Two supersolid phases in hard-core extended Bose-Hubbard model
TTwo supersolid phases in hard-core extended Bose-Hubbard model
Yung-Chung Chen and Min-Fong Yang
Department of Applied Physics, Tunghai University, Taichung 40704, Taiwan (Dated: March 20, 2018)The effect of the next-nearest-neighbor (nnn) tunneling on the hard-core extended Bose-Hubbardmodel on square lattices is investigated. By means of the cluster mean-field theory, the ground-statephase diagrams are determined. When a modest nnn tunneling is introduced, depending on its sign,two distinct supersolid states with checkerboard crystal structures are found away from half-filing.The characters of various phase transitions out of these two supersolid states are discussed. Inparticular, for the case with kinetic frustration, the existence of a half supersolid phase possessingboth solid and unconventional superfluid orders is established. Our work hence sheds light on thesearch of this interesting supersolid phase in real ultracold lattice gases with frustrated tunnelings.
PACS numbers: 03.75.Nt, 67.80.kb, 67.85.-d
I. INTRODUCTION
Ultracold atomic gases in optical lattices provide an ex-tremely versatile and well-controlled experimental plat-form for exploring complex phenomena in quantum mat-ters [1–3]. Owing to remarkable control over micro-scopic model parameters, they can simulate the physics ofstrongly correlated systems in regimes which are not eas-ily accessible in condensed matters. The rapid advancesin experimental techniques open the door to the searchfor exotic phases and in probing their quantum criticalbehaviors. One of the most striking experiments is theobservation of the superfluid-Mott transition of ultracoldbosons in an optical lattice [4], which can be described bythe seminal Bose-Hubbard model [5, 6]. This quantumphase transition is driven by the competition between thekinetic energy of particles hopping between lattice sitesand the on-site repulsive interaction.Recently, by using the lattice shaking technique, the ef-fective hopping parameters can be tuned to be negativeor even complex [3, 7–10]. As a result, lattice bosons withfrustrated hoppings and/or coupled to artificial gaugefields have now been realized, which can give rise to un-conventional superfluid (SF) phases [11, 12]. Besides,intriguing supersolid (SS) states have been predicted tooccur as well in the Bose-Hubbard model with hoppingfrustration induced by staggered flux [13].The SS phases are fascinating quantum states thathave superfluid and solid long-range orders simultane-ously. Such phases usually come out around the phaseboundary of commensurate solid phases and behave asintermediate states between the superfluid and the crys-talline ones. In addition to hopping frustration inducedby the gauge fields [13], ultracold lattice gases withdipole-dipole interaction [14–16] or the spin-orbit cou-pling [17–19] also provide possible candidates in simulat-ing such quantum phases. In particular, the existenceof the SS phases has been established in extended Bose-Hubbard models (i.e., Bose-Hubbard models with addi-tional intersite interactions) [20–25], which can be nat-urally realized with dipolar bosons. Due to intersite re-pulsions among lattice bosons, insulating solid states at commensurate filling factors can be stabilized. Underdoping particles or holes into these insulating solid states,the SS states arises when dopants delocalize and undergoBose-Einstein condensation rather than a phase separa-tion. The late realization of the extended Bose-Hubbardmodel with nearest-neighbour repulsion for magnetic er-bium atoms lays the groundwork in searching for the SSstates [26]. Very recently, the SS phases are observedexperimentally in atomic quantum gases trapped in anoptical lattice inside a high-finesse optical cavity [27, 28].They are stabilized by the long-range interaction medi-ated by a vacuum mode of the cavity.More exotic states of matter are expected when oneincorporates the hopping frustration into dipolar latticegases. To uncover the interesting physics of such systems,an extended Bose-Hubbard model with a frustrated next-nearest-neighbor