster size we have used(thelargest cluster we have attempted has 24 × ××