Z2 topological liquid of hard-core bosons on a kagome lattice at 1/3 filling
ZZ topological liquid of hard-core bosons on a kagome lattice at / filling Krishanu Roychowdhury, Subhro Bhattacharjee,
1, 2 and Frank Pollmann Max-Planck-Institut f¨ur Physik komplexer Systeme, Dresden 01187, Germany International Centre for Theoretical Sciences, Tata Institute of Fundamental Research, Bangalore 560012, India
We consider hard-core bosons on the kagome lattice in the presence of short range repulsive interactions andfocus particularly on the filling factor / . In the strongly interacting limit, the low energy excitations can bedescribed by the quantum fully packed loop coverings on the triangular lattice. Using a combination of tensor-product state based methods and exact diagonalization techniques, we show that the system has an extended Z topological liquid phase as well as a lattice nematic phase. The latter breaks lattice rotational symmetry. Bytuning appropriate parameters in the model, we study the quantum phase transition between the topological andthe symmetry broken phases. We construct the critical theory for this transition using a mapping to an Isinggauge theory that predicts the transition to belong to the O (3) universality class. PACS numbers:
I. INTRODUCTION
Strong correlations in quantum many-body systems canyield novel phases of matter at low temperatures. Themost prominent example is the fractional quantum Hall ef-fect (FQHE) which is characterized by its fractionalizedexcitations and topological order that manifests the longrange quantum entanglement present in the underlying many-body ground state. While the FQHE requires strong magneticfields, fractionalized states can occur more generally in cor-related systems that preserve time-reversal symmetry. Fol-lowing the seminal works by Anderson we know that aclass of two and three dimensional frustrated spin systemscan realize paramagnetic ground states, dubbed quantum spin-liquids (QSL), that support deconfined fractionalized S = 1 / (spinon) excitations as well as various forms of topologicalorders. Over the past decades, there has been a very active searchfor model systems stabilizing different types of topologicallyordered states.
A fertile ground to realize such phasesare frustrated systems in which the geometry of the latticeand/or competing interactions prohibit a simultaneous min-imization of all the inter-particle interactions.
This canlead to the suppression of conventional (spontaneous sym-metry breaking) orders and favor more exotic ground states.An associated question of much interest is about quantumphase transitions between topologically ordered and conven-tional symmetry breaking phases. In this regard, systems ofhard-core bosons with mutual short range repulsions on var-ious frustrated lattices have attracted much attention becausethey present a set of rich phase diagrams.
A particularlyinteresting case is the kagome lattice where, for suitable fill-ings, a variety of numerical and analytical evidences nowpoint to the existence of a topologically ordered Z liquidphase over an extended parameter regime. The low en-ergy physics of such models in the strongly correlated limitis generically described by quantum dimer models (QDM) on triangular lattices. This latter class of Hamiltonians isknown to harbor points in their parameter space, the so calledRokhsar-Kivelson (RK) points, where Z topological or-der is present and strong numerical evidence for its stabil- ity exists. The intimate connection, on the other hand, be-tween such hard-core boson models and S = 1 / modelswith XXZ anisotropy also makes these models relevant toresearch in quantum magnetism. With this insight, perhapsit is not surprising that there are proposals that the isotropicnearest neighbor Heisenberg antiferromagnet can potentiallyrealize a Z topologically ordered ground state. In this paper, we explore the strong coupling physics of ahard-core boson model on kagome lattice with short range re-pulsive interactions particularly for the filling factor f = 1 / .We show that the effective low energy theory is given by aQDM Hamiltonian on a triangular lattice with two dimers em-anating from each site of the triangular lattice. This effectivetheory is thus equivalent to a quantum fully packed loop (FPL)model on a triangular lattice. Such kind of models in theclassical version have been extensively studied on some of thebipartite lattices. The quantum FPL model on a squarelattice has been studied by Shannon et. al. where they showeda correspondence to the XXZ model on the checkerboard lat-tice in certain easy-axis limits. Using a combination of nu-merical techniques (tensor product states formalism and ex-act diagonalization on clusters), we analyze this model on thetriangular lattice. Our numerical analysis strongly advocatesfor a rich phase diagram of the quantum FPL on the trian-gular lattice consisting of an extended Z topological liquidphase as well as a crystalline phase, known as lattice nematic(LN), that breaks the three-fold rotational symmetry of thelattice (but not the translation symmetry). Taking clue fromour numerical results, we then construct a critical theory fora continuous phase transition between the two phases. Un-like the usual theories of phase transition, this critical theoryis not written in terms of the low energy long wavelength fluc-tuations of the LN order-parameter, but naturally in terms of“fractionalized” Ising degrees of freedom sitting at the centersof the triangles of the kagome lattice. Mapping the problemto the language of Ising gauge theory, we can isolate thesecritical degrees of freedom– the so called visons (Ising mag-netic flux ), whose condensation then describes the transitionfrom the topological liquid to the LN. The order parameter isa bilinear in terms of the visons and hence, the above transi-tion consists of an example of quantum criticality beyond theconventional Landau-Ginzburg-Wilson paradigm. Our cal- a r X i v : . [ c ond - m a t . s t r- e l ] M a y culation predicts that the critical theory belongs to the O (3) universality class.The remainder of the paper is organized as follows. Wefirst introduce the model Hamiltonian and derive the effectivemodel in Section II. We then present the numerical results andconclude about the phase diagram in Section III. In SectionIV we derive an effective gauge theory description and dis-cuss the nature of the transition between the liquid and the LNphase. We conclude with a brief summary in Section V. II. MODEL HAMILTONIAN
