Inhomogeneous hard-core bosonic mixture with checkerboard supersolid phase: Quantum and thermal phase diagram
IInhomogeneous hardcore bosonic mixture with checkerboard supersolid phase:Quantum and thermal phase diagram
F. Heydarinasab ∗ and J. Abouie † Department of Physics, Institute for Advanced Studies in Basic Sciences (IASBS), Zanjan 45137-66731, Iran (Dated: October 16, 2018)We introduce an inhomogeneous bosonic mixture composed of two kinds of hardcore and semi-hardcore boson with different nilpotency conditions and demonstrate that in contrast with thestandard hardcore Bose Hubbard model, our bosonic mixture with nearest and next nearest neighborinteractions on a square lattice develops the checkerboard supersolid phase characterized by thesimultaneous superfluid and checkerboard solid orders. Our bosonic mixture is created from a two-orbital Bose-Hubbard model including two kinds of bosons: a single orbital boson and a two-orbitalboson. By mapping the bosonic mixture to an anisotropic inhomogeneous spin model in the presenceof a magnetic field, we study the ground state phase diagram of the model by means of cluster meanfield theory and linear spin wave theory and show that various phases such as solid, superfluid,supersolid and Mott insulator appear in the phase diagram of the mixture. Competition betweenthe interactions and magnetic field causes the mixture to undergo different kinds of first and secondorder phase transitions. By studying the behavior of the spin wave excitations we find the reasonsof all first and second order phase transitions. We also obtain the temperature phase diagram of thesystem using cluster mean field theory. We show that the checkerboard supersolid phase persists atfinite temperature comparable with the interaction energies of bosons.
PACS numbers: 03.75.-b, 05.30.-d, 67.80.kb
I. INTRODUCTION
Supersolids are characterized by the simultaneous pres-ence of a nontrivial crystalline solid order and super-fluid phase order in the context of quantum lattice gasmodels . Discussing the possibility of supersolidity,has attracted renewed interest in connection with ul-tracold Bose gases in optical lattices . The precisecontrollability of optical lattice systems has motivatedtheoretical explorations of supersolid phase in varioussystems, such as one dimensional chains , two di-mensional square , honeycomb , triangular and kagome lattice structures, two-dimensional spin-1 / , bilayer systems of dipolar lat-tice bosons and three dimensional cubic lattice .These extensive studies show that no supersolid phasescan exist in the ground state phase diagram of the hard-core Bose Hubbard model with nearest neighbor inter-action for bipartite lattices . In these systems,due to the formation of antiphase walls between or-dered domains , supersolid states are unstable towardsphase separation . In order to have stable supersolidphases, one has to modify the model by introducing re-pulsive dipole-dipole interaction where has the roleof increasing the energy cost of domain wall formations.Adding next nearest neighbor interaction ,correlated hoppings , or treating soft core bosons ,two-component Bose-Fermi and Bose-Bose mix-tures, and three component Bose-Bose-Fermi mixture result also stable supersolids.In this paper we introduce a different inhomogeneousbosonic model (IBM) which is composed of two kinds ofhardcore and semi-hardcore boson, a and b , with different nilpotency conditions : ( a † i ) = 0 for a and ( b † i ) = 0 for b bosons, and show that the model on a square latticewith nearest neighbor (NN) and next nearest neighbor(NNN) interactions is an appropriate ground for search-ing various supersolid orders. The nilpotency conditionfor b bosons signifies that one can put up two b particleson each lattice site. Our IBM is created from a Bose-Hubbard model including two kinds of bosons: a singleorbital boson and a two-orbital boson. By mapping theIBM to an anisotropic inhomogeneous spin-(1,1/2) modelin the presence of a magnetic field, we study the groundstate phase diagram of the model by means of clustermean field (CMF) theory and linear spin wave (LSW)theory and show that various phases such as solid, super-fluid, supersolid and Mott insulator appear in the phasediagram of the mixture. We demonstrate that in con-trast with the standard hardcore Bose Hubbard modelin which long range hopping terms are required for thesuperfluidity , or long range dipole-dipole interactionsare necessary to suppress quantum fluctuations for thestability of checkerboard supersolid (CSS) order on thesquare lattice , our IBM possesses an stable CSSphase even in the absence of long range interaction andlong range hopping terms. This stability is attributedto the difference in the nilpotency conditions of a and b bosons. The small amount of spin wave fluctuations alsoshow the stability of the CSS phase. Making use of LSWtheory and obtaining the excitation spectra of the IBM,besides the strength of quantum fluctuations around themean field ground states we find the boundaries of thestability of the mean field phases.In this paper we also study the effects of temperatureon the phase diagram of the system. We obtain the tem-perature phase diagram of the mixture and show that in a r X i v : . [ c ond - m a t . o t h e r] N ov the presence of temperature various phases emerge in thephase diagram. Our results show that the CSS order canpersist even at finite temperatures comparable with theinteraction energies.This paper is organized as follows. In section II we in-troduce our IBM and map the model onto a mixed spinmodel by making use of hardcore boson-spin transforma-tions. In section III we give a brief review on the CMFtheory and generalize the theory to the mixed spin model.By computing the diagonal and off diagonal order param-eters we present the CMF ground state phase diagram ofthe model in section IV. In order to investigate the sta-bility of these phases against quantum fluctuations wecompute the order parameters within CMF theory withlarger clusters. The strengths of quantum fluctuations foreach phase are also obtained by means of LSW theory insection V. In this section we investigate the behavior ofthe spin wave dispersions at phase transitions to figureout the reason of all first and second order phase tran-sitions in the phase diagram of the IBM. In the secondpart of the paper, in section VI, we obtain the thermalphase diagram of the model and show that the CSS ordersurvives, at finite temperatures. Finally, we summarizeour results and give the concluding remarks in sectionVII. II. INHOMOGENEOUS BOSONIC MODEL
Let us consider the two kinds of hardcore and semi-hardcore boson a and b which interact via the Hamilto-nian: H B = − t (cid:88) (cid:104) i,j (cid:105) ( a † i b j + b † j a i ) + U (cid:88) i n bi n bi + V (cid:88) (cid:104) i,j (cid:105) n ai n bj + V (cid:88) (cid:104)(cid:104) i,j (cid:105)(cid:105) ( n ai n aj + n bi n bj ) − (cid:88) i ( µ a n ai + µ b n bi ) , (1)where a † i ( a i ) and b † j ( b j ) are respectively the cre-ation(annihilation) operators of a and b particles at sites i and j , on a two dimensional (2D) bipartite square lat-tice. The first term represents a hopping between twonearest neighbor sites (cid:104) i, j (cid:105) where i and j are the latticepoints in subsystems I and II, respectively. U is localCoulomb attraction energy ( U <