tunneling on square lattices is investi-gated recently [29]. In the hard-core limit, the grand-canonical Hamiltonian reads H = − t (cid:88) (cid:104) i,j (cid:105) ( b † i b j + h.c. ) − t (cid:48) (cid:88) (cid:104)(cid:104) i,j (cid:105)(cid:105) ( b † i b j + h.c. )+ V (cid:88) (cid:104) i,j (cid:105) n i n j − µ (cid:88) i n i , (1)where (cid:104) i, j (cid:105) indicates the nearest- and (cid:104)(cid:104) i, j (cid:105)(cid:105) next-nearest neighbors. The destruction (creation) operatorsof hard-core bosons on site i are denoted by b i ( b † i ). Theparameters t and t (cid:48) stand for the nearest-neighbor (nn)and the next-nearest-neighbor (nnn) hopping integrals,respectively. The repulsion between the nn sites is repre-sented by V , n i = b † i b i is the number operator of bosons,and µ is the chemical potential. Here both t and V > t (cid:48) can be either positive ornegative. When t (cid:48) <
0, the hopping frustration appears.In Ref. [29] where the frustrated case with t (cid:48) < V and modest | t (cid:48) | . Itsstabilization results from the interplay of the frustratednnn hopping and the nn repulsion. This HSS state ischaracterized by the preferential superfluid flow along the a r X i v : . [ c ond - m a t . qu a n t - g a s ] N ov FIG. 1: Schematic pictures of (a) the checkerboard supersolid(CSS), (b) the half supersolid (HSS), and (c) the checkerboardsolid (CBS). The empty (dark blue) circles indicate the sitesof no (full or nearly full) occupation and the light blue circlesstand for partially filled sites. The signs in the circles showthe relative phases of Bose condensate among sites. nnn bonds and nonzero Bose-Einstein condensate mainlyon one sublattice. In addition, the particle density showsa checkerboard long-range order. The lattice’s transla-tional symmetry is thus spontaneously broken. A phaseseparation regime consisting of the uniform SF and theHSS phases is observed in their fixed-density models [29].This implies a first-order phase transition between thesetwo phases.Actually, the non-frustrated nnn hopping ( t (cid:48) >
0) cansupport as well a similar supersolid phase possessing boththe SF and the checkerboard crystalline long-range or-ders [33]. The appearance of such a checkerboard super-solid (CSS) state for the postive- t (cid:48) case has been demon-strated by means of quantum Monte Carlo (QMC) sim-ulations. In contrast to the the negative- t (cid:48) case, the SF-CSS phase transitions are however found to be of second-order [33], instead of being discontinuous.In this paper, we focus on the detailed ground-statephase diagrams of the model in Eq. (1) for both signsof t (cid:48) and elucidate the nature of the phase transitionstherein. The cluster mean-field theory is employed here,which has been applied to various spin [34–38] and bo-son [39–47] systems with success. We note that the maindifference between two supersolid states, the HSS and theCSS states, comes from the distinct momentum states atwhich bosons condenses. Depending on the sign of t (cid:48) ,Bose condensate in the HSS phase occurs at the statewith momenta k = (0 , π ) and ( π, k = ( π, π ) inthe CSS phase. Schematic pictures of these two phasesare illustrated in Figs. 1(a) and (b). By measuring thetypes of condensation, both supersolid states can thusbe identified. Besides the uniform SF and the supersolidphases, there exists as well an incompressible checker-board solid (CBS) state at half-filling [see Fig. 1(c)]. Thissolid state breaks as well the lattice’s translational sym-metry and is signaled by a plateau in the density profileversus chemical potential. Various transitions betweenthese quantum phases are summarized in Fig. 2, whichis the main result of this work.For weak nnn hopping t (cid:48) , direct first-order transitionsbetween the CBS and the SF phases are observed. Uponincreasing | t (cid:48) | , an intermediate supersolid state (either -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7-101234 m t' CBS
HSS SF CSS (a) m VCBSSF HSS(b) SF CSSCBS m V (c)
FIG. 2: Phase diagram calculated by the cluster mean-fieldtheory using 4 × V = 8, (b) t (cid:48) = − .
3, and(c) t (cid:48) = 0 .