We start by considering an extended Hubbard model ofhard-core bosons on the sites of a kagome lattice given bythe Hamiltonian H = H t + H V , (1)where H t = − t (cid:88) (cid:104) i,j (cid:105) ( b † i b j + H.c. ) (2)describes the nearest-neighbor hopping, with amplitude t , forthe hard-core bosons that are created (annihilated) by b † i ( b i )on the sites of the kagome lattice and H V = V (cid:88) (cid:104) i,j (cid:105) n i n j + V (cid:88) (cid:104)(cid:104) i,j (cid:105)(cid:105) n i n j + V (cid:88) (cid:104)(cid:104)(cid:104) i,j (cid:105)(cid:105)(cid:105) n i n j − ˜ µ (cid:88) i n i (3)denotes respectively the first ( V ), second ( V ) and third ( V )neighbor repulsive interactions among the bosons ( n i = b † i b i )along with a chemical potential ˜ µ that fixes the particle num-ber. Interesting physics emerges at rational fractional fillings p/q (with p and q being mutually prime and p < q ). We shallrestrict ourselves to the specific fractional fillings of bosonswhich are / , / and / .At integer filling 0 or 1 there are only trivial product statespossible (due to the hard-core constraint). At fractional fill-ing factors and the presence of longer-range interactions muchrich phase diagrams emerge. For example, at / fillingand strong nearest-neighbor repulsion ( V ), a plaquette or-dered state is formed, while for / filling a uniform su-perfluid persists for all values of V /t . When turning on fur-ther neighbor interactions given by V and V , additional in-sulating lobes emerge at different fractional fillings. Manyof these bosonic insulators at fractional fillings can host inter-esting quantum phases with or without spontaneously brokensymmetry. In the following, we focus on the strong couplingphases occurring in the / , / and / lobes, with a partic-ular focus on the / lobe and show that topological as wellas long ranged ordered phases can emerge.To this end we look at the strong coupling limit of the aboveHamiltonian. For t = 0 , and V = V = V = 2 V , the FIG. 1: (Color Online) (a) Mapping from the / filled bosonic prob-lem on kagome to FPL on triangular lattice. Each particle on thekagome lattice is mapped to a loop segment on the triangular latticethat is obtained by connecting the centers of the kagome hexagons.The strength of the 1st, 2nd and 3rd repulsive interactions are denotedas V , V and V respectively. The limit of V = V = V = 2 V isthe focus of this paper. Figure (b) demonstrates the allowed lowestorder processes in t/V (see the text). (c) Shown is the sublattice usedfor the gauge transformation that changes the sign of g in Eq. (5) asdescribed in the text. The sublattice is constructed by the b and c sites of the shaded unit cells. interaction term can be written as H V = V (cid:88) { (cid:55) } (cid:20)(cid:16) n (cid:55) − µ V (cid:17) − µ V (cid:21) , (4)where µ = ˜ µ + 2 V is the effective chemical potential and n (cid:55) is the number of particles in each of the hexagons ofthe kagome lattice. It is clear from Eq. (4) that for µ =4 V, V, V , H V is minimized by having , , bosons perhexagon respectively (or alternatively filling fraction of f =1 / , / and / respectively). Clearly there are many differ-ent configurations of bosons that satisfy the above constraint,however, we note that since the hexagons share sites, the con-figurations for different hexagons are not completely indepen-dent.An insight to the number of states spanning the groundstate sector of H V for the above commensurate fillings canbe obtained from the one-to-one correspondence between theground state configurations of the bosons and the hard-coredimer coverings on the triangular lattice obtained by joiningthe centers of the hexagons of the kagome lattice as shown inFig. 1(a). Thus each site of the kagome lattice lies on a bondof the triangular lattice and the presence (absence) of a bosonon that site can then be identified uniquely with the presence(absence) of a dimer on the corresponding bond of the tri-angular lattice. This immediately shows that at / filling,the number of ground state bosonic configurations allowedby H V is equivalent to the number of hard-core dimer cov-erings on the triangular lattice which is known to be exten-sive ( ∼ . N ) in the system size, N (an estimate based onPauling’s approximation gives ∼ . N ). Similarly / fill-ing can be cast into a 3-dimer (three non overlapping dimersemerging from each site) problem m on the triangular lattice where Pauling estimate suggests that the extensive degeneracyof the ground state is ∼ . N .In case of / filling, which is the specific interest ofthe present paper, we obtain the equivalent fully-packedloop (FPL) model on the triangular lattice with two non-overlapping dimers emanating from each site and the dimersform non-intersecting loops as shown in Fig. 1(a). Here thePauling estimate shows that the number of loop configura-tions scales with system size as ∼ . N . Thus, in all theabove cases, for t = 0 , as expected, the ground state is macro-scopically degenerate and has a finite zero temperature en-tropy. Throughout the rest of this work, we shall exploit theabove equivalence between the bosons and the dimers andshall mostly use the language of the dimers, translating backto the bosons whenever applicable. A. Effective model in the strong coupling limit
Small but non-zero hopping ( t ) induces quantum fluctua-tions that (e.g. in the form of local ring exchange around smallplaquettes ) can lift the extensive ground state degeneracyby quenching the entropy of the classical model ( t = 0) ei-ther by spontaneously breaking one or more symmetries of thesystem (quantum order by disorder ) or more interestingly,by generating a long ranged quantum entangled state that doesnot break any symmetry of the Hamiltonian (quantum disor-der by disorder ) but can have non-trivial “topological or-der”. In this work, we shall show examples of both the routestaken by the bosons on the kagome lattice.To derive the effective Hamiltonian in the strong couplinglimit, we treat t/V as a small perturbation parameter to obtain(to the leading order in t/V ) H eff = − g (cid:88) α ( b α † p b α † l b αm b αq + H.c ) , (5)where g = t /V and p , q , l and m are the corner sites of thebow-tie labeled by α referring to Fig. 1(b). The sum includesall bow-ties that are related by the C symmetry. The center ofthe bow tie may or may not be occupied by a boson in case of / and / filling. Furthermore the sign of g in Eq. (5) can bealtered by using a simple gauge transformation that multipliesall configurations with the factor ( − N sub where N sub is thenumber of particles in the sublattice shown in Fig. 1(c). B. Dimer representation of the effective Hamiltonian
We now recast Eq. (5) using the particle to dimer mappingmentioned in the previous section. In terms of the dimers, theeffective Hamiltonian corresponds to the kinetic term of the
FIG. 2: (Color online) The triangular lattice is put on a torus bysetting periodic boundary conditions along two independent direc-tions given by the lattice vectors a and a . A denotes the inter-section point (a link on the lattice) of the two non-contractible loopson the torus. There are four topological sectors characterized by thedoublets ( , ), ( , ), ( , ) and ( , ). The local Hamiltonian inEq. (7) can not mix configurations from different topological sectorshence, block diagonal in the full configuration basis.