0) between b bosonsoccupying the same site, V is the interaction energy be-tween a and b bosons, V denotes the interaction betweentwo a or two b bosons, and µ a and µ b are chemical po-tentials. (cid:104) . . . (cid:105) and (cid:104)(cid:104) . . . (cid:105)(cid:105) indicate the summations overnearest and next nearest neighbors on the square lattice,respectively.The a particles are canonical hardcore bosons and sat-isfy the canonical commutation relations. The numberof these bosons at site i is n ai = a † i a i , and the nilpotencycondition for them is ( a † i ) = 0. The b particles are how- ever semi-hardcore bosons and satisfy the nilpotency con-dition ( b † i ) = 0, which signifies that one can put up two b particles on each lattice site. This uncommon nilpo-tency condition leads to the following non-canonical al-gebra (see appendix B, for the detailed calculation):[ b i , b j ] = [ b † i , b † j ] = 0 , [ b i , b † j ] = δ ij (1 − n bi ) , [ n bi , b † j ] = δ ij b † j , (2)where n bj ( (cid:54) = b † j b j ) is the number of b bosons which pos-sesses the relation ( n bi ) † = n bi .Since the number operator n b in not equal to b † b , theHamiltonian in Eq. (1) does not have the standard formof a Bose Hubbard Hamiltonian. But, as we will showin appendix A, this Hamiltonian is created from an stan-dard two-orbital bosonic Hubbard model (see Eq. (A1)in appendix A) by reducing effective number of degrees offreedom. The three-body constraint of the semi-hardcorebosons b , and consequently their non-canonical statisticsalgebra arise inevitably from the transformations in Eq.(A2) which are employed for mapping the two-orbitalHamiltonian in Eq. (A1) to the one in Eq. (1). At thefirst glance it may seem that the non-canonical statisticsof b particles makes the model complicated, but as we willshow in next sections, using a simple boson-spin trans-formation, the Hamiltonian (1) maps to an standard spinHamiltonian with rich phase diagrams.It is worth to mention that two independent physicalproperties are responsible for the quantum statistics ofparticles. The first one is exchange or permutation statis-tics which concerns braiding of particles and the secondone is exclusion statistics which concerns number of parti-cles allowed to occupy the same site . It should be notedthat, although the commutation relations in Eq. (2) havefractional exclusion statistics, they obey canonical ex-change statistics, and should not be confused with theanyonic particles with fractional exchange statistics .The anyonic algebra can be created and manipulated byusing the so-called conditional hopping terms whichis not the case in our paper.Fractional exchange statistics in bosonic systemscauses the system to experience different new phaseswhich are not seen in the standard system with canonicalbosons. For example, the one dimensional optical lat-tice of semi hardcore bosons with the constraint ( b † i ) =0 and fractional statistics, proposed by Greschner, et.al. , shows a novel two-component superfluid of holonand doublon dimers, characterized by a large but finitecompressibility and a multipeaked momentum distribu-tion, which is not seen in the one dimensional canoni-cal model . Moreover, including such an statistics inthe dipolar system with hardcore bosons results in anstriped supersolid phase . In Eq. (1) we have intro-duced an inhomogeneous system of hardcore and semi-hardcore bosons which could be realized in a two-orbitalbosonic system. The non-canonical statistics of the semi-hardcore bosons causes the model to be mapped to a 𝑎 𝑎 𝑎 𝑎 𝑉 , 𝐽 𝑉 𝜎 = 1 𝜏 = 1 FIG. 1. (Color online) The schematic illustration of a 2D fer-rimagnetic spin-( σ, τ ) system on square lattice. Left: beforethe translational symmetry breaking of the Hamiltonian (5),Right: after the translational symmetry breaking in subsys-tem with spin σ . Each unit cell contains two spins σ and τ .Before symmetry breaking the primitive vectors are (cid:126)a and (cid:126)a . In the symmetry breaking phase the primitive vectors are (cid:126)a (cid:48) = (cid:126)a + (cid:126)a and (cid:126)a (cid:48) = (cid:126)a − (cid:126)a . The right panel shows thecheckerboard solid phase in which the translational symmetryof the subsystem with spin σ is broken. mixed spin model which possesses the CSS phase, in ad-dition to the superfluid, Mott insulating and various solidphases.Using the Matsubara-Matsuda transformations for a hardcore bosons: σ zi = n ai − , σ + i = a † i , σ − i = a i , (3)and also the generalized transformations for b bosons: τ zj = n bj − , τ + j = √ b † j , τ − j = √ b j , (4)the Hamiltonian H B transforms to the following spinHamiltonian: H = − J (cid:88) (cid:104) i,j (cid:105) ( σ xi τ xj + σ yi τ yj ) + U (cid:88) i ( τ zi ) + V (cid:88) (cid:104) i,j (cid:105) σ zi τ zj + V (cid:88) (cid:104)(cid:104) i,j (cid:105)(cid:105) ( σ zi σ zj + τ zi τ zj ) − (cid:88) i ( h σ σ zi + h τ τ zi ) , (5)with the parameters J = √ t , h σ = µ a − V − V and h τ = µ b − U − V − V . This Hamiltonian is nothing butthe frustrated anisotropic mixed spin-(1 , /
2) XXZ modelon a bipartite square lattice, with on-site anisotropy, inthe presence of longitudinal magnetic fields h σ and h τ .Since the hardcore boson-spin transformations (3) and(4) are isomorphic, the symmetries and physical prop-erties of the IBM (1) and the mixed spin Heisenbergmodel (5) are identically the same. Throughout this pa-per we consider h σ = h τ = h , which results the relation µ b − µ a = 4 V − V − U , between the chemical po-tentials of the two species. A schematic illustration ofthe ferrimagnetic model (5) is depicted in Fig. 1. The -0.4 -0.2 0 0.2 0.4-1-0.500.51 0 1 2 3 4 5 6 a CSS a CSS Full MI(4/6) SF SF a CSS a CSS a a CS(5/6) a CS (5/6) b CS(4/6) MI(4/6) CS(3/6) b CS(4/6) a CSS CSS a CSS SF b CSS b CSS CS(3/6)
Full SF a CS(5/6) /VJ / V h / V h b CSS SF b CSS b CS(4/6)
CS(3/6) CS(3/6) SF a CS(5/6)
Full SF MI(4/6) SF a CS(5/6)
Full
MI(4/6) V /V =0.2 V /V =0.6 /VJ /Vh 𝑀 𝑣 𝑚 𝐴𝑧 𝑀 𝐵𝑧 𝑀 𝐷𝑧 𝑚 𝐶𝑧 b FIG. 2. (Color online) Top (a and b): Ground state phasediagram of the IBM in the absence of U , and for the two dif-ferent strengths of frustration V V , 0.2 (a) and 0.6 (b). Orderparameters are computed using CMF-2 × M v = (( M xT ) + ( M yT ) ) / with M x ( y ) T , the total magneti-zation in x ( y ) direction. The red(black) dotted lines showfirst(second) order phase transitions. Right: Schematic illus-trations of solids and Mott insulator. Different orders aredefined as in table I. Bottom: The sublattices longitudinalmagnetization and the total transverse magnetization versus J/V at h = 0, and versus h/V at J = 0. small(large) filled circles are the spins σ ( τ ). In the pres-ence of the translational symmetry of the Hamiltonian(5) the primitive vectors are (cid:126)a and (cid:126)a . When the trans-lational symmetry breaks (at least in one of the subsys-tems) a phase transition occurs to a checkerboard solidphase in which the lattice structure is given by the prim-itive translational vectors (cid:126)a (cid:48) and (cid:126)a (cid:48) with four basis. Asan example we have illustrated in the right panel of Fig.1 a checkerboard pattern where the translational symme-try of the subsystem with spin σ is broken.In anisotropic spin-1/2 models on square lattice withNN and NNN interactions, due to the strength of frus-tration, quantum fluctuations in spin direction are largeenough to destroy the CSS order. In contrast, we willdemonstrate that the anisotropic ferrimagnetic spin-(1,1/2) model in Eq. (5) possesses an stable CSS phase.This is in part due to the fact that each spin-1/2 is sur-rounded by four spins 1 which causes decreasing of quan-tum fluctuations. Besides the CSS phase, different solidorders and Mott insulating phase emerge in the phasediagram of the system which are not seen in the homoge-neous spin models. In following sections utilizing CMFapproach we study the phase diagrams of the model (5)on a square lattice. III. CLUSTER MEAN FIELD THEORY