5. Here t = 1 is assumed. The solid and theopen symbols indicate the second-order and the first-ordertransitions, respectively. The dashed (solid) lines connectingopen (solid) symbols are only a guide to the eye. The relativeerrors of the determined phase boundaries are set at the orderof 10 − and thus the error bars (not shown) for all the datapoints are smaller than the sizes of the symbols. CSS or HSS) emerges. In accordance with previous re-sults [29, 33], we find that the SF-HSS transitions areof first order, while the SF-CSS transitions are continu-ous. In contrast, both transitions from each supersolidphase to the CBS phase are shown to be continuous andbelong to the superfluid-insulator universality class [5].According to our CMF results, the stability of the twosupersolid phases is enhanced when the nn repulsion V and the magnitude of the nnn hopping t (cid:48) are increased.Since the model under consideration lies within the reachof present experimental setup, hopefully our predictionsmay be realized in the near future.This paper is organized as follows: In Sec. II, the clus-ter mean-field theory is briefly described and the mea-sured order parameters are introduced. Our numericalresults are presented and discussed in Sec. III. We sum-marize our work in Sec. IV. II. CLUSTER MEAN-FIELD THEORY
Our calculations are performed by using the clustermean-field (CMF) theory [34–47]. It has several differentnames in the literature: cluster Gutzwiller approach [43,47], hierarchical mean-field approach [35, 36, 45], andcomposite boson mean-field approach [44]. This methodhas been shown to be an extremely efficient way of ex-ploring vast uncharted territory in the phase diagram.Because short-range correlations within the cluster are
FIG. 3: A cluster of 4 × taken into account exactly, the CMF method can give farmore precise results than conventional single-site mean-field approaches. Under a scaling of the cluster sizes,even the accurate phase boundaries for infinite latticesmay be achieved [40, 41, 43].In the first subsection, we explain how the groundstates are obtained under the CMF approach. In thesecond subsection, the ways in determining the order pa-rameters and thus the ground-state phase diagrams aredescribed. The method of the cluster-size scaling is dis-cussed at the end. A. ground states in the CMF theory
In the CMF theory, the whole system is decomposedinto a cluster of lattice sites and its surroundings, as il-lustrated in Fig. 3. The interaction between these twoparts is approximated by the mean fields at the bordersof the cluster. The effective Hamiltonian of the cluster isthen given by H eff C = H C + H ∂C , (2)where H C is the exact Hamiltonian of the cluster C and H ∂C contains the effects of the mean fields acting only onsites at the boundary ∂C . By using the standard mean-field decoupling, the latter takes the form for our modelin Eq. (1), H ∂C = − t (cid:88) (cid:48)(cid:104) i,j (cid:105) (cid:16) b † i (cid:104) b j (cid:105) + h.c. (cid:17) − t (cid:48) (cid:88) (cid:48)(cid:104)(cid:104) i,j (cid:105)(cid:105) (cid:16) b † i (cid:104) b j (cid:105) + h.c. (cid:17) + V (cid:88) (cid:48)(cid:104) i,j (cid:105) n i (cid:104) n j (cid:105) , (3)where the prime over (cid:80) indicates that the site i locateson the boundary ∂C and the site j is outside the cluster.Such mean-field decoupling amounts to the approxima-tion of the many-body wave function being written as aproduct of the cluster and the environment wave func-tions.After diagonalizing the effective Hamiltonian H eff C , themean fields of the surroundings, (cid:104) b j (cid:105) and (cid:104) n j (cid:105) , can be ob-tained self-consistently from the