Competing supersolids of Bose-Bose mixtures in a triangular lattice
aa r X i v : . [ c ond - m a t . s t r- e l ] F e b Competing supersolids of Bose-Bose mixtures in a triangular lattice
Fabien Trousselet, Pamela Rueda-Fonseca, and Arnaud Ralko Institut N´eel, Universit´e Grenoble Alpes and CNRS, F-38042 Grenoble, France CEA, INAC-SP2M, F-38054 Grenoble, France (Dated: August 15, 2018)We study the ground state properties of a frustrated two-species mixture of hard-core bosonson a triangular lattice, as a function of tunable amplitudes for tunnelling and interactions. Bycombining three different methods, a self-consistent cluster mean-field, exact diagonalizations andeffective theories, we unravel a very rich and complex phase diagram. More specifically, we discussthe existence of three original mixture supersolids: (i) a commensurate with frozen densities andsupersolidity in spin degrees of freedom, in a regime of strong interspecies interactions; and (ii) whenthis interaction is weaker, two mutually competing incommensurate supersolids. Finally, we showhow these phases can be stabilized by a quantum fluctuation enhancement of peculiar insulatingparent states.
I. INTRODUCTION
In nowadays condensed matter physics, common fasci-nating collective behaviors and novel quantum phases arereported in various domains of physics, more specificallyin bosonic systems encountered in quantum magnetism,ultracold atoms on optical lattices and strongly corre-lated materials. Thanks to their versatility, an importantamount of exotic phases has been reported in the litera-ture these last years, both experimental and theoretical,such as different types of superfluids , insulators , Bosemetals or supersolid phases . In the latter, the sys-tem enters a phase combining crystalline order and super-fluidity, and typically arising from the quantum meltingof a Mott insulator; this phenomenon is at the origin ofintense scientific activities and debates; Experimentally,indications for supersolidity were supposed to be foundin He . However, necessary conditions for continuoussymmetry breaking questioned this interpretation andrecent experiments have clearly ruled out this scenario .Meanwhile, supersolids are easier achieved on lat-tice systems; whereas on a square geometry withnearest neighbor interactions a soft-core description isrequired , they can also be found in hard-core bosonicmodels when frustration is induced by either furtherneighbor interactions and/or lattice geometry, e.g. triangular . An other interesting direction to stabi-lize supersolidity is to increase the number of degrees offreedom in frustrated systems. In that respect, bosonicmixtures with several species of bosons, usually encoun-tered in optical lattices , either heteronuclear orhomonuclear , as well as bilayer systems with interac-tions but no hopping between layers are very promis-ing. Such systems allow for an even broader variety ofquantum phases than their single-species counterparts,depending on the intra- and inter-species interactions,and on the dimension and lattice connectivity.Theoretically, compared to an already rich literatureon mixtures in 1D only few works have focused on 2Dcases for instance on square and triangular lat-tices. In all those systems, when contact and dipolar interactions are taken independently, interaction-induced insulators, e.g. density-homogeneous or den-sity wave are favored. Hence, their competition can,along with quantum fluctuations, trigger various uncon-ventional phases , especially in systems with ki-netic or interaction frustration. FIG. 1: Triangular lattice hosting bosons of two species la-belled a (blue) and b (brown). the bosons can hope fromtwo neighboring sites with the energy − t , and interact via anintra-species interaction V and a point contact inter-speciesinteraction U . In this paper, we provide a theoretical study of a two-species bosonic mixture on a triangular lattice as de-picted in Fig. 1. We aim to focus on the most sim-ple model with competing interactions in frustrated ge-ometry in order to study possible mechanisms for sta-bilizing exotic phases. The richness of the phase di-agram of a related one-species model with hard coreconstraint allows us to expect even more exotic physicsfor a mixture of two mutually interacting species. Eventhough such interactions can be encountered in cold atomsystems with polarized dipoles , the presentwork describes the maximally frustrating cases for whichall interactions are repulsive. We address the followingissues: (i) how on-site and nearest neighbor interactionscompete and (ii) how lattice frustration