Density dependent tunneling in the extended Bose-Hubbard model
Michal Maik, Philipp Hauke, Omjyoti Dutta, Jakub Zakrzewski, Maciej Lewenstein
DDensity dependent tunneling in the extendedBose-Hubbard model
Michał Maik , , Philipp Hauke , , Omjyoti Dutta , , MaciejLewenstein , and Jakub Zakrzewski , Instytut Fizyki imienia Mariana Smoluchowskiego, Uniwersytet Jagielloński, ulicaReymonta 4, PL-30-059 Kraków, Poland ICFO – Institut de Ciències Fotòniques, Mediterranean Technology Park, E-08860Castelldefels (Barcelona), Spain Institute for Quantum Optics and Quantum Information of the Austrian Academyof Sciences, A-6020 Innsbruck, Austria ICREA – Institució Catalana de Recerca i Estudis Avançats, E-08010 Barcelona,Spain Mark Kac Complex Systems Research Center, Uniwersytet Jagielloński, Kraków,PolandE-mail: [email protected]
Abstract.
Recently, it has become apparent that, when the interactions between polarmolecules in optical lattices becomes strong, the conventional description using theextended Hubbard model has to be modified by additional terms, in particular adensity-dependent tunneling term. We investigate here the influence of this termon the ground-state phase diagrams of the two dimensional extended Bose–Hubbardmodel. Using Quantum Monte Carlo simulations, we investigate the changes of thesuperfluid, supersolid, and phase-separated parameter regions in the phase diagramof the system. By studying the interplay of the density-dependent hopping withthe usual on-site interaction U and nearest-neighbor repulsion V , we show that theground-state phase diagrams differ significantly from the ones that are expected fromthe standard extended Bose–Hubbard model. However we find no indication of pair-superfluid behaviour in this two dimensional square lattice study in contrast to theone-dimensional case.PACS numbers: 67.85.-d, 37.10.Jk, 67.80.kb, 05.30.Jp a r X i v : . [ c ond - m a t . qu a n t - g a s ] J un ensity dependent tunneling in the extended Bose-Hubbard model
1. Introduction
In the last decade, the physics of ultra-cold atoms in optical-lattice potentials hasundergone extensive developments due to the extreme controllability and versality ofthe realizable many-body systems (for recent reviews see [1, 2]). The tight-bindingdescription predicted in 1998 [3], termed Bose–Hubbard model (BHM) for bosonicatoms with contact s -wave interactions, was soon after verified via the experimentalobservation of the superfluid (SF) – Mott insulator (MI) transition [4]. For particlesinteracting via a long-range (e.g., dipole-dipole) potential, the original model has tobe modified, typically including a density–density interaction between different sites.The simplest approximation, taking into account only the interaction between nearestneighbours, is termed the extended Bose-Hubbard model (EBHM). As compared to theBHM, the extended model allows for the existence of novel quantum phases such ascheckerboard solids, supersolid phases [5, 6, 7, 8, 9, 10], exotic Haldane insulators [11],and more.Recently, however, it has been realized that even in the simpler case of contact s -wave interactions, in certain parameter regimes, carefully performed tight-bindingapproximations lead to an additional correlated tunneling term in the resultingmicroscopic description. This term, known in the case of fermions as bond-chargecontribution [12], is even more important for bosons [13, 14, 15]. It is found thatsuch tunneling terms along with the effect of higher bands can provide an explanation[14, 16] of the unexpected shift in the MI–SF transition point for Bose–Fermi [17, 18]and Bose-Bose mixtures [19] as well as shifts in absorption spectra for bosons in opticallattices [15].One may expect that similar bond-charge (or density-dependent tunneling) effectsmay play a similarly important role in the presence of dipolar interactions. Thisassumption has been verified by some of us [20] in a recent study, where it has beenshown that the additional terms in the Hamiltonian may destroy some insulating phasesand can create novel pair-superfluid states. That study [20] has been restricted toa one-dimensional (1D) model due to the numerical methods used. Here, we useQuantum Monte Carlo (QMC) methods to study soft-core dipolar gases trapped intwo-dimensional square optical lattices, where we assume a tight confinement in theremaining z direction (which is also the polarization direction of the dipoles). A similartwo-dimensional model without density-dependent tunneling terms was analyzed before[8], providing us with a benchmark against which we may test the importance of density-dependent tunneling. In Ref. [8], a supersolid phase was observed in the EBHM at halffilling. Such a supersolid is characterized by the coexistence of superfluid and crystal-likedensity–density diagonal long-range order [5, 6, 7, 8, 9, 10]. Experimental evidence ofthis counter-intuitive quantum phase is still missing, since the claim of an experimentalrealization of supersolidity in He [21, 22] could not be reproduced in later experiments[23, 24]. As we shall see, in the present model, the sign of the additional tunneling(or, more precisely, the relative sign between the standard tunneling and the density- ensity dependent tunneling in the extended Bose-Hubbard model