corresponding expecta-tion values inside the cluster. We note that the strategy in determining the values of mean fields is not unique.The implementation suggested in Refs. [40] and [41] isadopted here. A sublattice structure expected to emergein the parameter range is first imposed, into which thecluster of N C sites is embedded. To be compatible withall the expected phases in our system (cf. Fig. 1), stateswith a four-sublattice structure are considered in thepresent work. Assuming the mean fields (cid:104) b j (cid:105) to be real,there are thus eight mean-field parameters, (cid:104) b j (cid:105) and (cid:104) n j (cid:105) ,in total. Under this four-sublattice pattern, a cluster of N C = 4 × t , t (cid:48) , V , and µ , the self-consistent procedure for determining the ground stateincludes the following key steps:(i) Substitute a set of initial mean-field values of (cid:104) b j (cid:105) and (cid:104) n j (cid:105) into Eq. (3), and calculate the groundstate | GS (cid:48) (cid:105) of the effective Hamiltonian H eff C inEq. (2).(ii) Calculate the new mean-field values (cid:104) b (cid:48) j (cid:105) = (cid:104) GS (cid:48) | b j | GS (cid:48) (cid:105) and (cid:104) n (cid:48) j (cid:105) = (cid:104) GS (cid:48) | n j | GS (cid:48) (cid:105) for theground state | GS (cid:48) (cid:105) .(iii) Compare two sets of mean fields, {(cid:104) b j (cid:105) , (cid:104) n j (cid:105)} and {(cid:104) b (cid:48) j (cid:105) , (cid:104) n (cid:48) j (cid:105)} . If the net deviation is smaller than thegiven tolerance, that is, (cid:80) j |(cid:104) b (cid:48) j (cid:105)−(cid:104) b j (cid:105)| + (cid:80) j |(cid:104) n (cid:48) j (cid:105)−(cid:104) n j (cid:105)| < (cid:15) (in the present work, we take (cid:15) = 10 − ),then output | GS (cid:48) (cid:105) as the ground state, (cid:104) b (cid:48) j (cid:105) and (cid:104) n (cid:48) j (cid:105) as the self-consistent mean fields. Otherwise,set (cid:104) b j (cid:105) = (cid:104) b (cid:48) j (cid:105) , (cid:104) n j (cid:105) = (cid:104) n (cid:48) j (cid:105) and return to step (i). B. order parameters and phase diagram
To characterize different phases, several order param-eters are calculated. Nonzero Bose condensate indicatesthe presence of the SF long-range order. The condensatedensity ρ at a momentum k can be evaluated throughthe Fourier transform of the mean fields (cid:104) b j (cid:105) , ρ ( k ) ≡ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) N C (cid:88) j ∈ C (cid:104) b j (cid:105) e i k · r j (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , (4)where N C is the total number of sites in the cluster C . In a uniform SF phase, we have Bose condensateat k = (0 ,
0) with ρ (0 , (cid:54) = 0 and ρ ( k ) = 0 for allother k ’s. Instead, unconventional SF orders are foundin the CSS and the HSS phases. From the inspection oftheir condensation patterns depicted in Fig. 1, one ex-pects that both ρ (0 ,
0) and ρ ( π, π ) are nonzero in theCSS phase, while only ρ ( π,
0) = ρ (0 , π ) can be finite inthe HSS phase.On the other hand, long-range solid order appears ifthere exists density modulation with a commensuratewave vector k . This can be detected by nonzero valuesof the Fourier transform of the local densities (cid:104) n j (cid:105) ,˜ n ( k ) ≡ N C (cid:88) j ∈ C (cid:104) n j (cid:105) e i k · r j . (5)Its value at k = (0 ,