impacts on thestabilization of non-conventional phases of such a mix-ture.To achieve these goals, we use mainly a cluster meanfield theory (CMFT) and exact diagonalizations (ED)on periodic clusters; both methods are detailed, alongwith the model we consider, in Section II. Note that pre-liminary Quantum Monte-Carlo (QMC) simulations havebeen performed to support our findings (see text). Wepresent in Section III an overview of the phase diagramobtained this way, before focusing on the most interest-ing phases. First, we describe in Section IV some com-mensurate spin-like phases. These are characterized withmeans of perturbative approaches; they have frozen den-sities and, for one of them, spin-like supersolidity. Next,we analyze in Section V incommensurate phases foundin this study: these include two original two-species su-persolid (SS) phases, which belong to our main findings.Eventually we address in Section VI the nature of phasetransitions involving these peculiar phases, before someconcluding remarks in Section VII. II. MODEL AND METHOD
We study a two-species (spin) extended Bose-Hubbardmodel on a triangular lattice: H α = X h i,j i h − t α ( b † iα b jα + h.c. ) + V α n iα n jα i − µ α X i n iα H ab = X α H α + U X i n ia n ib . (1) H α is the one-species Hamiltonian ( α = a, b ), with t α and V α respectively the nearest neighbor hopping andinteraction amplitudes; µ α is the chemical potential forbosons of species α created by operators b † iα at site i .In this work, we focus on the limit of hard core bosons( U α,α ≫ | t α | , V α , µ α ) with an implicit onsite intra-speciesrepulsion U αα P i n iα ( n iα − /
2. Together with the re-pulsive interspecies coupling U , this allows us to maxi-mize the effects of frustration. We have studied the moregeneral case in function of independent µ a and µ b andfound that the richest physics was found in the symmet-ric case µ a = µ b , including the novel supersolid regimes.Hence, when the system is ( a, b )-symmetric ( t α = t , V α = V and µ α = µ ), as considered in this work since themost original two-spin phases arise there, the particle-hole transformation b ′ iα = b † iα allows for a mapping be-tween µ ∗ > µ ∗ <
0, with µ ∗ = µ − V − U/ µ ∗ = 0amounts to changing the sign of U . This U < .We compute the ground states of H ab on periodic clus-ters with up to N = 12 sites (see Fig.2), using eitherCMFT (if not precised) or ED methods. Note that theformer method has been employed successfully in manyone-species bosonic systems on various lattices such as triangular , pentagonal and hexagonal . It isthus expected to be also very efficient in the presentmodel. N = Λ= (cid:144) = Λ= (cid:144) = Λ= (cid:144) FIG. 2: Clusters considered in our CMFT analysis with N = 3, 6 and 12 sites. Internal and external bonds correspondrespectively to continuous and dashed lines. For each cluster,the cluster scaling parameter λ is given. In the former case, the N i ( N e ) internal (external)bonds are treated exactly (at the mean field level) and correlations are better taken into account as N in-creases. As in Ref. , we thus define the scaling pa-rameter λ = N i / (3 N ) which quantifies finite boundaryeffects; the Thermodynamic Limit (TL) is achieved for λ → n iα = h n iα i and the superfluid fractions (SF) φ iα = h b iα i . In ad-dition, to evidence 3-fold symmetry breaking, we de-fine the order parameter M α = | ¯ n α ( k ) | , the diagonal S dab = |h ( n a − n b )( − k )( n a − n b )( k ) i| and off-diagonal S odab = |h P i,j b ia b † ib b jb b † ja i| /N correlation functions. Weused the Fourier transform n α ( k ) = N P s n sα e ik · r s atpoint k = (4 π/ , III. OVERVIEW OF THE PHASE DIAGRAM
Let us first focus on the ( µ ∗ , U ) phase diagram in whichtwo-species phases - detailed along the paper - emerge, il-lustrated here for t/V = 0 .
15. For these parameters, theone-species Hamiltonian H α is known to present a richphase diagram with either empty/full, homogeneous su-perfluid, √ ×√ / /
3, or supersolidphases . When U is switched on, the correlations be-tween the two species increase, and the resulting GS canbe either a product of two one-species phases or a species-entangled state as depicted in the phase diagram shownin Fig. 3. (i) As µ ∗ < φ a = φ b = 0 ( φ α = N P i φ iα ). When n a and n b arelarge enough, V and U terms drive the system into a 3-fold ordered insulator characterized by φ a = φ b = 0 anda total density n = 2 / n a = n b = 1 / √ × √ (cid:144) (cid:144) (cid:144) (cid:144) - - Μ * (cid:144) V U (cid:144) V FIG. 3: Phase diagrams as obtained from CMFT calculationson a N = 3 cluster at t = 0 .