QDM. H eff = − g (cid:88) α (cid:16)(cid:12)(cid:12)(cid:12) (cid:115) (cid:115)(cid:115) (cid:115) (cid:20)(cid:20) (cid:69) (cid:68) (cid:115) (cid:115)(cid:115) (cid:115) (cid:12)(cid:12)(cid:12) + H . c . (cid:17) , (6)where the sum is over all rhombus shaped plaquettes ( α ) onthe triangular lattice.In addition to the kinetic term (above) the generic QDMalso includes a potential term (known as Rokhsar-Kivelson(RK) potential), which energetically favors configurationswith plaquettes having parallel dimers. Such potential termsare representative of higher order (four boson) terms that canbe generated in the strong coupling expansion in t/V of theunderlying boson model. However, here we simply use thisterm as a free tuning parameter in the model. The full Hamil-tonian of the QDM is written as H RK = − g (cid:88) α (cid:16)(cid:12)(cid:12)(cid:12) (cid:115) (cid:115)(cid:115) (cid:115) (cid:20)(cid:20) (cid:69) (cid:68) (cid:115) (cid:115)(cid:115) (cid:115) (cid:12)(cid:12)(cid:12) + H . c . (cid:17) + V RK (cid:88) α (cid:16)(cid:12)(cid:12)(cid:12) (cid:115) (cid:115)(cid:115) (cid:115) (cid:20)(cid:20) (cid:69) (cid:68) (cid:115) (cid:115)(cid:115) (cid:115) (cid:20)(cid:20) (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) (cid:115) (cid:115)(cid:115) (cid:115) (cid:69) (cid:68) (cid:115) (cid:115)(cid:115) (cid:115) (cid:12)(cid:12)(cid:12)(cid:17) , (7)where positive (negative) V RK denotes repulsive (attractive)interaction between the parallel dimers in a given plaquette.This generic form of the above Hamiltonian was first pro-posed on a square lattice by Rokhsar and Kivelson (RK) incontext of high temperature superconductivity. At a specialpoint when g = V RK , known as RK point, the spectrum con-tains zero energy ground state with the wave function givenby the equal weight superposition of all allowed dimer con-figurations. The RK point features a U (1) resonating valencebond (RVB) liquid phase with algebraic decay of dimer cor-relations.However, on non-bipartite lattices, the scenario changesdrastically. The RVB phase in this case fosters a gapped Z dimer liquid in an extended parameter regime (includingthe RK point) with exponentially decaying dimer correlationsand is characterized by a topological order in the form of four-fold ground state degeneracy for a system on a torus (as de-scribed in Fig. 2). The solid phases, on the other hand, spon-taneously break various lattice symmetries of the Hamiltonianand thus have long range dimer-dimer correlations. FIG. 3: (Color online) The phase diagram of the dimer models atdifferent fractional fillings f . The important details about each ofthem are given in the text. The focus of the present paper is at f =1 / . In context of the present work, as noted in the last sec-tion, we mention that the low energy effective Hamiltonian inthe strong coupling limit generally assumes the RK form irre-spective of the filling fraction of the bosons (or the equivalentdimer models: QDM, FPL or the 3-dimer model respectivelyfor f = 1 / , / and / ) considered, albeit with importantimplications for the stability and nature of both the liquid andthe solid phases which is summarized in Fig. 3. With this gen-eral formulation we now specialize to the physics of the Mottlobe for / filling. III. NUMERICAL CALCULATIONS AT / FILLING
We use a combination of tensor product states (TPS) for-malism and exact diagonalization (ED) methods to obtain thephase diagram (Fig. 3 middle line) of the FPL model on thetriangular lattice. Necessary details about implementing thenumerical methods for different clusters are furnished in thefollowing subsections facilitating a systematic analysis of ourmodel. We take the value of g to be 1 which is a convenientchoice for further numerical calculations. A. Entanglement entropy at the RK point: the Z liquid One of the most interesting features of the Hamiltonian inEq. (7) is the existence of the exactly solvable RK point ( g = V RK ). We start by showing that at this point the ground stateof our model (FPL) indeed has a topological order.In order to characterize the topological order, an instructivequantity to look at is the entanglement entropy ( S ) for a bipar- FIG. 4: (Color online) The plot shows the finite size dependenceof the topological entanglement entropy γ as a function of L , theperimeter of the subsystem A . The green dashed line corresponds tothe saturated value of ln 2 in the thermodynamic limit. The lattice isconstructed on an infinite cylinder assuming periodic boundary con-dition in one direction. The black dashed line indicates the bipartitionof the cylinder separating the subsystem A from the rest B . The insetshows the linear growth of the entanglement entropy S with L . tition of the system into two parts A and B . The entanglemententropy of the reduced density matrix ρ A of subsystem A isdefined as S = − Tr [ ρ A log ρ A ] where ρ A is obtained fromthe full density matrix by tracing out all the degrees of free-dom in the rest ( B ). For a gapped and topologically orderedgapped ground state, S satisfies the “area law” which goes as, S ( L ) = αL − γ + O ( L − ) + · · · , (8)where α in the leading term is a non-universal coefficient and L is the perimeter of the subsystem A. The sub-leading term γ , also known as the topological entanglement entropy, is, however, universal bearing the anyonic content of the statethat reflects the topological order. This is directly related tothe total quantum dimension (D) of the underlying topologicalfield theory as γ = log D . Since D = 2 for a gapped Z liquid(described by the Z gauge theory), the quantity γ in Eq. (8)saturates to log 2 in the thermodynamic limit i.e. L → ∞ .The ground state at the RK point can be exactly repre-sented by tensor networks using the framework of projectedentangled-pair states (PEPS). We use this construction ona cylindrical triangular lattice (Fig. 4) to calculate S as a func-tion of L and obtain γ (Fig. 4) using Eq. (8). The subsystem A is constructed by bipartitioning the semi-periodic triangularlattice with the dashed line as shown in Fig. 4. The circum-ference L of the cylinder enters Eq. (8) as the perimeter of thesubsystem and should be much larger than the maximum cor-relation length associated with the state. The inset of Fig. 4shows the expected linear growth of S with L that is predictedby the leading term in Eq. (8). The topological entanglemententropy ( γ ) is extracted from the intersection of the function S on the y -axis when extrapolated backward and plotted againstdifferent values of L . The tendency of γ to saturate at thevalue of log 2 for large L indicates to the fact that the RKpoint for the FPL model on the triangular lattice representsthe ground state of a topologically ordered Z liquid akin tothe other dimer models at / and / fillings. This is oneof the main results of this work.