0) is nothing but the averaged den-sity. In the thermodynamic limit, this quantity has adirect relation, | ˜ n ( k ) | = S ( k ) /N C , to the static struc-ture factor S ( k ) ≡ (1 /N C ) (cid:80) i,j ∈ C (cid:104) n i n j (cid:105) e i k · ( r i − r j ) . Forsystems with the CBS order, ˜ n ( π, π ) (cid:54) = 0 is expected.In particular, the classical picture of the incompressibleCBS state at half-filling predicts ˜ n ( π, π ) = 1 / e = 1 N C (cid:18) E G − (cid:104) H ∂C (cid:105) (cid:19) , (6)where E G is the lowest energy eigenvalue of the effectiveHamiltonian H eff C . We note that double counting termscoming from the mean-field decoupling of two-cluster in-teractions must be subtracted. This explains the occur-rence of the second term in Eq. (6).The phase diagrams presented in Fig. 2 are obtained bymeans of the CMF approach using 4 × λ = N B / ( N C × z/ N B denotes thenumber of the nn bonds within the cluster and z is thecoordination number for the lattice ( z = 4 for the presentcase of square lattice). Because the denominator repre-sents the number of the nn bonds of the original latticeper N C sites, the parameter λ provides an indication ofhow much the correlation effects between particles areconsidered in the cluster. The infinite-size limit thus cor-responds to λ = 1. In the next section, linear extrapo-lations toward λ = 1 for some typical phase transitionpoints are performed to illustrate the cluster-size depen-dence of our results. III. NUMERICAL RESULTS ANDDISCUSSIONSA. non-frustrated case of t (cid:48) > In order to provide benchmarks on the performance ofthe CMF method, we begin with the non-frustrated caseof t (cid:48) >
0, where some QMC data are available in the lit-erature [33]. Fig. 4(a) shows our CMF results of various m / t ' o r de r pa r a m e t e r m/ t'(a) l CSS CBS CSS SF t/t ' t/t' r (0,0) r ( p , p ) 0.5|n( p , p )| (b) l FIG. 4: Dependence of various order parameters (a) on thechemical potential µ for V = 3 . t (cid:48) and t = 0 . t (cid:48) ; (b) on thenn hopping t for V = 3 . t (cid:48) and µ = 2 . t (cid:48) , which are calcu-lated by using 4 × t (cid:48) = 1 is taken as energyunit. In both cases, ρ ( π,
0) = ρ (0 , π ) = 0. Insets in thepanels (a) and (b) show the cluster-size scalings of the CMFdata for the phase transition points µ c and t c , respectively.The scaling parameter λ = 0 .
5, 0.625, 0.75, 0.79 correspondrespectively to the clusters of rectangular geometry with sizes N C = 2 ×
2, 2 ×
4, 4 ×
4, and 4 ×
6. The lines are the linear fitsof the values for the largest three sizes. The red open symbolsindicate the QMC results [33]. order parameters as functions of the chemical potential µ for V /t (cid:48) = 3 . t/t (cid:48) = 0 .
01 obtained by using 4 × t (cid:28) t (cid:48) , upon decreas-ing µ , the incompressible CBS state with ˜ n ( π, π ) = 1 / ρ (0 ,
0) = ρ ( π, π ) = 0 transits continuously into theCSS state with coexistence of the CBS and the SF orders.This continuous melting transition occurs at µ c /t (cid:48) (cid:39) . µ c /t (cid:48) (cid:39) . λ = 1. As a fur-ther benchmark, the CMF results using 4 × V /t (cid:48) = 3 . µ/t (cid:48) = 2 . t , the uniform SF statewill be eventually stabilized, such that ˜ n ( π, π ) vanishesand ρ ( k ) is nonzero for k = (0 ,
0) only. The CSS-SFtransition is found to be as well continuous and it occursat t c /t (cid:48) (cid:39) .
34. They are consistent with the QMC re-sults (see the lowest panel of Fig. 4 in Ref. [33]), where t c /t (cid:48) (cid:39) .
27 is found. Again, excellent agreement can beobtained after a linear extrapolation toward the scalingparameter λ = 1, as shown in the inset of Fig. 4(b). Allthese results validate the CMF approach to the presenthard-core boson problem. mm mm (d)(c) (a) m l p , p )| r (0,0) r ( p , p ) m (b) l SF CBS SF CSS CBS
CSS-CBSSF-CSS1.50 1.550.000.050.100.15 n( p , p ) r (0,0) r ( p , p ) FIG. 5: Various order parameters as functions of the chemicalpotential µ calculated by using 4 × V = 8for (a) t (cid:48) = 0 . t (cid:48) = 0 .
