Entanglement Hamiltonian of the quantum Néel state
EEntanglement Hamiltonian of the quantum N´eel state
Didier Poilblanc Laboratoire de Physique Th´eorique, CNRS, UMR 5152 and Universit´e de Toulouse, UPS, F-31062 Toulouse, France (Dated: September 10, 2018)Two-dimensional Projected Entangled Pair States (PEPS) provide a unique framework giving ac-cess to detailed entanglement features of correlated (spin or electronic) systems. For a bi-partitionedquantum system, it has been argued that the Entanglement Spectrum (ES) is in a one-to-one corre-spondence with the physical edge spectrum on the cut and that the structure of the correspondingEntanglement Hamiltonian (EH) reflects closely bulk properties (finite correlation length, criticality,topological order, etc...). However, entanglement properties of systems with spontaneously brokencontinuous symmetry are still not fully understood. The spin-1/2 square lattice Heisenberg anti-ferromagnet provides a simple example showing spontaneous breaking of SU(2) symmetry down toU(1). The ground state can be viewed as a “quantum N´eel state” where the classical (N´eel) stag-gered magnetization is reduced by quantum fluctuations. Here I consider the (critical) ResonatingValence Bond state doped with spinons to describe such a state, that enables to use the associatedPEPS representation (with virtual bond dimension D = 3) to compute the EH and the ES for apartition of an (infinite) cylinder. In particular, I find that the EH is (almost exactly) a chain ofa dilute mixture of heavy ( ↓ spins) and light ( ↑ spins) hardcore bosons, where light particles aresubject to long-range hoppings. The corresponding ES shows drastic differences with the typical ESobtained previously for ground states with restored SU(2)-symmetry (on finite systems). I. INTRODUCTION
It is known from early Quantum Monte Carlo (QMC)simulations that the ground state (GS) of the spin-1/2 Heisenberg antiferromagnet (AFM) on the bipar-tite square lattice is magnetically ordered [1] and, hence,breaks the hamiltonian SU(2) symmetry. The GS canbe viewed as a “quantum N´eel state” (QNS) where themaximum classical value m stag = 1 / criticalstate [3].Recently, a number of new powerful tools based on en-tanglement measures have emerged. The entanglementspectrum (ES) and its associated Entanglement Hamil-tonian defined via the reduced density matrix (RDM)of a bi-partioned quantum system (see definitions later)provides new insights. In particular, it has been ar-gued that the ES is in a one-to-one correspondence withthe physical edge spectrum on the cut for topologicalground states [4] and low-dimensional quantum antifer-romagnets [5] and that the structure of the correspond-ing Entanglement Hamiltonian (EH) reflects closely thebulk properties (holographic principle) [6]. However,new interesting features might arise in the entanglementproperties of systems with spontaneously broken con-tinuous symmetry, such as the QNS for which SU(2)symmetry is broken down to U(1). First, the entan-glement entropy (the entropy associated to the RDM)have revealed anomalous additive (logarithmic) correc-tions [7, 8] to the area law – i.e. the linear (asymp-totic) scaling of the entropy with the length of the cut. It was proposed afterwards that the origin of such cor-rections may lie in the existence of Goldstone modes [9]associated to the spontaneously broken continuous sym-metry. Note however that, in any finite system (as inmost “exact” simulations), the SU(2) symmetry is re-stored by quantum fluctuations and one has a unique GSinstead of a degenerate manifold. In fact, recent state-of-the-art SU(2)-symmetric Density Matrix Renormal-ization Group (DMRG) studies established an interest-ing correspondence [10], in the (singlet) ground state oftwo-dimensional antiferromagnets in their magnetically-ordered phases, between the SU(2) tower of states andthe lower part of the ES below an “entanglement gap”(although DMRG does not provide