FIG. 5: (Color online) (a) The real space dimer-dimer correlation isplotted on a triangular lattice cluster at V RK = − . which indicatesthe parallel loop pattern deep into the LN phase. (b) Same is plottedfor V RK = 1 . showing the exponential decay of the dimer-dimercorrelation function in the liquid phase at the RK point. The refer-ence link, with respect to which correlations at all the other links aremeasured, is denoted by R . Red and blue links stand for positiveand negative correlation respectively while the width measures thecorrelation strength. (c) Plotted is the same correlator as a functionof | r i − r j | at different RK potential. The red dashed line representsan approximate exponential fit e − r/ξ for ξ ∼ . . However, away from the RK point in the accessible param-eter space, the ground state is no longer exactly known. Weadopt ED techniques to infer that the topological liquid is sta-ble even away from the RK point over a finite window (seeFig. 3) up till the system undergoes a quantum phase transi-tion into the LN phase beyond a critical value of V RK . In thefollowing subsections, we present the ED results containingthe information about the low-lying spectrum of the modeland measurements of various correlation functions that reflectthe salient features of the phase diagram depicted in Fig. 3middle line. B. Dimer-dimer correlation and the LN order
At the RK point, the loops are strongly fluctuating and thedimer-dimer correlations decay exponentially (as opposed tothe algebraic decay on the square lattice) like the one in theQDM at / filling. This indicates the lack of long rangedimer order, as expected, in the Z liquid phase. To checkthis, we calculate the dimer-dimer correlation function C ij ( | r i − r j | ) = (cid:104) n ( r i ) n ( r j ) (cid:105) − (cid:104) n ( r i ) (cid:105)(cid:104) n ( r j ) (cid:105) , (9)using ED on different clusters. Here n ( r i ) is the dimer oc-cupation number on i th link of the triangular lattice specified by the position vector r i . Each cluster has an extension of L x along the unit vector a = (1 , and L y along the unit vector a = ( , √ ) (see Fig. 2).Although the considered clusters are small, the numericalresults shown in Fig. 5(c) suggest that the correlator (as afunction of r ≡ | r i − r j | ) is indeed exponential at the RKpoint with the correlation length ξ ∼ one lattice spacing. Thisis a known result for the case of the QDM on a triangularlattice (or equivalently the / filling). As mentioned ear-lier, at the RK point the model is exactly solvable and allthe quantum correlations can be calculated classically (usingsome classical numerical techniques ). The real space real-ization of the dimer correlation function at the RK point isshown in Fig. 5(b) which points to an exponentially decayingnature of the correlation function as expected. The red andblue bonds correspond to the positive and negative correlationrespectively with the width of the bond being proportional tothe absolute value of the correlation function.The exponential decay disappears as V RK is taken to largenegative values till we get into a phase with long range dimer-dimer correlations with C ( r ) (cid:39) ( − r (up to an offset) deepinside this phase signaling the onset of long range dimer order.To explore the nature of this long range ordered phase we plotthe real space dimer-dimer correlation function in Fig. 5(a).The plot suggests that the ordered phase is characterized byparallel alignments of loops on the triangular lattice whichdoes not break any translation symmetry but the three-foldrotational symmetry (corresponding to the three-fold quasi-degeneracy in the ground sate of the spectrum). Since theparallel loops do not have an orientation along the directionof their alignment, we refer to this as the LN phase. C. The phase transition between the Z liquid and the LN Having characterized the Z liquid phase (by non-trivial en-tanglement entropy and short range dimer-dimer correlations)and the LN phase (by long range dimer-dimer correlation), wenow focus on the quantum phase transition between the twophases. First we study the excitation gap in the liquid withED and observe that the gap closes only at a finite distancefrom the RK point (see Fig. 6). Thus the Z topological liquidpersists over the whole region from V RK = 1 to V RK ∼ − . which suggests that alone the kinetic term in Eq. (7) can po-tentially stabilize the liquid phase even for V RK = 0 . The con-tinuous vanishing of the gap near V RK = − . gradually de-stroys the liquid state driving the system into a charge-orderedLN phase as suggested from the behavior of the density-density correlation function shown in Fig. 5(c).A generic way to locate the critical point within ED is tolook at the response function of the system as a function of theparameters that define the Hamiltonian in Eq. (7). In our case,an equivalent quantity can be framed using the second deriva-tive of the ground state energy with respect to the RK poten-tial: − δ E g /δV RK . This is plotted in Fig. 7 where the singlepeak in the response is visible approximately at V RK ∼ − . .This shows that a single transition separates the topologicalliquid and the LN. While it is impossible to rule out the possi- FIG. 6: (Color online) In the left panel, the gap to the first excitedstate tends to vanish at V RK ∼ − . for different symmetric andstripped clusters. The ground state always lies in the zero-flux sector(0,0) for all of them. In the right panel, the derivative of the gap isplotted as a function of V RK to locate the transition point at V RK ∼− . .FIG. 7: (Color online) The ground state susceptibility − δ E g /δV RK ,in the (0,0) sector indicates a continuous phase transition betweentwo phases for different clusters. The transition point is close to V RK ∼ − . as evident from the right panel of Fig. 6. bilities of a first order transition from our finite cluster results,the smooth and gradual increase of the response along withthe absence of shoulders suggests that the transition betweenthe liquid and the LN may be continuous or weakly first order.The full energy spectrum of each of the clusters (not shown)suggests that at large negative V RK , the ground state becomesnearly 3-fold