5. Here t = 1 is taken asenergy unit. In both cases, ρ ( π,
0) = ρ (0 , π ) = 0. Insetsin the panels show the cluster-size scalings of the CMF datafor the phase transition points µ c . The scaling procedure isdiscussed in Fig. 4. Lower panels show the details of the orderparameters around the SF-CSS and the CSS-CBS transitionpoints in panel (b) with enlarged scales. Lines in these panelsare (c) the square-root and (d) the linear fits of the data. The requirement of t < t (cid:48) studied in Ref. [33] is hardto be satisfied in most cold atom experiments. One maywonder if similar physics remains in the more practicalsituation such that the nn hopping is larger than the nnnone (i.e., t > t (cid:48) ). In the rest of this paper, only weak t (cid:48) cases are considered, and we take t = 1 as the energyunit.In Fig. 5, the CMF results using 4 × t (cid:48) = 0 . V =8 are presented. For the t (cid:48) = 0 . ρ (0 ,
0) and ˜ n ( π, π ),change discontinuously. This first-order transition occursat µ c (cid:39) .
14. As seen from the inset, the transition pointapproaches to µ c (cid:39) .
49 in the infinite-size limit (i.e., λ → t (cid:48) case is expected. In the t (cid:48) = 0 limit, it hasbeen known that the hard-core boson model [49–51] (orthe equivalent spin-1/2 XXZ model [52, 53]) does notstabilize the supersolid phase on a square lattice and thedirect SF-CBS transition is of first order. Turning on asmall t (cid:48) would not change the whole story, as shown byour results.According to the Ginzburg-Landau-Wilson paradigm,a direct transition between the SF and the CBS states,which break different symmetries, must be of first order.Otherwise, an intermediate supersolid phase with coex-istence of SF and solid long-range orders will emerge in between. The occurrence of the CSS phase between theSF and the CBS phases at modest nnn hoppings has beenestablished in Fig. 5(b). For t (cid:48) = 0 .
5, within our CMFresults using 4 × . (cid:46) µ (cid:46) .
80. The transitions at these twophase boundaries are both continuous. As seen from theinset, the stability region of the CSS phase shrinks butremains finite in the infinite-size limit (i.e., λ → n ( π, π ) showa square-root dependence on µ , ˜ n ( π, π ) ∝ ( µ − µ c ) / ,as presented in Fig. 5(c). That is, the critical exponenthas a mean-field value β = 1 /
2. This reflects the mean-field character of our approach. On the other hand, theCSS-CBS transition should belong to the SF-insulatoruniversality class [5], such that the condensate fractionvanishes linearly around the transition point. Our resultsof both condensate fractions ρ (0 ,
0) and ρ ( π, π ) shownin Fig. 5(d) are in agreement with this prediction.The phase diagrams for the extended Bose-Hubbardmodel in Eq. (1) with t (cid:48) > t (cid:48) (cid:29) t limit investigated in Ref. [33],this phase can appear as well for weak t (cid:48) cases as long asstrong nn repulsion V is present. B. frustrated case of t (cid:48) < Now we turn to the frustrated case of t (cid:48) <
0. Due to thenegative-sign problem caused by the hopping frustration,QMC simulations are no longer applicable. Therefore, toobtain quantitative predictions of the present frustratedsystems, other theoretical approaches, such as the tensornetwork state method employed in Ref. [29], are neces-sary. With success in the t (cid:48) > t (cid:48) < × t (cid:48) = − . − . V = 8 are presentedin Fig. 6. Similar to the cases of weak positive t (cid:48) , a directfirst-order transition between the SF and the CBS phasesis observed for the t (cid:48) = − . µ c (cid:39) .
82 and the valueof the transition point approaches to µ c (cid:39) .
09 in theinfinite-size limit (i.e., λ → t (cid:48) , anintermediate HSS phase with coexistence of SF and solidlong-range orders can emerge in between. Similar to theCSS phase in the non-frustrated case, the HSS state ischaracterized as well by the preferential superfluid flowalong the nnn bonds. Nevertheless, Bose condensate inthe HSS phase occurs at the state with momenta k =(0 , π ) and ( π, k = ( π, π ) in the CSS phase. (d)(c) (a) (0,0) ( ,0) (b) SF CBS SF HSS CBS
HSS-CBSSF-HSS ( ,0) FIG. 6: Various order parameters as functions of the chemicalpotential µ calculated by using 4 × V = 8for (a) t (cid:48) = − . t (cid:48) = − .