information on themomenta of the ES). This suggests strongly that theabove-mentioned corrections in the entropy should beassociated with the tower of states structure, while thearea law arises from ES levels above the entanglementgap [10]. A priori important differences may occur inthe entanglement properties of a N´eel-like wave-functionbreaking the continuous SU(2) symmetry explicitly i.e.with a finite staggered magnetization. In particular, oneexpects the ES (and the EH) of a symmetry-broken QNSto differ qualitatively from the ones associated to the GSwith restored SU(2) symmetry, computed on finite sys-tems [10, 11]. Computing the entanglement properties ofa (variational) state with a finite order parameter is themain goal of this paper.The formalism of Projected Entangled Pair States(PEPS) [6, 12] enables i) to easily construct symmetrybroken variational states and ii) to compute the corre-sponding EH. Note that, for a given variational stateand system size (one uses infinite cylinders with a finiteperimeter), the calculation of the EH is fundamentally ex-act and provides a complete analytic expansion in termsof N-body interactions whose amplitudes are numerically a r X i v : . [ c ond - m a t . s t r- e l ] S e p FIG. 1. (a) The N´eel state is represented as a spinon-dopedRVB state : Singlets are oriented from the A to the B sub-lattice and doped spinons are polarized along ˆ z ( − ˆ z ) on theA (B) sites. Implicitly, a sum over all singlet/spinon config-urations is assumed, the average spinon density being con-trolled by a fugacity. (b) Under a π -rotation around ˆ y on allthe B-sites, all spinons become oriented along ˆ z and singletstransform into √ {| ↑↑ (cid:11) + | ↓↓ (cid:11) } on every NN bonds. computed. Here I therefore make use of a simple PEPSansatz of the QNS in order to calculate its EH associ-ated to a bi-partition of an infinite cylinder. The vari-ational wave function used here is in fact the simplestPEPS (i.e. with the smallest bond dimension D = 3) onecan construct to capture the physics of the symmetry-broken N´eel state. Ans¨atze with a larger bond dimensionwill not allow to consider a cylinder with a large enoughperimeter. Note that the PEPS formalism provides alsothe momentum-resolved ES. This is to be contrasted toDMRG that also gives easily the ES but without thecorresponding momenta of the Schmidt states. Also ananalytic form of the EH cannot be obtained in DMRG.As shown recently using PEPS, a EH with local in-teractions is expected in a gapped bulk phase (withshort-range entanglement), whereas a diverging interac-tion length of the EH is the hallmark of critical behaviorin the bulk [6]. One therefore expects to see fingerprintsof the critical behavior of the QNS in its EntanglementHamiltonian. II. DOPED-RVB ANSATZ FOR THE N´EELSTATE
I start with the square lattice Resonating Valence Bond(RVB) wavefunction defined as an equal-weight superpo-sition of nearest-neighbor (NN) hardcore singlet cover-ings [13, 14]. The sign structure of the wave function isfixed by imposing that the singlets | ↑↓(cid:105) − | ↓↑(cid:105) are alloriented from one A sublattice to the other B sublat-tice. Such a wave function is a global spin singlet – i.e. aSU(2)-invariant state – with algebraic (i.e. critical) dimercorrelations (and short-range spin correlations) [15, 16]. To construct a simple ansatz for the QNS, let us nowassume that one breaks SU(2) symmetry down to U(1)by doping the NN RVB state with on-site spinons (i.e.spin-1/2 excitations) with opposite orientations on thetwo sublattices. For simplicity, I choose hereafter thestaggered magnetization pointing along the ˆ z -axis. Sucha simple ansatz is schematically shown in Fig. 1(a). Theaverage density of spinons – identical on the two sublat-tices – directly gives the staggered magnetization m stag ( ×