CMF theory is an extension of the standard mean field(MF) theory in which ” clusters ” of multiple sites are usedas an approximate system instead of single sites. Treat-ing exactly the interactions within the cluster and in-cluding the interaction of spins outside the cluster as aneffective field, one can partially take into account fluctu-ations around classical ground state as well as the effectsof correlations of particles. We have generalized the CMFapproach of Yamamoto, et al which is an extensionof Oguchi’s method to multiple-sublattice problems, tothe inhomogeneous mixed-spin model in Eq. (5). Weassume a background with four-sublattice structure (Aand C for spins σ , and B and D for spins τ ) and embeda cluster of N C sites into this background. The four-sublattice structure is expected to be emerged due to theNN and NNN interactions. Now, instead of treating themany-body problem in the whole system, we consider theeffective cluster Hamiltonian: H effC = H C + (cid:88) i ∈ C ( (cid:126)h effi · (cid:126)σ i + (cid:126)g effi · (cid:126)τ i ) , (6)where the interaction within cluster is given by H C , theHamiltonian in Eq. (5) with i, j ∈ C , while the inter-actions of spins inside the cluster with the rest of thesystem are included via the effective fields: (cid:126)h effi = (cid:88) (cid:104) i,j (cid:105) ,j ∈ ¯ C [ − J ( M xj ˆ x + M yj ˆ y ) + V M zj ˆ z ]+ V (cid:88) (cid:104)(cid:104) i,j (cid:105)(cid:105) ,j ∈ ¯ C m zj ˆ z,(cid:126)g effi = (cid:88) (cid:104) i,j (cid:105) ,j ∈ ¯ C [ − J ( m xj ˆ x + m yj ˆ y ) + V m zj ˆ z ]+ V (cid:88) (cid:104)(cid:104) i,j (cid:105)(cid:105) ,j ∈ ¯ C M zj ˆ z, (7)with ¯ C part of the system outside the cluster. The mag-netizations (cid:126)m j (= (cid:104) (cid:126)σ j (cid:105) CMF ) and (cid:126)M j (= (cid:104) (cid:126)τ j (cid:105) CMF ) are the expectation values within the CMF method which actas mean fields on the spins σ and τ . The order param-eters m x,y,zj and M x,y,zj are calculated self-consistentlyas the expectation values of the spins inside the clus-ter. This method reduces to the conventional MF theoryfor N τC = N σC = 1 and becomes exact in the limit of N C → ∞ . IV. GROUND STATE PHASE DIAGRAM
According to the relations between the sublatticesmagnetizations various kinds of solid and supersolid or-ders are observed in the ground state phase diagramof the IBM (See Fig. 2 for U = 0 and Fig. 5 for U = − . V ).In the absence of magnetic field, at h = 0, for largevalues of hopping energy | J | and any strength of frus-tration, the IBM is in a superfluid (SF) phase wherethe U(1) symmetry of both subsystems is broken andeach boson is spread out over the entire lattice, withlong range phase coherence. By decreasing | J | the IBMhowever, behaves differently for strong and weak frus-trations (see Fig. 2, bottom panels, the behavior of thesublattices longitudinal magnetizations versus J/V at h = 0). For V /V < .
4, at the first order transition line: V ≈ − . J + 0 . V (not shown) the off diagonal longrange order are suddenly destroyed and a quantum phasetransition occurs from SF to the MI(4/6) Mott insulatingphase where both the U(1) and the translational symme-tries are preserved. In this phase the average number ofbosons in each unit cell is 4 /
6. Increasing V /V , destroysthis Mott insulating phase. For V /V ≥ .
4, by decreas-ing | J | the translational symmetry of the subsystem b also breaks and a phase transition from SF to the b CSSsupersolid phase occurs at the first order transition line: V ≈ . J + 0 . V , where the checkerboard solid orderemerges in the subsystem b in addition to the off-diagonalone (see table I for the definition of supersolid phases).By further decreasing of | J | , the off diagonal order disap-pears at the transition line: V ≈ . J + 0 . V , and thetranslational symmetry of subsystem a also breaks andthe mixture enters the CS(3/6) solid phase. In this phasethe spins 1 as well as the spins 1 / / h (cid:54) = 0, depend-ing on the strength of frustration, various kinds of solidorder appear in the phase diagram of the IBM. We haveplotted in Fig. 2 the phase diagram of the mixture forthe two strengths of frustration, V /V = 0 . . V /V = 0 .
2, in a symmetricregion around J = 0, at small and moderate magneticfields the system prefers to be in the MI(4/6) phase. Byincreasing the magnetic field, the translational symme-try of the subsystem a is broken and the spins 1 / a CS(5/6) solidphase (For the definitions of solid orders, see table I and
TABLE I. Definitions of various orders. TS is the abbreviation of the translational symmetry of the Hamiltonian.Order parameters broken symmetriesPhases sublattices magnetizations total magnetization M v subsystem a subsystem b SF m zA = m zC , M zB = M zD (cid:54) = 0 U(1) U(1)MI(4/6) m zA = m zC , M zB = M zD m zA = m zC = 1 / M zB = M zD = 1 0 - - a CS(5/6) m zA = − m zC , M zB = M zD b CS(4/6) m zA = m zC , M zB = − M zD m zA = − m zC , M zB = − M zD a CSS m zA (cid:54) = m zC , M zB = M zD (cid:54) = 0 TS, U(1) U(1) b CSS m zA = m zC , M zB (cid:54) = M zD (cid:54) = 0 U(1) TS, U(1)CSS m zA (cid:54) = m zC (cid:54) = M zB (cid:54) = M zD (cid:54) = 0 TS, U(1) TS, U(1) the schematic pictures in the right column of Fig. 2).In this phase the average number of bosons on each unitcell is 5 /
6. By increasing V /V , the antiferromagnetic V interactions try to make the spins 1 antiparallel aswell as spins 1 /
2. For V /V = 0 . J = 0, thetranslational symmetry of both subsystems breaks andthe CS(3/6) solid order emerges in the system. By in-creasing the magnetic field, the translational symmetryof the subsystem a is restored and a phase transition oc-curs to the b CS(4/6) where the average number of bosonson each unit cell is 4/6. By further increasing of the mag-netic field, the a CS(5/6) solid also appears in the phasediagram of the model just below the saturation field. Asthe solid phases possess different broken symmetries, weexpect the transitions between solid phases to be firstorder which are illustrated with red dotted lines in thephase diagrams.Besides the superfluid, solids and Mott insulator, var-ious supersolids also appear in the phase diagram of themixture (see the definition of supersolid orders in tableI). For the whole range of V /V , in the two narrow re-gions on the top and bottom sides of the a CS(5/6) solidphase, the spins tend to lie in the plane perpendicular tothe magnetic field. In these regions, the system exhibitsthe a CSS supersolid phase in which both diagonal (solid a CS(4/6)) and off diagonal long range orders coexist inthe system. Increasing the hopping parameter | J | , thetranslational symmetry of the subsystem a restores andthe a CS(5/6) solid order disappears where a phase transi-tion occurs from a CSS to the SF phase. For larger valuesof V /V , two other supersolid orders, the b CSS and theCSS phases, also appear in the phase diagram at smallmagnetic fields around the CS(3/6) solid phase (see Fig.2-b). Phase transitions from the a CSS and b CSS to theSF are of first or second order, depending on the valuesof h and | J | . All these first and second order phase tran-sitions, are attributed to the behavior of the low energyspin wave excitation which will be discussed in sectionV.In order to see the effects of quantum fluctuationswe investigate the behavior of both the diagonal andoff diagonal order parameters considering clusters withlarger sizes in CMF theory. Employing clusters of 8 spins(CMF-2 × =0.2, J/V /V V 5=0.16 =0.6, J/V /V V h/V h/V M I( / ) SF a C SS a CS(5/6) a CSS SF F u ll C SS C S ( / ) b C SS b CS(4/6) a CS(5/6) a CSS SF F u ll M I( / ) SF C SS CS(5/6) CSS SF F u ll C SS C S CS(4/6) CS(5/6)
CSS SF F u ll z M v M FIG. 3. Diagonal and off diagonal order parameters, com-puted using CMF theory with 2 × × V /V = 0 . .