degenerate (the quasi-degeneracy attributes tofinite-size effect) where the three states are formed by super-position of the three loop patterns allowed by the C symme-try. We conclude on our numerical results from ED by notingthat as the system gradually enters the ordered phase crossingthe transition point, a set of three states in the bottom of thespectrum starts separating from the rest. Deep inside the LNphase, these three states become nearly degenerate (up to thefinite size effects) with a finite excitation gap which is muchhigher in magnitude than the liquid gap and scales linearlywith V RK . This quasi-degeneracy is exact in the thermody-namic limit where the C symmetry is spontaneously broken. Having established the two phases and the possibilities ofa continuous phase transition between them, we now explorethe critical theory for the predicted critical point (at V RK ∼− . ). We note that such a continuous transition would bevery interesting in the sense that it describes the destructionof a topologically ordered phase. IV. CONTINUOUS TRANSITION BETWEEN THE Z LIQUID AND LN
To construct a theory for the continuous phase transition be-tween the Z liquid and the LN phases, we now introduce analternative spin representation of the hard-core boson model(or the equivalent dimer model). A. Spin representation and the gauge theory
To obtain such a description, we first identify a spin / degree of freedom on each site of the kagome lattice by virtueof the well known mappings: b † i = σ + i , b i = σ − i and n i = ( σ zi + 1) / where an up(down) spin represents pres-ence(absence) of a boson at the lattice site. Eq. (4), then, be-comes H V = V (cid:88) { (cid:55) } (cid:0) σ z (cid:55) + h (cid:1) − µ N (cid:55) / V , (10)where N (cid:55) is the total number of kagome hexagons and h =3 − f . The sum of all spin moments in a given hexagonis denoted by S z (cid:55) ≡ σ z (cid:55) where σ z (cid:55) = (cid:80) i ∈ (cid:55) σ zi . Clearlyin the spin description, different fillings of bosons ( f ) corre-spond to different integer values of h which essentially playsthe role of an external magnetic field. Lowest energy con-figurations of the spin system specified by Eq. (10) satisfythe constraint that sum of the moments in every hexagon isexactly opposite to h which depends on the filling factor f .Thus for f = 1 / , / , / ; h = 2 , , and hence poten-tial term is satisfied if the total magnetization per hexagon atthese fillings are S z (cid:55) = − , − , respectively, correspondingto having one, two or three up spins (which means presenceof one, two or three bosons as expected) per hexagon. B. Effective spin model
In terms of the spins, the effective Hamiltonian (in Eq. [5])representing the dynamics within the degenerate ground statemanifold of H V is given by, H eff = (cid:88) α ˆ P α (cid:34) − g (cid:89) α σ x + V RK (cid:35) . (11)Each term in Eq. (11) involves a product of four spins thatform a bow-tie, as shown in Fig. 10(a). The projector which FIG. 8: (Color online) The two-vison correlator as a function of | r I − r J | at different values of V RK for a symmetric cluster of L = 4 .The red dashed line represents an exponential decay: e −| r I − r J | /ξ for ξ ∼ . . selects out the flippable bow ties is expressed as ˆ P α = (cid:88) ξ ± (cid:89) a ∈ α (cid:18)
12 + ξ ( − a σ za (cid:19) , (12)and we have also added the potential term in Eq. (11) to re-cover the RK Hamiltonian given in Eq. (7).When the first term in Eq. (11) dominates, the systemprefers to align all the spins in the σ x direction and hence theboson number per site fluctuates. This is indeed the salientfeature of a Z liquid phase (see below). On the other handwhen the second term dominates, the system prefers to choosea pattern to order in the σ z direction and we have a longrange order in the boson density which turns out to be the LNphase that we discussed before. The actual magnitude of thecoupling constants for which the transition between the twophases takes place depends on the details of the microscopicmodel. C. Ising gauge theory, visons and vison correlator
Each spin sitting at the site of the kagome lattice is a partof two hexagonal plaquettes. We can define the Ising vari-ables (cid:15) h = ± for each such hexagonal plaquette. Clearlythe Hamiltonian in Eq. (11) is invariant under the Z gaugetransformation σ xi → (cid:15) h σ xi (cid:15) h (cid:48) , (13)where h and h (cid:48) denotes the two hexagons of which the site i is a part of. Such an Ising gauge structure, generated by thegauge transformations G h = exp[ iπS z (cid:55) ] , is an emergent prop-erty of the low energy subspace of the original microscopicbosonic Hamiltonian (in Eq. [1]) in the strong coupling limit( V (cid:29) t ). Indeed the above Hamiltonian represents an Isinggauge theory where the plaquette term F (cid:52) = ( σ x σ x σ x ) and F (cid:53) = ( σ x σ x σ x ) , shown in Fig. 10(a), measures the Isingmagnetic flux through each triangle of the kagome lattice.The operator σ zi creates two such magnetic fluxes on thetwo triangles of the kagome lattice of which it is a part of. Such Ising flux excitations have been dubbed as visons . Onecan create a single vison excitation by applying a productof σ z operators along any path C starting from any site of agiven triangle of the kagome lattice to infinity: v zI = I →∞ (cid:89) C σ zi , (14)where i is a kagome lattice site encountered on the path C which runs from the corresponding triangular kagome plaque-tte I , where the vison resides, to spatial ∞ . Since the path op-erators commute with each other, it is straightforward to showthat the two-vison wave function is symmetric under the ex-change of the visons, or in other words, the visons are bosonsthemselves. With reference to the dimer covering, the aboveoperator is nothing but the number of dimer variables encoun-tered along the path. This immediately implies that the vison-vison correlator is given by (cid:104) v zI v zJ (cid:105) = (cid:104) I → J (cid:89) C σ zi (cid:105) ≡ v IJ , (15)(where the path C runs from I to J ) translates to (cid:104) v zI v zJ (cid:105) = (cid:104) ( − N IJ (cid:105) , (16)where the operator N IJ counts the total number of dimers en-countered on the path C form I to J in