2) and, as one will see later on, can be controlled by afugacity γ .Before going further, it is convenient to rewrite theN´eel-RVB state in a translationally invariant form. In-deed, under a (spin) π -rotation around ˆ y on the B-sites,B-spinons transform as | ↓ (cid:11) → | ↑ (cid:11) and | ↑ (cid:11) → −| ↓ (cid:11) .Under such a (unitary) transformation, the new N´eel-RVB state acquires the same (average) polarizationon the A and B sublattices as shown in Fig. 1(b).The original NN singlets are also transformed into √ {| ↑↑ (cid:11) + | ↓↓ (cid:11) } dimers which are now symmetricw.r.t. the bond centers. III. PEPS CONSTRUCTION ANDENERGETICS
Such a state can in fact be represented by a PEPS | Ψ PEPS (cid:11) with bond dimension D = 3, where each lat-tice site is replaced by a rank-5 tensor A sα,α (cid:48) ; β,β (cid:48) labeledby one physical index, s = 0 or 1, and by four virtualbond indices (varying from 0 to 2) along the horizon-tal ( α, α (cid:48) ) and vertical ( β, β (cid:48) ) directions, as shown inFig. 2(a). Physically, the absence of singlet on a bondis encoded by the virtual index being ”2” on that bond.I define : A = R + γ S , (1)where R is the original RVB tensor [17, 18], S is a polar-ized spinon tensor and γ ∈ R is a fugacity controlling theaverage spinon density. To enforce the hardcore dimerconstraint, one takes R sα,α (cid:48) ; β,β (cid:48) = 1 whenever three vir-tual indices equal 2 and the fourth one equals s , and R sα,α (cid:48) ; β,β (cid:48) = 0 otherwise. The spinon tensor has onlyone non-zero element, S , , = 1. The wave functionamplitudes are then obtained by contracting all virtualindices (except the ones at the boundary of the system).Note that the above PEPS ansatz for the N´eel state baressimilarities with the one used to describe the honeycombRVB spin liquid under an applied magnetic field [19].However, a crutial difference is that this new ansatz is,by construction, fully U(1)-invariant in contrast to thespinon-doped RVB state of Ref. [19].Following a usual procedure, I now place the squarelattice of tensors on infinite cylinders with N v sites in theperiodic (vertical) direction as shown in Fig. 2(b) and usestandard techniques (involving exact tensor contractionsand iterations of the transfer operator) to compute rele-vant observables. In the PEPS formulation the boundary FIG. 2. (Color online) (a) Local (rank-5) PEPS tensor. (b)Tensors are placed on a square lattice wrapped on a cylinderof perimeter N v and (quasi-) infinite length N h (cid:29) N v . B L and B R boundary conditions are realized by fixing the virtualvariables going out of the cylinder ends. A bipartition of thecylinder generates two L and R edges along the cut. m stag -0.7-0.65-0.6-0.55-0.5 NN ene r g y ( pe r s i t e ) N v =4N v =6 m stag NNN ene r g y ( pe r s i t e ) exact (QMC) RVB Neel NeelRVB (a) (b)
FIG. 3. (Color online) NN (a) and next-NN (b) correlators2 (cid:10) S i · S j (cid:11) – corresponding to the energies per site in units ofthe coupling constants – plotted as a function of m stag . Com-putations are done on infinite cylinders of perimeter N v = 4and N v = 6. conditions B L and B R can be simply set by fixing the vir-tual states on the bonds “sticking out” at each cylinderend. E.g. open boundary conditions are obtained bysetting the boundary virtual indices to “2”. Generalizedboundary conditions can be realized as in Fig. 2(b) bysetting some of the virtual indices on the ends to “0” or“1”.I have computed the (staggered) magnetization m stag and the expectation values of the spin-1/2 Heisenberg ex-change interactions S i · S j between NN and next-NN sites,varying γ from zero to large values (to approach the clas-sical N´eel state). The data (normalized as the energy persite of the corresponding Heisenberg model) are displayedas a function of m stag in Fig. 3(a,b). The NN energyshows a broad minimum around m stag ∼ .
35, a value a bit larger than the QMC extrapolation ∼ .
307 [2] forthe pure NN quantum AFM. However, (i) the variationalenergy curve is rather flat around the minimum and (ii)the minimum energy is only within ∼ .