6, and the two values of hopping param-eter:
J/V = 0 .
125 and 0 .
165 where all phases appear inthe phase diagram by increasing h . According to the CMF-2 × a CSS and b CSS phases into the CSS phase. The CSS, SF and MI(4/6)phases are not changed by quantum fluctuations. dinal and transverse magnetizations for different valuesof h and J . We found out that the quantum fluctuationsconvert the a CSS and b CSS phases to the CSS phase. Ac-tually, competition between NN and NNN interactionscauses the a CS(5/6) and b CS(4/6) solids transform re-spectively to the CS(5/6) and CS(4/6) solids in whichthere is no relation between the sublattices longitudinalmagnetizations, but the occupation number of each unitcell is conserved. These effects are clearly seen in Fig.3-bottom, in the behavior of the total magnetizations for V /V = 0 . J/V = 0 . V /V = 0 . J/V = 0 . h . The MI(4/6) insulatorand the CSS supersolid are however stable and quantumfluctuations cannot destroy these orders. This is in con- U/V = 0U/V = - 0.4U/V = - 1.4 h/V FIG. 4. The order parameter Q z versus magnetic field fordifferent values of on-site interaction, at V /V = 0 . J/V = 0 . CS(3/6) b CSS CSS b CS(4/6) SF SF a CS(5/6) a CSS a CSS Full h / V J/V b CSS
FIG. 5. (Color online) Ground state phase diagram of theiBH mixture for V /V = 0 . U/V = − . trast with the standard V − V hardcore Bose Hubbardmodel on square lattices in which the CSSphase is unstable against quantum fluctuations, and thepresence of long range dipole-dipole interactions betweenhardcore bosons or long range hopping terms are neces-sary for the stability of the CSS phase . Actually,due to the intrinsic difference in the nilpotency conditionbetween a and b bosons (( a † i ) = 0 and ( b † j ) = 0, where i and j are nearest neighbor sites) quantum fluctuationsare not strong enough to destroy the MI(4/6) and CSSphases. The stability of the CSS phase can be attributedto the large roton-like energy gap in the low energy spinwave excitation spectrum which will be discussed in Sec.V. In this section we will obtain the amount of quantumfluctuations in terms of the number of spin waves andshow that the results of CMF-2 × b bosons satisfy the condition ( b † j ) = 0, one canput up two b particles on each lattice site. This motivatesus to investigate the behavior of the pair superfluid or-der parameter (cid:104) ( b † j ) (cid:105) in all phases. In the spin language this parameter is equivalent to (cid:104) ( τ + j ) (cid:105) . According toour CMF results (not shown) we find that the pair su-perfluidity order parameter is zero in the whole range ofparameter space. This means that although ( b † j ) = 0,but no pairing occurs in the system. We have also com-puted the order parameter Q z = (cid:104) ( τ zB ) + ( τ zD ) (cid:105) indifferent phases. Actually, investigation of the behaviorof Q z gives more intuitions on the properties of solid andMott insulating phases appeared in the phase diagram.This order parameter is almost 1 in all solid and Mottinsulating phases (see Fig. 4). This implies that the localHilbert space basis for b particles is given by the states: | (cid:105) and | (cid:105) , and consequently the effective Hilbert spacedimensions for b particles is two. This fact has been al-ready shown in the schamtic pictures of solid and Mottinsulating phases in the right panel of Fig. 2.In the presence of the attractive on-site interaction,at U (cid:54) = 0, although the superfluidity order parameter de-creases with increasing the on-site interaction, there is noconsiderable changes in the nature of the phases in com-parison with the cases of U = 0 (see Fig. 4 and 5). Theattractive on-site interaction between b particles causesthese particles prefer to be at the same site to minimizethe interaction energy. This leads to the stability of allthe phases at larger values of | J | , and therefore to theshift of the phases’ boarders to the larger values of | J | inthe presence of U . V. LINEAR SPIN WAVE THEORY
In this section, utilizing LSW theory we obtain theexcitation spectra of the mixed spin model in Eq. (5).From these spectra besides the strength of quantum fluc-tuations around the MF ground states, one can find theboundaries of the stability of the mean field phases. Fur-thermore, by investigating the behavior of the low energyspin wave dispersions at phase transitions one can figureout the reason of all first and second order phase transi-tions. Before starting the spin wave approach we imple-ment a unitary transformation on the spin Hamiltonianin Eq. 5 and perform the following rotations on all σ and τ spins; ˜ σ xi ˜ σ yi ˜ σ zi = cos θ i cos φ i − cos θ i sin φ i − sin θ i − sin φ i cos φ i θ i cos φ i sin θ i sin φ i cos θ i σ xi σ yi σ zi , ˜ τ xj ˜ τ yj ˜ τ zj = cos ϑ j cos ϕ j − cos ϑ j sin ϕ j − sin ϑ j − sin ϕ j cos ϕ j ϑ j cos ϕ j sin ϑ j sin ϕ j cos ϑ j τ xj τ yj τ zj , where cos θ i = (cid:104) σ zi (cid:105) /σ , tan φ i = (cid:104) σ yi (cid:105) / (cid:104) σ xi (cid:105) , cos ϑ j = (cid:104) τ zj (cid:105) /τ and tan ϕ j = (cid:104) τ yj (cid:105) / (cid:104) τ xj (cid:105) . Here, (cid:104) . . . (cid:105) denotes theexpectation value on the MF ground state of the Hamil-tonian in Eq. (5). The rotated spin Hamiltonian is ex-pressed in terms of the new bosonic operators ˆ a , ˆ b with aCS(5/6) h/V =4.1, J/V =0.165, aCS(5/6) h/V =4.1 J/V1=0.165 bCS(4/6) h/V =1.65, J/V =0.165, bCS(4/6) h/V =1.65 J/V1=0.165 CS(3/6) h/V =0.1, J/V =0.165,CS(3/6) h/V =0.1 J/V1=0.165 h/V =2.0,J/V =0.125,MI(4/6) h/V =2.0 J/V1=0.125 MI(4/6) h/V =6.0, J/V =0.165, Full h/V1=6.0 J/V1=0.165 Full (-1,-1) (0,0) (1,1) (-1,-1) (0,0) (1,1) (-1,-1) (0,0) (1,1) (-1,-1) (0,0) (1,1) (-1,-1) (0,0) (1,1)