any given dimer con-figuration. It is easy to verify that the explicite expressions ofthe vison operators in terms of the spins given in Eq. (18) areindependent of the contour C up to an overall sign that canbe fixed by measuring the correlator with respect to a fixedreference dimer configuration.In the Z liquid phase, where F (cid:52) = F (cid:53) = +1 , the vi-son excitations have a finite gap and the ground state does notcontain free visons. On the other hand, the phase character-ized by (cid:104) σ z (cid:105) (cid:54) = 0 , which we shall show is the LN phase, is avison condensate. Thus we expect that the transition betweenthe Z liquid and the LN is described by the closing of the vi-son gap leading to the vison condensation. Hence, the groundstate expectation value of the vison correlator should also de-cay exponentially in the liquid phase with a length scale pro-portional to the inverse of the vison gap while it should haveasymptotically reached a constant value in the LN phase.In Fig. 8 we show the two-vison correlation function atdifferent values of V RK . The data at V RK = 1 . fits wellwith the exponential curve e −| r I − r J | /ξ for ξ ∼ . . As V RK is decreased further, the system gradually enters the orderedphase and the two-vison correlator becomes asymptoticallyconstant. This behavior is expected, as we shall show below.Note that in Fig. 8, close to V RK = − . the value of v IJ changes by an order of magnitude much in the similar way asthe density-density correlation does in Fig. 5. D. Lattice description for the visons
Our numerical results indicate that the phase transition be-tween the Z liquid and the LN phase, driven by the conden-sation of vison excitations, is possibly continuous. We shallnow derive an effective critical theory for such a continuoustransition. This would then compliment our numerical under-standing of the phase diagram of the microscopic model.In spirit of the universality of continuous phase transitions,to this end, we perform a series of mappings to isolate thevison degrees of freedom which we use to describe the criticaltheory for the transition. The effective Hamiltonian in Eq. (11)can be obtained from ˜ H eff = − g eff (cid:88) α ( F ( α ) (cid:52) + F ( α ) (cid:53) ) + V eff (cid:88) (cid:55) ( S z (cid:55) + h ) + u eff (cid:88) α P α (17)in the limit of g eff /V eff , u eff /V eff → g eff , V eff > wherethe leading term is obtained in the second order perturbationtheory with g ∼ g /V eff and u eff ∼ V RK . We immediatelynote that the last two terms in Eq. (17) commute with eachother as expected and hence, in the regime | u eff | (cid:29) g eff (and V eff being the largest energy scale), the above model is ren-dered classical and the u eff term chooses an appropriate orderthat is consistent with the filling ( i.e. allowed by V eff ). For u eff < , this favors the order as depicted in Fig. 10(c). Onthe other hand, when | u eff | (cid:28) g eff , this order is expected tomelt due to the quantum fluctuations in the spins. In particu-lar, a state which allows flipping of spins in closed loops whilemaintaining the filling constraint becomes favorable. This isindeed consistent with the picture of the Z liquid of the mi-croscopic model which is exact at the RK point.To arrive at the critical theory describing the vison conden-sation, we exploit the gauge structure, described in the lastsub-section, of the effective theory to isolate the vison degreesof freedom. At this point, we introduce the honeycomb latticewhich is the medial lattice of the kagome lattice as shown inthe left most panel in Fig. 9. On the sites of this medial lattice,we define the Ising variables v z = ± and on its links we de-fine the Ising gauge fields ρ z = ± . The mapping from the σ to the v and ρ variables is defined as follows, v xJ = (cid:89) (cid:52) σ xa σ xb σ xc ; σ za = v zI ρ zIJ v zJ . (18)Clearly, with the insight of Eq. (14), in the above mapping the v zI s are nothing but the visons (with conjugate momenta givenby v xI ). Since they carry Ising magnetic charge, they cou-ple to the dual Ising gauge potential, ρ zIJ and transform undera projective representation (that forms a projective symmetrygroup) of the symmetries of the underlying spin Hamiltonian.We note that the product of the dual gauge fields around thehexagonal plaquette is given by (cid:89) (cid:55) ρ zIJ = (cid:89) (cid:55) σ za = ± , (19)where the first product is over each honeycomb hexagon andthe second one is on each kagome hexagon.Though not directly relevant to this work, we would liketo point out that the presence of the dual magnetic flux, isnothing but the Ising electric charges ( spinons ): (cid:89) (cid:55) ρ zIJ = (cid:89) (cid:55) σ za = − , (20) FIG. 9: (Color online) The medial honeycomb lattice is shown alongwith the original kagome lattice. Removal of the blue-colored bo-son creates two spinon excitations sitting at the centers of the ad-jacent shaded hexagons (the “defect hexagons”) which violate theconstraint of having two bosons. By virtue of the single particle hop-ping term, these two defects can separate further apart, however theircreation is energetically suppressed in the strong coupling limit. which sit at the center of the hexagonal plaquettes. In terms ofthe original bosons, these represent hexagons where the con-straint of having two bosons is violated (see Fig. 9). Evidently,such “defect hexagons”, are created in pairs and are energeti-cally very costly in the strong coupling limit. It is clear fromthe above equation that the visons see the spinons as a sourceof π -flux and hence the dual gauge potential naturally cap-tures the mutual semionic statistics between the visons and thespinons. Now note that at / filling, we have two bosonsper hexagon. Hence, we must have (cid:89) (cid:55) ρ zIJ = (cid:89) (cid:55) σ za = 1 . (21)This just means that there are no spinons at low energies be-cause they are too costly. Hence, on circling the plaquettes ofthe medial honeycomb lattice, the “flux” seen by the v z -spinsis zero. This type of Ising gauge theory (IGT) are called evenIGT as opposed to the odd IGT that arises in case of / or / filling when the v z -spins sees a π -flux in each honeycombplaquette.With the mapping in Eq. (18) the