5% of the QMCestimate, a remarkable result considering the simplicityof the one-dimensional family of D = 3 PEPS. Note alsothat the minimum energy agrees very well with optimized D = 3 iPEPS [21] and finite PEPS up to D = 6 [22].For completeness, I also show the next-NN energy inFig. 3(b). In fact, the pure (critical) RVB state providesthe lowest next-NN exchange energy, suggesting the ex-istence of a transition, upon increasing the next-NN cou-pling, from the N´eel state to a gapless spin liquid [23, 24].Note that a direct transition from the N´eel state to a Va-lence Bond Crystal – with no intermediate gapless spinliquid phase – is also a realistic scenario [3, 25]. IV. ENTANGLEMENT HAMILTONIAN ONINFINITE CYLINDERSA. Bipartition and reduced density matrix
To define an Entanglement Hamiltonian associated tothe family of N´eel-RVB wavefunctions, I partition the N v × N h cylinder into two half-cylinders of lengths N h / N h → ∞ as before.The reduced density matrix of the left half-cylinder ob-tained by tracing over the degrees of freedom of the righthalf-cylinder, ρ L = Tr R {| Ψ PEPS (cid:11)(cid:10) Ψ PEPS |} , can be sim-ply mapped, via a spectrum conserving isometry U , ontoan operator σ b acting only on the D ⊗ N v edge (virtual)degrees of freedom, i.e. ρ L = U † σ b U [6]. The Entangle-ment (or boundary) Hamiltonian H b introduced above isdefined as σ b = exp ( − H b ). As σ b , H b is one-dimensionaland its spectrum – the entanglement spectrum (ES) – isthe same as the one of − ln ρ A . Note that the left and theright half-cylinders give identical EH. For further detailson the derivation and the procedure, the reader is kindlyasked to refer to Ref. 6.For a topological state, such as the γ = 0 RVB state,the Entanglement Hamiltonian depends on the choice ofthe B L and B R cylinder boundaries that define “topo-logical sectors” [18, 20]. Adding any staggered magneti-zation m stag in the PEPS immediately breaks the gaugesymmetry of the tensors which is responsible for the dis-connected topological sectors, as also happens in the caseof field-induced magnetized RVB states [19]. Therefore,all topological sectors are mixed and H b become inde-pendent of the boundary conditions B L and B R provided N h → ∞ . Note also that H b inherits the U(1) symmetry(associated to rotations around the direction of m stag ) ofthe N´eel state. N-body H b (cid:97) =0.1 H b (cid:97) =0.6 H b (cid:97) =1.5 H b (cid:97) =3 N v =6 w e i gh t s o f E n t ang l e m en t H a m il t on i an FIG. 4. (Color online) Weights of the Entanglement Hamilto-nian H b expended in terms of N-body operators. Data of sev-eral N´eel-RVB wavefunctions (whose γ values are mentionedon the plot) are shown. Calculations are done on an infinitecylinder with perimeter N v = 6. As seen e.g. in Ref. [18],finite size effects for such integrated quantities are typicallyquite small. B. Expansion in terms of N-body operators
To have a better insight of the Entanglement Hamil-tonian, I expand it in terms of a basis of N -body op-erators, N = 0 , , , · · · [6, 18]. For this purpose, Iuse a local basis of D = 9 (normalized) ˆ x ν opera-tors, ν = 0 , · · · , i ) configurations {| (cid:11) , | (cid:11) , | (cid:11) } , where | (cid:11) is the vac-uum or “hole” state and | (cid:11) and | (cid:11) can be viewed asspin down and spin up particles, respectively. Moreprecisely, ˆ x = I ⊗ , ˆ x = (cid:113) ( | (cid:11)(cid:10) | − | (cid:11)(cid:10) | ) andˆ x = √ ( | (cid:11)(cid:10) | + | (cid:11)(cid:10) | − | (cid:11)(cid:10) | ), for the diagonal ma-trices, complemented by ˆ x = ˆ x † = √ | (cid:11)(cid:10) | actingas (effective) spin-1/2 lowering/raising operators, andˆ x = ˆ x † = √ | (cid:11)(cid:10) | and ˆ x = ˆ x † = √ | (cid:11)(cid:10) | actingas particle hoppings. In this basis H b reads [18], H b = c N v + (cid:88) ν,i c ν ˆ x iν + (cid:88) ν,µ,r,i d νµ ( r ) ˆ x iν ˆ x i + rµ + (cid:88) λ,µ,ν,r,r (cid:48) ,i e λµν ( r, r (cid:48) ) ˆ x iλ ˆ x i + rµ ˆ x i + r (cid:48) ν + · · · , (2)where site superscript indices have been added and onlythe first one-body, two-body and three-body terms areshown.The total weights corresponding to each order of theexpansion of H b in terms of N-body operators are shownin Fig. 4 as a function of the order N using a semi-logarithmic scale. The data reveal clearly a fast decayof the weight with the order N . This decay is compat-ible with an exponential law although more decades inthe variation of the weights (i.e. larger N v ) would be distance r distance r distance r
2 1 34 43 11 22 57 75 68 86 12 21 (b) (cid:97) =0.6 (c) (cid:97) =3 (a) (cid:97) =0.1 weights FIG. 5. Largest weights | c ν | and | d νµ ( r ) | of the one-body(i.e. r = 0) and two-body operators in the expansion of H b ofthe N´eel-RVB PEPS as a function of distance r , for increasing γ values (corresponding to staggered magnetizations m stag ∼ . , .