CSS h/V =1.15,J/V =0.165,CSS h/V =1.15 J/V1=0.165 aCSS h/V =5.23,J/V =0.08,aCSS h/V =1.4 J/V1=0.165 bCSS h/V =1.4, J/V =0.165, bCSS h/V =5.23 J/V1=0.08 h/V1=5.41 J/V1=0.165 h/V =5.41, J/V =0.165, SF SF k a /π k a /π k a /π k a /π k a /π Δ FIG. 6. (Color online) Excitation spectra in various phases of the IBM. Number of excitation modes reflects the number ofsublattices in each phase. Top-left: 2D lattice with primitive vectors (cid:126)a = a ˆ x and (cid:126)a = a ˆ y for the MI(4/6), SF and Full phases.Top-center: 2D lattice with primitive vectors (cid:126)a = a (ˆ x + ˆ y ) and (cid:126)a = a ( − ˆ x + ˆ y ) for the solid and supersolid phases wherethe original lattice symmetry is broken. Top-right: the unfolded and folded Brillouin zones. Middle and bottom: spin waveexcitations in all phases in k x = k y direction of the unfolded Brillouin zone. The roton gap (∆) varies in each supersolid phase.All plots are for V /V = 0 . V /V = 0 . the following Holstein-Primakoff (HP) transformations:˜ σ zi = σ − ˆ a † i ˆ a i , ˜ σ + i = (cid:113) σ − ˆ a † i ˆ a i ˆ a i ≈ √ σ ˆ a i , ˜ σ − i = ˆ a † i (cid:113) σ − ˆ a † i ˆ a i ≈ √ σ ˆ a † i , (8) and ˜ τ zj = τ − ˆ b † j ˆ b j , ˜ τ + j = (cid:113) τ − ˆ b † j ˆ b j ˆ b j ≈ √ τ ˆ b j , ˜ τ − j = ˆ b † j (cid:113) τ − ˆ b † j ˆ b j ≈ √ τ ˆ b † j . (9)The spin wave Hamiltonian has the following form;˜ H = E + H (cid:48) , (10)where E is the classical MF energy and H (cid:48) consists ofbilinear terms in HP boson operators. This part yields, h/V =3.59,J/V =0.22,bCS h/V =3.38, J/V =0.165, aCSS h/V =1.57, J/V =0.165, bCSS h/V =1.29,J/V =0.165,CSS h/V =0.955,J/V =0.165,CS(3/6) CS(3/6) → CSS nd order CSS → b CSS nd order b CSS → b CS(4/6) nd order a CSS → a CS(5/6) nd order b CS(4/6)→ a CSS st order, J/V =0.22 b CSS → SF st order, J/V =0.35 a CSS → SF h/V =5.1765,J/V =0.135,aCSS h/V =2.415, J/V =0.35, SF h/V =2.41, J/V =0.35, bCSS h/V =4.9756, J/V =0.185, SF h/V =1.495, J/V =0.52, bCSS (-1,-1) (0,0) (1,1) k a /π (-1,-1) (0,0) (1,1) k a /π (-1,-1) (0,0) (1,1) k a /π (-1,-1) (0,0) (1,1) k a /π (-1,-1) (0,0) (1,1) k a /π (-1,-1) (0,0) (1,1) k a /π h/V =4.9755,J/V =0.185,aCSS a CSS → SF st order, J/V =0.185 b CSS → SF nd order nd order h/V =5.1765 J/V =0.135 h/V =0.955 J/V =0.165 h/V =1.57 J/V =0.165 h/V =1.29 J/V =0.165 h/V =3.38 J/V =0.165 h/V =3.595 h/V =3.59 h/V =4.9755 h/V =4.9755 h/V =1.495 J/V =0.52 h/V =2.41 h/V =2.415 Ɛ ( k ) Ɛ ( k ) h/V =3.595,J/V =0.22,aCSS FIG. 7. (Color online) Excitation spectra at various first and second order phase transitions for V /V = 0 .
6. In the secondorder supersolid-solid phase transition (phase transition from a CSS to a CS(5/6), from b CSS to b CS(4/6) and from CSS toCS(3/6)) the linear dispersions around k = (0 ,
0) and k = ( π/a, π/a ) softens to quadratic ones and become gapped in solidphases. In second order phase transitions from a CSS to SF, and from b CSS to SF number of excitations changes and theamount of the roton gap varies continuously. At first order b CSS-SF phase transition the roton gap decreases suddenly. At firstorder b CS(4/6)- a CSS phase transition the lowest gapped mode abruptly touches zero and the quadratic dispersion changes toa linear one around k = (0 ,
0) and k = ( π/a, π/a ). after diagonalization, the excitation spectra in each phase(For the details of diagonalization see the appendix C).From general symmetry analysis, the off diagonal orderparameter manifold has U(1) freedom to rotate the trans-verse spin order around the magnetic field direction. Inthe superfluid phase, the U(1) symmetry is spontaneouslybroken and a gapless Goldstone mode with a roton-likeminimum appears in the excitation spectra. The slopeof the line connecting the origin of the ε − k plane withthis minimum is proportional to the critical velocity ofthe superfluid and the energy of this minimum is theroton energy gap. Upon approaching the transition (sec-ond order) from the superfluid side the roton energy andconsequently the critical velocity decrease to zero. Atthe same time the superfluid order parameter remainsfinite through the supersolid transition. Inside super-solid phases due to the translational symmetry break-ing the spatial periodicity is doubled, and the Brillouinzone becomes smaller. So half of the excitation spec-trum is folded back to the point k = (0 , v aCSS < v bCSS < v CSS .We have plotted in Fig. 6 the spin wave excitationspectra of all phases. Number of excitation modes andtheir behavior depend on the number of sublattices aswell as their longitudinal and transverse magnetizations.According to the Brillouin zone folding, the k x = k y di-rection in the unfolded zone corresponds to the k x di-rection in the folded one (See Fig. 6), and so the points k = (0 ,
0) and ( π/a, π/a ) are equivalent. In the SF phaseand all the supersolid phases the lower excitation has lin-ear dispersion around the points k = (0 , a CSS and b CSSphases causes fluctuations annihilate low energy rotonsand convert the a CSS and b CSS phases to the CSS one.In a solid phase there is no Goldstone zero mode andall excitations are gapped. The lowest gapped exci-tation spectrum has quadratic dispersion ( k ) around k = (0 , h/V h/V M I( / ) a CSS SF a C SS SF Full a C S ( / ) a C S ( / ) V /V =0.2 𝑑 † 𝑑 𝑐 † 𝑐 𝑎 † 𝑎 𝑏 † 𝑏 h/V C S ( / ) CSS b CSS b C S ( / ) b CS(4/6) a CS(5/6) a C SS a C S ( / ) SF Full V /V =0.6 h/V h/V FIG. 8. (Color online) Number of HP bosons around mean field ground state. Top: V /V = 0 .
2, at line
J/V = 0 . V /V = 0 .
6, at line
J/V = 0 . phase the two energy bands are related by: | ε ( k ) − ε ( k ) | = − hV + 1 . , < hV < . , | ε ( k ) − ε ( k ) | = hV − . , . < hV , (11)and in the CS(3/6) phase, the four energy bands havethe relation | ε ( k ) − ε ( k ) | = | ε ( k ) − ε ( k ) | = hV , where ε and ε , and ε and ε are the two branches with thesame energy behaviors. By increasing magnetic field thetwo lowest energy bands repel each other causing theenergy of the lowest mode decreases and touches zero at k = (0 , k = (0 ,
0) disappearsand the low energy mode softens around this point andbecomes gapped in solid phases.Quantum fluctuations around MF ground state are given by (cid:104) a † a (cid:105) = (cid:104) σ zA (cid:105) MF − (cid:104) σ zA (cid:105) SW , (cid:104) c † c (cid:105) = (cid:104) σ zC (cid:105) MF − (cid:104) σ zC (cid:105) SW , (cid:104) b † b (cid:105) = (cid:104) τ zB (cid:105) MF − (cid:104) τ zB (cid:105) SW , (cid:104) d † d (cid:105) = (cid:104) τ zD (cid:105) MF − (cid:104) τ zD (cid:105) SW , (12)which are the number of ˆ a , ˆ b , ˆ c , and ˆ d HP bosons. Here, (cid:104) σ zA,C (cid:105) MF and (cid:104) τ zB,D (cid:105) MF , and (cid:104) σ zA,C (cid:105) SW and (cid:104) τ zB,D (cid:105) SW are the MF and LSW sublattices’ magnetizations, re-spectively. We have plotted in Fig. 8 the number ofHP bosons versus magnetic field, for the two differentstrengths of frustration V /V = 0 . .