Hamiltonian in Eq. (17)becomes (we put u eff = 0 for the moment) H dual = − g eff (cid:88) I v xI + V eff (cid:88) (cid:55) (cid:88) (cid:104) I,J (cid:105)∈ (cid:55) v zI ρ zIJ v zJ + h , (22)expanding which we get, H dual = − g eff (cid:88) I v xI + hV eff (cid:88) (cid:104) I,J (cid:105) v zI ρ zIJ v zJ + V eff (cid:88) (cid:104)(cid:104) II (cid:48) (cid:105)(cid:105) v zI ρ zII (cid:48) v zI (cid:48) + (cid:88) I,J,I (cid:48) ,J (cid:48) ∈ (cid:55) v zI ρ zIJ v zJ v zI (cid:48) ρ zI (cid:48) J (cid:48) v zJ (cid:48) (23)up to a constant. Until now we have ignored the potential termin the RK Hamiltonian. This term, in the spin language, hasthe form V RK (cid:88) α O α (24)where α refers to all bow-ties and has the typical form, for thebow-tie in Fig. 10(a), of O α = 18 [ − σ z σ z − σ z σ z − σ z σ z − σ z σ z + σ z σ z + σ z σ z − σ z σ z σ z σ z ] . (25)This, under the mapping to the v I variables, augments the sec-ond neighbor as well as the four-spin terms along with provid-ing the interactions between four spins, two each on adjacenthexagons. As remarked earlier, the v z fields transform under aprojective symmetry group (PSG) and in general a third neigh-bor Ising term for the v zI fields would also be allowed by thePSG of an even-IGT. These term, would arise, for instance,on integrating out the four spin interactions which would alsorenormalize the nearest and the second neighbor interactions.In addition, a Maxwell term for the ρ zIJ fields of the form (cid:81) (cid:55) ρ zIJ is also allowed and they renormalize the energy ofthe spinons, but due to the constraint in Eq. (19), such termsare trivial and hence left out. Now, Because of the constraintsin Eq. (19) ( i.e. no spinons), it is possible to choose a gaugewhere ρ zIJ = +1 , ∀ I, J . (26)Hence the general form of the model is given by H dual = J (cid:88) n.n v zI v zJ + J (cid:88) n.n.n v zI v zJ + J (cid:88) n.n.n.n v zI v zJ − Γ (cid:88) J v xI . (27)Thus, the minimal gauge theory for the FPL model is dual tothe ferromagnetic transverse field Ising model on the honey-comb lattice with first, second and third neighbor Ising inter-actions.In the large Γ limit we can neglect the J terms and the v -spins are polarized in the x direction with the finite vison gap ∼ . The underlying σ z spins are thus fluctuating allowingfor the boson number per site to fluctuate. This paramagneticphase is nothing but the Z liquid in the dual description.On increasing the Ising couplings ( J , , ), the visons gaindispersion and if the minima of the dispersion touches zero,they can condense leading to (cid:104) v zI (cid:105) (cid:54) = 0 . The nature of theordering depends on the relative signs and magnitude of theIsing couplings and the phase diagram in the limit of large J , , / Γ is shown in Fig. 10(d). Setting J /J = t and J /J = t and Γ /J = ˜Γ with J = 1 , in Eq. (27), the softvison modes can be obtained from the Fourier transformof the Ising terms. This is given by H dual = (cid:88) k J ( k )Ψ † k Ψ k , (28)where J ( k ) is the Fourier transform of the adjacency matrixof the Ising terms in Eq. (27): J ( k ) = (cid:18) t δ γ + t ηγ ∗ + t η ∗ t δ (cid:19) (29) FIG. 10: (Color online) (a) A flippable bow tie. (b) The first Brillouinzone of the honeycomb lattice showing M points. (c) Loop orderingat M point. A loop segment is placed on the link of the triangularlattice whenever it crosses a honeycomb bond joining antiparallelspins which represent the vison degrees of freedom. (d) The yellowregion qualitatively shows the allowed ranges of t and t for whichthe minima of J ( k ) lie in the vicinity of the M points within anenergy window of − . The vertical phase boundary indicates tothe case when the minima are exactly on the M points correspondingto the LN phase for V RK (cid:28) . and Ψ k = ( v x k v x k ) T and Ψ † k = ( v x − k v x − k ) . The momen-tum dependence of the parameters goes as δ = cos k + cos k + cos ( k + k ) γ = 1 + e ik + e − ik η = 2 cos ( k + k ) + e i ( k − k ) , (30)where k i = (cid:126)k · (cid:126)s i with (cid:126)s = ( √ / , − / and (cid:126)s =( √ / , / being the basis vectors of the honeycomb lat-tice shown in Fig. 10(c). For an extended range of positive t and t , as highlighted by the yellow region in Fig. 10(d),the minima of the energy dispersion J ( k ) occur at the threeinequivalent M points; M , M and M in the Brillouinzone [see Fig. 10(b)] whose coordinates are ( π/ √ , − π/ ),( π/ √ , π/ ) and ( , π/ ) respectively. This extended re-gion of the phase diagram, where the ordering occurs at the M points of the Brillouin zone of the medial honeycomb lattice,yields to a v zI ordering pattern which is shown in Fig. 10(c).Translating back (using Eq. (18) and n i = (1 + σ zi ) / ), thisgives rise to the ordering pattern for V RK (cid:28) . There are threesuch patterns (for the three M points) which corresponds tothe three LN phases related by the C symmetry as obtainedfrom our previous numerical calculations.While it is not possible to calculate the values of the Isingcouplings in terms of the couplings of the underlying RKHamiltonian in Eq. (7), we observe that, the effective modelallows phases that are observed in the microscopic model to-gether with the possibilities of a direct continuous quantumphase transition from the Z liquid to the LN phase. Sincethe symmetries of the actual QDM and the effective gaugetheories are identical, we expect that the transitions, which isattributed to the condensation of the visons, in both the mod-els belong to the same universality class. We now attempt tofind the structure of the critical theory which can predict thenature of this transition.0 E. Critical theory for the transition