327 and 0 . needed to draw a definite conclusion. In any case, H b isdominated by two-body contributions in addition to thenormalization constant and subleading one-body terms.The quantum N´eel state is believed to be critical withpower-law decay of spin-spin correlations [3]. Therefore,according to Ref. [6], one expects H b to be long-rangedto some degree. So, one still needs to refine the analysisand investigate further the r-dependence of the leadingtwo-body contributions. In the next Subsection, I showthat H b indeed possesses long-range two-body terms thatI characterize. C. Entanglement Hamiltonian: an effectiveone-dimensional t–J model
It is known that the EH of the γ = 0 RVB PEPSbelongs to the 1 / ⊕ | (cid:11) and | (cid:11) states ( | (cid:11) states) as ↓ and ↑ spins (holes). In the presence of a finite(staggered) magnetization in the bulk, the SU(2) symme-try is broken but H b keeps the unbroken U (1) symmetrycorresponding to spin rotations around the direction ofthe staggered magnetization.The (largest) non-zero real coefficients in (2) computedon an infinitely-long cylinder of perimeter N v = 6 areshown in Fig. 5(a-c) for small (a), intermediate (b) andlarge (c) (staggered) magnetizations. At large and inter-mediate values of m stag , one finds a dominant one-body(diagonal) term which can be interpreted as a chemicalpotential term (up to a multiplicative factor) : H = c (cid:88) i ˆ x i = 3 √ c (cid:88) i ( n i − / , (3)where n i counts the number of particles (i.e. “0” and“1” states) on site i . The subleading one-body operatortakes the form of a Zeeman coupling : H = √ c (cid:88) i S zi , (4)where S zi is an effective spin-1/2 component (along ˆ z )and c (cid:39) c / atall distances for the majority “spins” ( | (cid:11) states) : H ( r ) = d ( r ) (cid:88) i (ˆ x i ˆ x i + r + ˆ x i ˆ x i + r )= 3 d ( r ) (cid:88) i ( b † i + r, b i, + b † i, b i + r, ) , (5)where b † i,s ( b i,s ) are the canonical bosonic creation (anni-hilation) operators of the virtual s = 0 , | (cid:11) states) only hop at even distances (weights at odd distances are negligable) withmuch weaker amplitudes, d = d (cid:28) d = d . Thenext subleasing corrections are diagonal 2-body density-density interactions H ( r ) = d ( r ) (cid:88) i ˆ x i ˆ x i + r = 92 d ( r ) (cid:88) i ( n i − / n i + r − / , (6)which become dominant when γ, m stag → U (1) symmetry,like the anisotropic XXZ chain ( d (cid:54) = d = d ) ormixed operators of the form H ∝ (cid:80) i S zi ( n i ± r − / H b (approximately) conserves thehole “2-charge” and, hence, does not contain pair-fieldoperators with sizable amplitudes, in contrast to previousstudies of D = 3 PEPS [18, 19]. If one restricts to thedominant contributions (3) and (5), H b is exactly a chainof a dilute mixture of heavy ( ↓ spins or | (cid:11) states) andlight ( ↑ spins or | (cid:11) states) hardcore bosons, where lightparticles are subject to long-range hopping. V. ENTANGLEMENT SPECTRUM
It is also of high interest to examine the ES in theQNS and compare it to ES obtained for GS where SU(2)-symmetry is restored on finite size systems [10]. By def-inition the ES is the spectrum of − ln ρ L . Since ρ L and σ b = exp ( − H b ) are related by an isometry, it is alsothe spectrum of the Entanglement Hamiltonian H b . ESare shown in Fig. 6(a-c) for 3 values of the fugacity γ ,as a function of the momentum along the cut. Since σ b conserves the total S z of the chain ( U (1) symmetry),it can be block-diagonalized using this quantum num-ber and the eigenvalues of − ln σ b are displayed in each S z sector separately. It can be seen from Fig. 6(a) that - (cid:47) /2 (cid:47) /2 (cid:47) momentum K - (cid:47) /2 (cid:47) /2 (cid:47) momentum K - (cid:47) /2 (cid:47) /2 (cid:47) momentum K S z =0S z =+1S z =-1S z =+2S z =-2S z =+1/2S z =-1/2S z =+3/2S z =-3/2 E n t ang l e m en t ene r g i e s (a) (cid:97) =0.1 (c) (cid:97) =3 (b) (cid:97) =0.6 FIG. 6. Entanglement spectrum of a bipartitioned N v = 6RVB-N´eel cylinder as a function of the momentum along thecut, for different values of the spinon fugacity γ = 0 . , . m stag ∼ . , .