6. Largerfrustration in the case of V /V = 0 . V /V = 0 . V /V = 0 .
2. For all strengths of frustrationthe number of spin waves increases in the vicinity of thetransition points which is a result of the strong quantumfluctuations at phase boundaries. Strong quantum fluc-tuations in the a CSS and b CSS phases is the reason ofthe converting of these phases to the CSS phase in theCMF-2 × b and ˆ d HP bosons in comparisonwith the number of ˆ a and ˆ c is an indication of the weakerquantum fluctuations in the sublattices B and D . Themaximum values of (cid:104) a † a (cid:105) (= (cid:104) c † c (cid:105) ) and (cid:104) b † b (cid:105) (= (cid:104) d † d (cid:105) ) in0CSS phases reach respectively about 30 and 5 percentof the classical values of the spin lengths σ = 1 / τ = 1. This means that the prediction for the groundstates within CMF-2 × which ismuch larger than the case of IBM, and cause the CSSphase of the standard Bose Hubbard model, predictedby MF to be unreliable. VI. FINITE TEMPERATURE PHASEDIAGRAM
At zero temperature, in the superfluidity state each bo-son is spread out over the entire lattice, with long rangephase coherence. At finite temperature, the superfluiddensity is suppressed and the system undergoes a tran-sition to a thermal insulating phase with varying fillingfactor. The thermal insulator (TI) is a weak Mott insu-lator in the sense that it preserves both the translationaland the U(1) symmetries. In the presence of tempera-ture, at T (cid:54) = 0, in the solid phases the plateaus’ widthon longitudinal magnetization curves decreases graduallyby increasing temperature and disappears eventually ata transition temperature which depends on the strengthof frustration, the hopping energy J and the magneticfield h . For example, for the strength of V /V = 0 . J/V = 0 .
22, as it is clearly observedfrom the T − h phase diagram of the IBM (see the top ofFig. 9), the two solid phases a CS(5/6) and b CS(4/6) withconstant average number of bosons 5/6 and 4/6 surviveat low temperature however, by increasing temperaturethese regions become narrower and finally disappear atthe critical temperature T c ∼ . V and T c ∼ . V ,where the mixture has a phase transition to the a CS and b CS phases, respectively. In these phases number of par-ticles are not fixed in each unit cell. In the CSS phase,both the diagonal and the off diagonal long range orderstend to be destroyed by thermal fluctuations, however incomparison with the superfluidity order the CS order ismore robust. By increasing temperature the superfluid-ity order parameter vanishes at a transition temperaturewhere the CSS-CS phase transition occurs. (The value ofthis transition temperature for V /V = 0 . J − h phase diagram of our IBM, we have also plotted in Fig.9 the J − h phase diagram of the mixture for the frus-tration parameter V /V = 0 .
6, at the finite temperature
T /V = 0 .
1. In the presence of temperature, in addi-tion to the ground state phases, several solid orders likeCS, a CS and b CS also appear in the phase diagram of
Full SF a CS a CS(5/6) a CSS SF b CS(4/6) b CS TI b CSS CSS CS(3/6) CS CS h / V T/V CS(3/6) Full TI SF SF a CS a CSS a CSS a CS(5/6) CS b CS b CS(4/6) a CS a CSS a CSS CSS CSS b CSS b CSS h / V J/V CS FIG. 9. (Color online) Top: CMF T − h phase diagram of theIBM for V /V = 0 . J/V = 0 .
22, where all phasesemerge by increasing the magnetic field. The SF-TI transitiontemperature for the two values of magnetic fields h = 3 . V and h = 5 . V are respectively 0 . V and 0 . V . The CSS-CS transition temperature at h/V = 1 .
30 is 0 . V . Bottom:CMF J − h phase diagram for the frustration strength V /V =0 . T /V = 0 . the mixture. Moreover in the region below the Full solidphase the mixture experiences the TI phase which boththe U(1) and the translational symmetries of the originalHamiltonian are preserved. This phase is not seen in theground state phase diagram and is a result of thermalfluctuations. VII. SUMMARY AND CONCLUSION
We have introduced an inhomogeneous hardcorebosonic model composed of two kinds of boson with dif-ferent nilpotency conditions and have shown that themodel is an appropriate ground for searching various su-persolid orders. By generalizing the cluster mean fieldtheory to the IBM, we have studied both the groundstate and the temperature phase diagram of the modelon a 2D square lattice. We have found that in addition tothe superfluidity phase, various kinds of solid and super-solid emerge in the phase diagram of the inhomogeneousmixture. We have also found that for small strengths offrustration the system possesses a Mott insulating statewhich preserves both the U(1) and the translational sym-1metries of the Hamiltonian. In order to see the effectsof quantum fluctuations on the stability of the groundstate phases, we have obtained the diagonal and off diag-onal order parameters using the cluster mean field theorywith larger clusters. Furthermore, using linear spin wavetheory we have studied the behaviors of spin wave exci-tations and the amount of quantum fluctuations aroundthe mean field ground state to see the stability of theground state phases of the model. We have demonstratedthat in contrast with the standard Bose Hubbard modelin which dipole-dipole interactions or long range hop-pings are necessary to have an stable checkerboard su-persolid phase, our inhomogeneous bosonic mixture withnearest and next nearest neighbor interaction possessesthe checkerboard supersolid phase. This stability is at-tributed to the difference in the nilpotency conditionsbetween a and b bosons.We have also studied the behavior of the excitation en-ergies in each phase to get more insights on the groundstate phases of the model. The excitation modes of thesolid phases are gapped and the lowest energy modehas quadratic dispersion around the ordering vectors k = (0 , U (1) symmetry break-ing due to the superfluidity long range order.We have also investigated the behavior of the excita-tion energies around all phase transition points to figureout the reason of first and second order phase transitions.We found out that the abrupt and smooth changes in thebehavior of the low energy excitation modes and the ro-ton gap responsible of all first and second order phasetransitions, respectively. For example, softening of thelinear gapless mode which is accompanied by vanishingof the roton minimum at k = (0 , ACKNOWLEDGMENTS
The authors would like to thank Stefan Wessel for in-sightful comments on the manuscript. We also thankMarcello Dalmonte for reading the manuscript and intro-ducing some references. Useful discussions with RosarioFazio, Alexander Nersesyan, Sebastiano Pilati and San- dro Sorella are acknowledged. JA also thanks ICTPwhere the initial form of this paper for submission wasprepared.