15 and V = 1. Top: wide range of µ ∗ showing two-spin superfluids, 2/3 and 4/3 plateaus, XXZphysics domain and original collective two-spin physics (whiterectangle). Bottom: zoom of the white rectangle and all newtwo-spin phases (see text for acronyms). Solid (dashed) linesare phase boundaries subsisting (vanishing) in the Thermo-dynamic Limit (TL). density, various supersolids with 3-sublattice structuresare stabilized; they will be discussed in detail in SectionV. Finally, an insulating regime with n = 1 is stabilizedfor U > . V in the vicinity of µ ∗ = 0; it will be theobject of Section IV. For µ ∗ >
0, we obtain an equivalentphase diagram thanks to the particle-hole symmetry.
IV. COMMENSURATE SPIN-LIKE PHASES
For strong inter-species repulsion U ≫ V, t, | µ ∗ | (tri-angular uppermost domain on Fig. 3-up) the system isMott insulating, with φ α = 0 and n = 1. Indeed, U im-poses the local constraint of single occupancy defined as¯ n ia + ¯ n ib = 1. This is reflected by n = 1 plateaus in bothED (Fig. 4-a) and CMFT (Figs. 4-b and 7) results. Inorder to better describe this regime, we define spin 1 / σ zi = ( n ia − n ib ) / σ + i = b † ia b ib . At secondorder of the perturbation theory, we obtain an effectiveXXZ model H XXZ = − J ⊥ X h i,j i ( σ + i σ − j + σ − i σ + j ) + J z X h i,j i σ zi σ zj , (2)where J ⊥ = 4 t /U and J z = 2 V + 4 t /U .Interestingly, this model predicts a (2 m z , − m z , − m z )SS with diagonal h σ z i 6 = 0 and off-diagonal h σ + i 6 = 0order parameters when J z /J ⊥ > . . In the present context,these phases correspond respectively to a spin supersolid (SSS) and a spin superfluid (SSF). While the density isuniform, the SSS has a (2 m z , − m z , − m z ) structure where æ ææææææ ææææ ææ ææ æ ææ ææà àààà àà àààà àà àà à àà àà àààà àà àààà àæ ææææææ ææææ ææ ææ æ ææ ææà àààà àà àààà àà àà à àà àà àààà àà àààà àì ìììììì ìììì ìì ìì ìì ìì ìììì ìììììì ì - - à à à à à à àààà à à à àæ æ æ æ æ æææ æç ç ç ç ç ççç çá á á á á á á á á á á á á ááááá - - - - - E D n , Φ SSS Μ * (cid:144) V Μ * (cid:144) U H a LH b L abod n S abd t (cid:144) U = V (cid:144) U = à n, á Φ V (cid:144) U = æ n, ç Φ FIG. 4: Strong U regime characterized using either N = 12(ED: a) or N = 6 (CMFT: b) clusters. (a) n , S dab and S odab asa function of µ ∗ /V for U = 2 V and t = 0 . V . The n = 1plateau and both finite S dab and S odab evidence the SSS; (b) n and φ = φ a + φ b as a function of µ ∗ /U distinguishing the SSF( V = 0) and the SSS ( V = 0 . U ) regimes. m z quantifies the spin disproportion on each sublattice.As depicted in Fig. 4 and Fig. 5(a), for | µ ∗ | ≪ V, U ,both ED and CMFT approaches confirm the existenceof the SSS; indeed the finite structure factors S dab , S odab and M = ( M a + M b ) / t/U < .
17 signal long-rangecorrelations and the corresponding XXZ couplings verify J z /J ⊥ = 1 + U V / t ≥ .