327 and 0 . S z sectorsof the edge. the γ = 0 SU(2) spin multiplets are split by a smallspinon density. For increasing γ (i.e. staggered magneti-zation), the splittings of the Kramer’s multiplets increase(see Fig. 6(b,c)) due to the relative increase of the am-plitudes of the SU(2)-symmetry breaking terms like (5)in the EH. In the limit of large γ where the classicalN´eel state is approached, one finds separated bands ofenergy levels. It may be that the ES is gapped for all γ but finite size effects remain too large to reach a def-inite conclusion. In any case, the ES of Fig. 6 are tobe contrasted to the ES obtained in DMRG for GS withrestored SU(2)-symmetry (due to the use of finite sizesystems). Obviously the two types of ES are very differ-ent with a SU(2) tower of states structure at low energyfor the ES of the singlet GS [10] and a U(1) symmetricES in the (variational) symmetry-broken N´eel state. VI. SUMMARY AND DISCUSSION
In this paper, I have investigated entanglement prop-erties of a simple one-dimensional family of PEPS de-signed to describe qualitatively the GS of the square lat-tice AFM. These ans¨atze exhibit a finite staggered mag-netization i.e. they break explicitly the SU(2) symmetrydown to U(1) and can be studied on infinite cylinderswith a finite perimeter. The goal of this study is thereforeto examine the effects of such a finite order parameter onvarious entanglement properties and compare them to(QMC or SU(2)-symmetric DMRG) studies where sym-metry is restored in a finite system. Thanks to the PEPSstructure, the Entanglement Hamiltonian associated to abipartition of the cylinder can be derived exactly (for afixed perimeter). It is found that the EH inherits theU(1) symmetry of the N´eel state and possesses a verysimple structure : (i) its Hilbert space is the same as theone of a one-dimensional bosonic t–J model, interpretingthe 3 virtual states on the edge as a ↑ spin, a ↓ spin anda hole, (ii) when expended using a local basis of opera-tors, it shows dominant two-body interactions and (iii)higher-order operators (three-body terms and beyond)represent less than 10% of its total weight. Examiningin details the form of the two-body interactions, I findthat the dominant ones are long-range hoppings of themajority (let say ↑ ) spins. It is however not possible todistinguish a power-law versus an exponential decay ofthese hopping terms. In any case, the associated Entan-glement Spectrum is found to be qualitatively very dif-ferent from the ones obtained in GS with restored SU(2)-symmetry [10] (no tower of states structure is found, assuspected). Whether the entropy exhibits additive loga-rithmic corrections as in Refs. [7, 8] is difficult to answer. The absence of the tower of states in the ES suggestsa negative answer. However, an hypothetical power-lawdecay of the hopping terms in the EH (instead of expo-nential) might lead to some additive corrections to theentropy. It would be interesting to complement our cal-culation of the ES using “conceptually exact” numericalmethods (such as QMC or SU(2)-symmetric DMRG) onlarge but finite systems, adding a small external stag-gered field (to produce a finite order parameter), takingthe limit of infinite system size first.I acknowledge fundings by the “Agence Nationale de laRecherche” under grant No. ANR 2010 BLANC 0406-0and support from the CALMIP supercomputer center(Toulouse). I thank Claire for her patience and I amindebted to Fabien Alet, Ignacio Cirac, Nicolas Lafloren-cie, Roger Melko, Anders Sandvik, Norbert Schuch andFrank Verstraete for numerous discussions and insightfulcomments. [1] J. D. Reger and A. P. Young, Monte Carlo Simulationsof the Spin-1/2 Heisenberg Antiferromagnet on a SquareLattice , Phys. Rev. B , 5978 (1988).[2] A. W. Sandvik and H. G. Evertz, Loop updates for varia-tional and projector quantum Monte Carlo simulations inthe valence-bond basis , Phys. Rev. B , 024407 (2010).[3] A. W. Sandvik, Lecture notes for course given at the 14thTraining Course in Physics of Strongly Correlated Sys-tems, Salerno (Vietri sul Mare), Italy, in October 2009 ,AIP Conf.Proc. 1297:135 (2010).[4] Hui Li and F. D. M. Haldane,
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