Appendix A: Origin of the IBM Hamiltonian
In this appendix we will explain how the IBM Hamilto-nian introduced in Eq. (1), originates from an standardtwo-orbital Bose-Hubbard model . Let us consider a bi-partite model of hard core bosons including two kinds ofbosons: a single orbital boson a and a two-orbital boson b α ( α = { , } ), interacting via the Hamiltonian: H B = − t (cid:88) (cid:104) i,j (cid:105) ,α ( a † i b αj + h.c. ) + U (cid:88) i n i n i + V (cid:88) (cid:104) i,j (cid:105) n ai ( n j − V (cid:88) (cid:104)(cid:104) i,j (cid:105)(cid:105) [ n ai n aj + ( n i − n j − − (cid:88) i (cid:0) µ a n ai + µ b n i (cid:1) , (A1)where, n ai = a † i a i and n i = n i + n i with n αi = b † αi b αi .The local Hilbert space of this bipartite system is a prod-uct of the local Hilbert spaces of the subsystems I and II.The dimension of the local Hilbert space of the subsys-tem I is D I = 2, since per lattice site i we can only havethe states {| (cid:105) I , a † i | (cid:105) I } where, | (cid:105) I is the vacuum statein subsystem I. The dimension of the local Hilbert spaceof the subsystem II is D II = 4, since the states per latticesite j are {| , (cid:105) II , b † j | , (cid:105) II , b † j | , (cid:105) II , b † j b † j | , (cid:105) II } where | , (cid:105) II is the vacuum state in subsystem II.By defining the following new operators:˜ b † j = 1 √ b † j − b † j ) , b † j = 1 √ b † j + b † j ) , (A2)and applying them to the vacuum state | , (cid:105) II one canget the following antisymmetric and symmetric states: (cid:12)(cid:12)(cid:12) ˜Ψ j (cid:69) = ˜ b † j | , (cid:105) II = 1 √ | , (cid:105) − | , (cid:105) ) , | Ψ j (cid:105) = {| , (cid:105) II , b † j | , (cid:105) II , b † j b † j | , (cid:105) II } = {| , (cid:105) , √ | , (cid:105) + | , (cid:105) ) , | , (cid:105)} . (A3)These states generate another basis for the local Hilbertspace of the subsystem II with D II = 4 ( n j = ˜ b † j ˜ b j + b † j b j ).By writing the Hamiltonian (A1) in terms of the abovesymmetric and antisymmetric operators, the Hamilto-nian decouples into two symmetric and antisymmetricparts, which would be a very helpful step for studyingthe ground state properties of the model. Before doingthis step, some remarks on the local algebra of these op-erators are in order. Following we will show that the local2algebra satisfied by ˜ b † j and b † j is not the same as the onesatisfied by b j † and b j † .Let us label the symmetric states as {| (cid:105) , | (cid:105) , | (cid:105)} ≡{| , (cid:105) II , b † | , (cid:105) II , b † b † | , (cid:105) II } . Action of the opera-tors b † , b and n b on these states leads to the followingrelations: b † | (cid:105) = | (cid:105) , b | (cid:105) = 0 , n b | (cid:105) = 0 ,b † | (cid:105) = | (cid:105) , b | (cid:105) = | (cid:105) , n b | (cid:105) = | (cid:105) ,b † | (cid:105) = 0 , b | (cid:105) = | (cid:105) , n b | (cid:105) = 2 | (cid:105) . (A4)From the above relations one can conclude that the states | (cid:105) , | (cid:105) and | (cid:105) , respectively contain 0, 1 and 2 numbers of b particles, and n b is the number operator of b -particles.Moreover the action of the antisymmetric operators ˜ b † , ˜ b on the antisymmetric state is given by˜ b † (cid:12)(cid:12)(cid:12) ˜Ψ (cid:69) = − | (cid:105) , ˜ b (cid:12)(cid:12)(cid:12) ˜Ψ (cid:69) = | (cid:105) . (A5)Writing down the operators b † , b , n b , ˜ b † , and ˜ b in termsof the symmetric and antisymmetric states as: b † = | (cid:105) (cid:104) | + | (cid:105) (cid:104) | , b = | (cid:105) (cid:104) | + | (cid:105) (cid:104) | ,n b = (cid:12)(cid:12)(cid:12) ˜Ψ (cid:69) (cid:68) ˜Ψ (cid:12)(cid:12)(cid:12) + | (cid:105) (cid:104) | + 2 | (cid:105) (cid:104) | , (A6)and˜ b † = (cid:12)(cid:12)(cid:12) ˜Ψ (cid:69) (cid:104) | − | (cid:105) (cid:68) ˜Ψ (cid:12)(cid:12)(cid:12) , ˜ b = | (cid:105) (cid:68) ˜Ψ (cid:12)(cid:12)(cid:12) − (cid:12)(cid:12)(cid:12) ˜Ψ (cid:69) (cid:104) | , (A7)we can obtain the following unusual commutation rela-tions: [ b, b ] = [ b † , b † ] = 0 , [ b, b † ] = | (cid:105) (cid:104) | − | (cid:105) (cid:104) | = − n b , [ n b , b † ] = | (cid:105) (cid:104) | + | (cid:105) (cid:104) | = b † , where is the identity matrix for the subsystem II whichis written as = | (cid:105) (cid:104) | + | (cid:105) (cid:104) | + | (cid:105) (cid:104) | + (cid:12)(cid:12)(cid:12) ˜Ψ (cid:69) (cid:68) ˜Ψ (cid:12)(cid:12)(cid:12) . Alsoin the subsystem I we have: a † = | (cid:105) I (cid:104) | I , a = | (cid:105) I (cid:104) | I , n a = | (cid:105) I (cid:104) | I (A8)Now let us go back to the Hamiltonian (A1). Substitut-ing the transformations (A2) into the Hamiltonian (A1),and then using the relations (A6), (A7), and (A8), theHamiltonian is transformed to (apart from a constant): H = −√ t (cid:88) (cid:104) i,j (cid:105) ( a † i b j + h.c. ) + U (cid:88) i n bi ( n bi − V (cid:88) (cid:104) i,j (cid:105) n ai ( n bj − V (cid:88) (cid:104)(cid:104) i,j (cid:105)(cid:105) ( n ai n aj + ( n bi − n bj − − (cid:88) i ( µ a n ai + µ b n bi ) . (A9) Using the following rescalings √ t → t, U → U, V → V , V → V , (A10) µ a → µ a + 4 V , µ b → µ b + 2 U + 8 V the Hamiltonian (A9) is simplified to: H = − t (cid:88) (cid:104) i,j (cid:105) ( a † i b j + h.c. ) + U (cid:88) i n bi n bi + V (cid:88) (cid:104) i,j (cid:105) n ai n bj + V (cid:88) (cid:104)(cid:104) i,j (cid:105)(cid:105) ( n ai n aj + n bi n bj ) − (cid:88) i ( µ a n ai + µ b n bi ) , (A11)which is the Hamiltonian we have introduced in Eq. (1) ofthe manuscript. As it is clearly seen the resulted Hamil-tonian is obtained in terms of the symmetric operators b † and b , which means that the local Hilbert space di-mension of the subsystem II is effectively D s = 3. Asthe dimension of the symmetric space is 3 we have em-ployed a simple boson-spin transformations and mappedthis subsystem to a system of spin one. Appendix B: Generalized boson-spintransformations for b bosons In this appendix we explain briefly how the uncommonnilpotency condition of the b bosons and the fractionalexclusion statistics in Eq. 2 lead to the boson-spin map-ping of b bosons in Eq. 4. From the commutation rela-tions in Eq. 2, we find that: 1) the number operator of b bosons possesses the relation (ˆ n bi ) † = ˆ n bi and 2) the num-ber operator of