Approaching the transition point from the liquid side ( Γ >J in Eq. [27]), we can write down the critical theory in termsof the vison modes that goes soft at the transition. These softmodes can be written as Ψ( (cid:126)r ) = (cid:88) j =1 ψ j ( (cid:126)r ) v j e i (cid:126)M j · (cid:126)r , (31)where ψ j are the amplitudes of the three soft modes occurringat the three M points of the Brillouin zone. To construct theLandau-Ginzburg action in terms of the soft modes we need tofigure out the transformation of ψ j ( j = 1 , , among them-selves under various symmetries of the Hamiltonian. Theseare: (1) T : lattice translation along (cid:126)s , (2) T : lattice trans-lation along (cid:126)s , (3) I : bond inversion or parity, (4) C : rota-tion of π/ about the center of a plaquette, and (5) global Z symmetry under which v x → − v x . On a point of specific co-ordinates { x , y } , the actions of the above symmetries are thefollowing: T : { x, y ; a, b } → { x + 1 , y ; a, b } T : { x, y ; a, b } → { x, y + 1; a, b } I : { x, y ; a, b } → {− x, − y ; b, a } C : (cid:40) { x, y ; a } → { x − y + 1 , x ; b } , { x, y ; b } → { x − y, x ; a } . (32)The transformation matrices of the three critical modes cor-responding to different lattice symmetry operations and theglobal Z are as follows: R T = − − ; R T = − − ; R I = − ; R C = −
11 0 00 1 0 ; R Z = − − − . (33)These five matrices generate a 24 elements finite subgroup of O (3) which is isomorphic to C ⊗ A . Respecting all theabove projective symmetry transformations, the most generalLandau Ginzburg (LG) functional in (2 + 1) dimensional Eu-clidian space-time assumes the following form, S = (cid:90) d r dτ L , (34)where the Lagrangian density (up to th order) is given by L = ∇ (cid:126)ψ · ∇ (cid:126)ψ + ∂ τ (cid:126)ψ · ∂ τ (cid:126)ψ + r (cid:126)ψ · (cid:126)ψ + ˜ u ( (cid:126)ψ · (cid:126)ψ ) + ˜ v ( (cid:126)ψ · (cid:126)ψ ) + a ( ψ + ψ + (cid:126)ψ ) + b ( ψ ψ ψ ) (35) with ψ = ( ψ , ψ , ψ ) T . If a = b = 0 , Eq. (35) representsthe usual soft spin O (3) action (or the N = 3 linear σ model).The a term introduces cubic anisotropy in the system.Evidently, for a < , the ordering occurs at one of the softmodes preferentially, i.e. , the functional is minimized whenone of the components (among ψ , ψ , and ψ ) takes a finitevalue while other two remain zero. Three such possibilitiesgive rise to three symmetry oriented loop orderings. For ex-ample, the order at M , which corresponds to ψ (cid:54) = 0 and ψ = ψ = 0 , can be read from the structure of the full eigen-vector v = (1 , T of J ( k ) in Eq. (29) set to the momentum M = ( π/ √ , − π/ . The resultant v z -spin configuration isshown in Fig. 10(c) and the loop covering of the dimers can beobtained using Eq. (18) which is equivalent to replacing theantiferromagnetic bonds of the honeycomb lattice by a loopsegment on the underlying triangular lattice. This gives backthe loop ordering which does not break any translation sym-metry (because M and − M are identical and related by themomentum space lattice vectors) but spontaneously breaks therotational symmetry. Similarly the ordering at other M pointsare related to the present one by the spontaneously broken C symmetry. This way the critical theory captures the orderingpatterns obtained in the numerical calculations of the micro-scopic model.For b = 0 , leading order (cid:15) = 4 − d expansion suggeststhat the cubic anisotropy is irrelevant and the critical point isof O (3) Wilson-Fisher type (however, higher order ex-pansion suggests that it may belong to the cubic critical pointwith critical exponents very close to the O (3) class ). The -th order anisotropy term (denoted by b (cid:54) = 0 ) is also irrelevantat such a point along with the O (3) invariant th order term, ˜ v in Eq. [35]. These considerations suggest that the phasetransition between the Z liquid and the LN phase, as seen inthe microscopic model, may belong to the O (3) universalityclass and the anisotropy terms are dangerously irrelevant atthis critical point.It is important to note that the critical theory is not written interms of the order parameter of the LN phase, as the conven-tional theory of phase transition would suggest. Instead,it is naturally written in terms of vison fields and the orderparameter is bilinear in terms of such vison fields. Hence weshould expect large anomalous dimensions in the scaling di-mension of the LN order parameter.
Such large anoma-lous dimensions are characteristics to these types of uncon-ventional phase transitions.
V. SUMMARY AND OUTLOOK
In summary, we have studied the strong coupling limit ofthe extended Hubbard model for hard-core bosons with shortrange repulsive interactions. Focusing on the particular frac-tional filling of / , we show that the low energy physics ofthe bosonic model is described by a quantum FPL model onthe triangular lattice. Using a combination of different numer-ical techniques we analyze the ground state phase diagram ofthis quantum FPL model. Our numerical calculations conclu-sively establish the presence of a topologically ordered Z liq-1uid phase over an extended parameter regime of the effectivelow energy Hamiltonian. On tuning appropriate parameters ofthe effective model, we find indication for a continuous phasetransition from the topological phase to a phase that breaksthe C rotational symmetry of the triangular lattice. Using amapping to an Ising gauge theory and PSG based arguments,we construct the effective field theory for a generic continuoustransition between the two phases, which we argue, belongs tothe O (3) universality class. We then show that such a transi-tion is naturally related to the condensation of the (bosonic)magnetic charges of the Ising gauge theory, the so called vi-sons. Hence, contrary to the conventional theory of continu-ous transition, the present critical theory is written in termsof the soft vison modes and not the LN order parameter whichis a bilinear in terms of the vison fields.The present calculations show that the model of stronglyinteracting hard-core bosons can harbor rich and interestingphase diagrams including conventionally as well as topolog- ically ordered states at different fractional fillings. It is in-teresting to note that considering the interacting particles tobe fermions would induce a non-trivial statistics in the dimerproblem. Interestingly, such kind of fermionic dimer mod-els can arise in context of describing the metallic state of thehole-doped cuprates at low hole densities. Whether a sta-ble liquid phase can still be realized in such fermionic modelsrequires more understanding regarding the underlying gaugetheory and constitutes an interesting direction for further stud-ies in future.
VI. ACKNOWLEDGMENTS
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