5. In contrast, for V = 0 and t/U ≤ ( t/U ) c ≃ . µ ∗ = 0 thespatially uniform SSF predicted in the XXZ model, asillustrated in Fig. 4(b). Finally, Fig. 5(b) shows that thekinetic energy gain ǫ ∗ ≃ − kt /U w.r.t. the electrostaticcontribution N ( V − µ ) is well reproduced by the XXZmodel in both regimes; this validates our approach. V. INCOMMENSURATE SUPERSOLIDS
In contrast to the above mentioned SSS in which su-persolidity comes from the species (spin-like) degrees offreedom, in SS phases discussed here n α may be incom-mensurate and have finite φ α . From the species-resolvedobservables shown in Fig. 6 and 7, we identify three suchSS phases. (i) For large U/V = 2 (right column) and µ ∗ /V about ± . M α indicates a 3-fold or-der (3FO for both species, while only one species has anon-zero φ α . This phase, dubbed 1SF/2CO in Fig. 3,is a rearrangement of two one-species phases of H α , the √ × √ . Two sublattices are (al-most) filled by a and b bosons respectively, while on theremaining, an incommensurate density for bosons of onespecies ( e.g. a ) accounts for superfluidity ( φ a >
0) and apopulation imbalance ( n a > n b ). Within this structure, àààààààààààààààààà ææææææææææææææææææ àààààààààààààààààà ææææææææææææææææææ àààààà à àááááááá æææææææ ççççççç òòòòòòò óóóóóóó - - - M , Φ Ε * (cid:144) t SSS BPB t (cid:144) Ut (cid:144) U H a LH b L à M æ Φ V =
0, t = à H ab (cid:144) á H XXZ V =
1, t = æ H ab (cid:144) ç H XXZ V =
1, t = ò H ab (cid:144) ó H XXZ
FIG. 5: Strong U regime characterized using either N = 6(CMFT: a) or N = 12 (ED: b) clusters. (a) Species-averagedorder parameter M and superfluid fraction Φ, evidencing thequantum fluctuation enhancement from the SSS to the BPBphase at V = 1 and t = 0 .
15; (b) comparison of the kineticenergy gain ǫ ∗ ≃ − kt /U for H ab (filled) and H XXZ (empty)for three ( t, V ) sets. - - - - - - n Α U = V (cid:144) M Α Φ Α Μ * (cid:144) V BPB3SS2 (cid:144) FIG. 6: Species-resolved observables ( n α , M α , φ α ). Eachspecies corresponds to either continuous or dashed line. Alldata come from a N = 6 CMFT with fixed t/V = 0 .
15 and U = 0 . V . The arrow points out a tiny region which disap-pears in the TL (see text). The shaded regions correspond tothe different phases of Fig. 3. the repulsion energy ∝ U is minimized thanks to the lo-calization of bosons of a single species. (ii) A distinctphase is found for small U/V = 0 . µ ∗ /V about ± . φ α for both species aswell as a 3-fold symmetry breaking.As shown in the typical snapshot obtained by CMFTin Fig. 8(c), it can be seen as a superposition of two one-species SS (these would be obtained at U = 0); both - - - - - - n Α U = M Α Φ Α Μ * (cid:144) V SSS1SF (cid:144) (cid:144)
FIG. 7: Same as Fig. 6, but for U = 2 V . species contribute symmetrically to superfluidity. Thisphase, called 3-fold SS (3SS) is the first example of a col-lective two-species supersolid and is obtained from theparent n = 2 / µ ∗ increases. As in this parent state, the weak a L b L c L d L e L FIG. 8: Examples of two-spin supersolids with 3-sublatticestructure obtained by CMFT density maps on the 12-site clus-ter (a,b,c) and by QMC correlations for 36 ×
36 sites cluster(d,e). (a) the n = 1 spin supersolid (SSS) at µ ∗ = 0 and U = 2 V ; (b) and (c) respectively the bosonic pinball (BPB)at µ ∗ = − . V and the 3-supersolid (3SS) at µ ∗ = − . V ,for t = 0 . V and U = 0 . V . n a − n b is 0 for the 3SS andfinite for the BPB for all CMFT clusters (Fig. 9 for the TLstudy). (d) and (e) are respectively QMC results for the BPBat µ ∗ = − . V and the 3SS at − . V . Note that the 3SSand BPB have distinct symmetries (under π/ inter-species coupling merely forces the localized a and b bosons to occupy distinct sublattices. (iii) The mostremarkable incommensurate SS phase is achieved, uponincreasing t/U , when quantum fluctuations become toostrong for the density-uniform n = 1 SSS phase. Thisoriginal two-species SS is called the bosonic pinball (BPB)due to a structure very similar to its fermionic coun-terpart with similar interactions, the pinball liquid , andis depicted in Fig. 8(b,d). The latter has almost onelocalized electron per site on one sublattice (pins) anda metallic behavior on the remaining hexagonal lattice(balls) . Here, the particles are bosonic and thespecies play the role of the spins. The structure is de-picted on Fig.1(b). One sublattice, forming a triangularsuper-lattice with ¯ n ia + ¯ n ib close to 1, is filled in majorityby one type of bosons. The two-spin superfluid characteris carried by the remaining bosons on the complementaryhexagonal lattice. This is shown in the BPB region onFig. 6 and Fig. 9, by a coexistence of solid ( M α = 0)and superfluid ( φ α = 0) orders in both species. This lat-tice symmetry breaking is reminiscent of the parent SSSphase, the partial quantum melting of which involves acondensation of two types of defects coming from doubly-occupied and empty sites. Finally, unlike the 3SS and1SF/2CO, the BPB can be stabilized at the n = 1 com-mensurability when µ ∗ = 0. VI. THERMODYNAMIC LIMITS AND PHASETRANSITIONS