b bosons is not equal to b † b , i.e. ˆ n bi (cid:54) = b † i b i .Actually, due to the non-canonical statistics this kind ofbosons are different from the canonical one, and the ac-tion of the creation(annihilation) operator b † ( b ) on thestate | n b (cid:105) does not lead to √ n b | n b + 1 (cid:105) ( √ n b | n b − (cid:105) ).As the local space of the b particle is isomorphic withthe Hilbert space of a spin 1, employing the followingcorrespondence of boson-spin basis | (cid:105) → | , − (cid:105) , | (cid:105) → | , (cid:105) , | (cid:105) → | , +1 (cid:105) , (B1)and τ + | , − (cid:105) = √ | , (cid:105) , τ + | , (cid:105) = √ | , (cid:105) , τ + | , (cid:105) = 0 , (B2)we find the following relations: τ z = n b − , τ + = αb † , (B3)where the coefficient α is readily obtained as follows: τ + | , − (cid:105) = √ | , (cid:105) , → τ + = √ b † , τ − = √ b. (B4)These kinds of mapping between fractional hard corebosons and spin operators are usually employed for differ-ent standard and non-standard boson Hamiltonian whichcould be seen in Ref. [56]3 Appendix C: Diagonalization of the spin waveHamiltonian
In order to diagonalize the bilinear part of the spinwave Hamiltonian H (cid:48) in Eq. (10), the first step in anstandard approach, is definition of a Fourier transfor-mation for boson operators ˆ a i and ˆ b i . Before going tothis step, we notice that since the phases: MI(4/6), SFand Full preserve the translational symmetry of the orig-inal Hamiltonian the classical background has a two-sublattice structure and the excitations of these phasesare achieved by defining the two HP bosons: ˆ a and ˆ b .However, in other solid and supersolid phases, accord-ing to the translational symmetry breaking, the classicalbackground has a four-sublattice structure and more HPbosons should be employed to attain the excitation spec-tra of these phases. In this respect, we consider a generalbackground and divide the subsystem with spin σ to twosublattices with bosons ˆ a and ˆ c , and the subsystem withspin τ to two sublattices with bosons ˆ b and ˆ d . Definingthe primitive vectors as in the top-center of Fig. 6 andutilizing the following Fourier transformations;ˆ a j = 1 (cid:112) N/ (cid:88) k e − i k · r j ˆ a k , ˆ c j = 1 (cid:112) N/ (cid:88) k e − i k · r j ˆ c k , ˆ b j = 1 (cid:112) N/ (cid:88) k e − i k · r j ˆ b k , ˆ d j = 1 (cid:112) N/ (cid:88) k e − i k · r j ˆ d k , where N/ H (cid:48) = (cid:88) k ψ † k H k ψ k , with ψ k , the following 8-component vector: ψ † k = (cid:0) ˆ a † k ˆ b † k ˆ c † k ˆ d † k ˆ a − k ˆ b − k ˆ c − k ˆ d − k (cid:1) , and H k = (cid:18) A BB ∗ A ∗ (cid:19) , (C1)where A and B are two 4-square matrices with complexfunctions. The general forms of the matrices A and B are given by A = α α ∗ α α ∗ α α α α α α ∗ α α ∗ α α α α , B = α ∗ α α ∗ α ∗ α ∗ α α α ∗ α ∗ α ∗ α α ∗ , where α = 2 w ab cos ( k x a (cid:48) / , α = 2 w ab cos ( k x a (cid:48) / ,α = 2 w cd cos ( k x a (cid:48) / , α = 2 w cd cos ( k x a (cid:48) / ,α = 2 w ad cos ( k y a (cid:48) / , α = 2 w ad cos ( k y a (cid:48) / ,α = 2 w cb cos ( k y a (cid:48) / , α = 2 w cb cos ( k y a (cid:48) / ,α = 4 V g ac cos ( k x a (cid:48) /
2) cos ( k y a (cid:48) / ,α = 4 V g bd cos ( k x a (cid:48) /
2) cos ( k y a (cid:48) / ,α = 2( w ab + w ad ) + 4 V g ac − he a ,α = 2( w ab + w cb ) + 4 V g bd − he b ,α = 2( w cb + w cd ) + 4 V g ac − he c ,α = 2( w ad + w cd ) + 4 V g bd − he d , (C2)with w αβmn = V g αmn − Jf βmn . (C3)Here, a (cid:48) is the length of the primitive vectors shown inthe top-center of Fig. 6, α and β are 1, 2, 3, 4 and m and n are the sublattices label: a, b, c and d . The coefficients f βmn , g αmn and e m are given in terms of θ m , θ n and φ m , φ n as follows: f mn = (cid:112) S m S n ((cos θ m cos θ n −
1) cos ( φ m − φ n ) , + i sin ( φ m − φ n )(cos θ n − cos θ m )) ,f mn = (cid:112) S m S n ((cos θ m cos θ n + 1) cos ( φ m − φ n ) , + i sin ( φ m − φ n )(cos θ n + cos θ m )) ,f mn = − S n sin θ m sin θ n cos ( φ m − φ n ) ,f mn = − S m sin θ m sin θ n cos ( φ m − φ n ) ,g mn = 12 (cid:112) S m S n sin θ m sin θ n ,g mn = − S n cos θ m cos θ n ,g mn = − S m cos θ m cos θ n ,e m = − cos θ m , (C4)where S m and S n are the spins of sublattices m and n ,respectively. Performing a paraunitary transformation T , the Hamiltonian H k in Eq. (C1) is diagonalized as: ψ † k H k ψ k = ψ † k T † ( T † ) − H k T − T ψ k = Ψ † k E k Ψ k , where E k is the para-diagonalized matrix containing theexcitation energies and Ψ k = T ψ k is a para-vector ofnew bosonic operators. The paraunitary transformationsatisfies the following relations T ˆ IT † = ˆ I, T † ˆ IT = ˆ I, T † ˆ I = ˆ IT − , (C5)with ˆ I × = (cid:18) I × − I × (cid:19) , where I × is the 4 × T we utilize the follow-ing procedure which introduced by Colpa for a positive-definite Hamiltonian . First we write the Hamiltonian4 H k as H k = κ † k κ k where the matrix κ k is the Choleskydecomposition of the Hamiltonian. Then, we find theunitary transformation matrix, υ k which diagonalizes thehermitian matrix κ k ˆ Iκ † k as L k = υ † k ( κ k ˆ Iκ † k ) υ k . The di-agonal matrix E k is readily obtained from the relation E k = ˆ I L k . Finally, by solving the equation υ k √E k = κ k T − row to row, we achieve the paraunitary transfor-mation T .The above diagonalization procedure is for the gen-eral case of four-sublattice structure which is employedfor the a CS(5/6), b CS(4/6) and CS(3/6) solid, and for a CSS, b CSS and CSS supersolid phases. For the MI(4/6),SF and Full phases where the translational symmetry inboth subsystems is preserved, the MF ground states aregiven by a two-sublattice structure and the 4 × A and B are simplified to two 2 × a j = 1 √ N (cid:88) k e − i k · r j ˆ a k , ˆ b j = 1 √ N (cid:88) k e − i k · r j ˆ b k , where N is the number of each HP boson and r j is givenin terms of the primitive vectors shown in the top-left ofFig. 6, the bilinear Hamiltonian is readily obtained as: H (cid:48) = (cid:88) k ψ † k H k ψ k , with ψ † k = (cid:0) ˆ a † k ˆ b † k ˆ a − k ˆ b − k (cid:1) , and H k = (cid:18) A BB ∗ A ∗ (cid:19) , (C6)where A and B are given by A = (cid:18) α α ∗ α α (cid:19) , B = (cid:18) α α ∗ α ∗ α (cid:19) , where α = 2 w ab cos( k x a/
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