3. Here t = 1 is takenas energy unit. In both cases, ρ ( π, π ) = 0 and ρ (0 , π ) = ρ ( π, µ c . The scalingprocedure is discussed in Fig. 4. Panel (c) shows the ground-state energies e as a function of µ calculated by Eq. (6) aroundthe SF-HSS transition point in panel (b). Two straight graylines connecting symbols are only a guide to the eye. Theydisplay the level crossing and the cusp in energy curve. Panel(d) gives the condensate fraction ρ ( π,
0) and its linear fitaround the HSS-CBS transition point in panel (b) with anenlarged scale.
The case of t (cid:48) = − . . (cid:46) µ (cid:46) .
05 within ourCMF results using 4 × λ → ρ ( π, t (cid:48) >
0. The discontinuousjumps in the order parameters at the SF-HSS transitionpoint is a clear indication that phase separation wouldoccur in a canonical system, as observed in Ref. [29].First-order quantum phase transitions come from en-ergy level crossing in the ground state and are thus sig-naled by a cusp in the ground-state energy curve [48]. InFig. 6(c), the energy curve in the ground state aroundthe discontinuous SF-HSS transition is shown, where thelocation of the cusp gives the transition point we want.In the present work, all the first-order transition points,including those of the direct SF-CBS transitions shownin Figs. 5(a) and 6(a), are determined by this way.Distinct characters of the SF-CSS transition for the t (cid:48) > t (cid:48) < k = ( π,
0) and (0 , π ) in the HSS stategives (cid:104) b j (cid:105) ∝ e i ( π, · r j + e i (0 ,π ) · r j = cos( πx )+cos( πy ). Thisleads to alternating phase change in π among sites withinthe sublattice on which bosons condense, as displayed inFig. 1(b). Thus a smooth evolution from the HSS state tothe uniform SF state should be unlikely. On the contrary,there is no phase difference in the condensate for the CSSstate [see Fig. 1(a)]. A continuous transition from theCSS state to the uniform SF state is thus expected.The phase diagrams for the extended Bose-Hubbardmodel in Eq. (1) with t (cid:48) < V and the magnitudeof the nnn hopping t (cid:48) are increased. As seen from thephase diagrams in Fig. 2, the HSS phase for t (cid:48) < t (cid:48) > t (cid:48) /t < − /
2, the uni-form SF phase becomes absent and the HSS phase canexist for all incommensurate fillings. This can be under-stood from the single-particle dispersion relation, whichhas local minima at k = (0 , π ) and k = ( π,
0) when t (cid:48) /t < − /
2. This thus favors the Bose condensationconsistent with the SF order in the HSS state, ratherthan the uniform SF one.
IV. SUMMARY AND CONCLUSIONS
Using the cluster mean-field theory, we explored in de-tail the formation of two supersolid phases, which arise inthe hard-core extended boson Hubbard model of Eq. (1).We find that, in the presence of a large nn repulsion,by introducing a modest nnn hopping t (cid:48) , either the CSSor the HSS phases can be stabilized away from half-filling. Both supersolid phases come out around thephase boundary of the CBS phase and behave as interme-diate states between the SF and the CBS states. Despitecarrying the same checkerboard crystalline long-range or-der, bosons condense at different momentum states inthese two supersolid phases, as shown in our calculations.The characters of various phase transitions out of theCSS or the HSS states are discussed. Their transitionsto the CBS state are found to be continuous and belongto the SF-insulator universality class [5]. However, forthe frustrated case with t (cid:48) <
0, the SF-HSS transitionis shown to be of first order, instead of being continuousobserved at the SF-CSS transition for the t (cid:48) > Acknowledgments
Y.C.C. and M.F.Y. acknowledge the support fromthe Ministry of Science and Technology of Taiwan un-der Grant No. NSC 102-2112-M-029-003-MY3, MOST105-2112-M-029-005, MOST 106-2112-M-029-003, andMOST 106-2112-M-029-004. We also thank Kwai-KongNg for the critical reading and useful discussions. [1] M. Lewenstein, A. Sanpera, and V. Ahufinger,
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