All the phases described above exist in the TL. In-deed, in Fig. 9 are evidenced the cluster dependencies of M and φ , shown as functions of µ ∗ and the examples ofthe BPB and the 3SS (c) prove their finiteness as λ → æ æ æà à à b L n a - n b Λ à à àæ æ æá á áç ç ç c L M, Φ (cid:144) Λ ææææææææææææææææææææææææææ ææææææææææææææææææææææææææ M Φ (cid:144) L N - - - Μ * (cid:144) V FIG. 9: M and φ as a function of µ ∗ for t = 0 . V and U =0 . V obtained by CMFT on clusters of N = 3 (continuous),6 (dashed) and 12 (symbols) site. See Fig. 2 for the clustershapes. Cluster scalings as function of λ (see text) are shownfor b) n a − n b with different possible fits giving either zero(continuous) or finite (dashed) TL values (see text) and c) M (filled symbols) and φ/ | µ ∗ | = t (squares) and the 3SS at | µ ∗ | = 3 t (circles). affected by size effects thus Fig. 3 is representative of theTL. In the strong correlation limit U, V ≫ t [39], pertur-bation theory allows to locate phase transitions where the defect energy vanishes (condensation), e.g. for the2 / → a ) inserted in the empty sub-lattice of the 2 / V . Viasecond order processes, such a defect can hop to secondneighbors with an amplitude − t eff = − t /U − t /V . Theterms − t /U and − t /V account respectively for pro-cesses where (i) the extra a boson hops via the b -filledsublattice, or (ii) a vacancy on the a -filled sublattice iscreated and then deleted. In this limit, the defects con-dense and lead to supersolidity for µ c = 3 V − t eff , e.g. µ c /V ≃ − .
65 for U = 0 . V in good agreement with theresults of Fig. 6 and 7. This defect condensation mecha-nism implies a gauge symmetry breaking, additionally tothe lattice symmetries already broken in the 2 / φ and M as function of µ (Fig. 9). This corresponds to the stan-dard picture of defect condensation accounting for an in-commensurate SS; the SF density (here inhomogeneous)is carried by defects on top of density modulations .In contrast, the transitions between (i) homogeneous SFand 2 / µ is, from one CMF cal-culation to another, either stepwise increased or stepwisedecreased (see [40]). In case (ii), n a − n b (Fig. 9(b)) hasdistinct size-dependence, but a vanishing value in the TLcannot be ruled out for the BPB, despite a non-linear be-havior. n a − n b is strictly 0 for the 3SS. For the BPB,the non-linear behavior makes difficult to extract n a − n b as λ →
1, and various fits can give either zero (linear)or finite ( e.g. weighted a + λ b ) value. However, BPBand 3SS have different symmetries as obtained by CMFT(Fig. 8(b,c)) and confirmed by preliminary QMC calcu-lations. Indeed, to check this point, we have performedStochastic Series Exchange (SSE) QMC simulations onclusters up to 36 ×
36 sites for various U in function of µ ∗ , for which real-space correlations h n sa n iα i (for site i far away from the reference site s to get rid of short-distance effects) confirm the existence and symmetries ofboth the 3SS and the BPB (Fig. 8(d,e)). The completeanalysis of this model by SSE-QMC, being beyond thescope of this paper, will constitute a separate work. VII. CONCLUSION
We study an interacting two-species bosonic mixtureon a triangular lattice by combining CMFT, ED and per-turbative methods, supported by SSE-QMC results. Wefocus on a region of parameter space in which peculiarphases arise due to the competition between frustrationand quantum fluctuations. Within a very rich and com-plex phase diagram, we evidence three original mixturesupersolids, a commensurate spin supersolid (SSS) andtwo mutually competing incommensurate phases arisingfrom the partial quantum melting of parent states. Themost interesting phase, dubbed bosonic pinball (BPB)due to an inner structure very reminiscent of its fermioniccounterpart , results from strong inter-species effects. Itis worth mentioning that this rich physics is found forattractive U (for specific parameters), a situation moredirectly connected to dipolar cold atom bilayer experi-ments. We hope this work will stimulate further investi-gations in this direction. Acknowledgments
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