2. The model
The appropriate tight-binding model to study interacting dipolar bosons occupying thelowest band in a lattice reads [20] H = − t (cid:88) (cid:104) i,j (cid:105) ( a † i a j + h . c . ) + U (cid:88) i n i ( n i −
1) + V (cid:88) (cid:104) i,j (cid:105) n i n j − T (cid:88) (cid:104) i,j (cid:105) ( a † i ( n i + n j ) a j + h . c . ) + P (cid:88) (cid:104) i,j (cid:105) ( a † i a † i a j a j + h . c . ) − µ (cid:88) i n i , (1)where a † i ( a i ) is the creation (annihilation) operator of a boson at site i and n i is thenumber operator; t is the regular hopping term, U the onsite repulsion, and µ thechemical potential. We assume a system of dipolar bosons in a 2D square lattice withdipolar moments polarized perpendicularly to the lattice, thus leading to dipole-dipolerepulsion. Then, the present model contains three terms that come from the dipolarinteractions, the nearest-neighbor repulsion V , the density-dependent hopping T , andthe correlated pair tunneling P . We restrict here the range of V to the nearest neighborsto allow for a direct comparison with the results of Ref. [8] and [20] Within the standardEBHM, both the T and P terms are neglected. However, the analysis presented inRef. [20] has shown that, although V is typically an order of magnitude larger thanboth T and P , the latter terms cannot be neglected in the presence of strong dipolarinteractions.The four parameters U , V , T , and P have the same physical origin, namelyinteractions, and are therefore correlated. However, in the two-dimensional model,changing the trapping frequency in the direction perpendicular to the plane affectsquite strongly only the on-site U term (for dipolar as well as for the contact part of theinteractions), while leaving the other three parameters practically unaffected [20]. Thus,we shall consider U as an independent parameter. To facilitate a comparison with earlierworks (e.g., [8]) that did not take T tunneling into account we span a similar parameterrange for U , V , and filling fractions. The values of T , V , and P are strongly correlatedas they originate from nearest-neighbour scattering due to long-range interactions. Fora broad range of optical-lattice depths, the parameters T and V are typically relatedas V ≈ | T | . The absolute value of P is almost another magnitude smaller than T (compare Fig. 1 of [20]). Thus, for simplicity, we will set V = | T | in the following andneglect the P term altogether. This will allow us to study in depth the effects due todensity-dependent tunnelings. The previous study [20] has shown that there is a broadtunability regarding the relationship of the two tunneling parameters T and t , allowinga regime where the two hopping terms have opposite signs, and even the exotic situationthat T dominates over t .Whether both hopping terms are of the same or of opposite sign has a majorinfluence on the phases that will appear in the system. Generally speaking, when both ensity dependent tunneling in the extended Bose-Hubbard model L = 8 and L = 16 ) and the Multiscale Entanglement Renormalization Ansatz (MERA)(system sizes up to L = 128 ) were used to study the phase diagram at zero temperature.The results, when both T and P are set to zero, show the existence of three phases. Atweak interaction, there is a superfluid phase (SF), while at stronger dipolar strength,two charge density wave (CDW) phases appear. The CDW phases are characterizedby a periodic, crystal-like structure where occupied and empty sites alternate in acheckerboard pattern. In the following, we denote cases where the populated sitesare occupied by a single atom (two atoms) as CDW I (II). In the one-dimensional caseof Ref. [20], the two observed CDW phases are a CDW I phase at half filling witha modulation of | ... ... (cid:105) and a CDW II phase at unit filling with a modulationof | ... ... (cid:105) . Now, when the extra terms T and P are incorporated into theHamiltonian, besides an overall deformation of the phase diagram, there appears also anovel pair-superfluid phase (PSF). This more exotic phase is characterized by a finitetwo-particle NN correlation function Φ i = (cid:80) { j } (cid:104) a † j a † j a i a i (cid:105) and a smaller non-vanishingone-particle correlation function φ i = (cid:80) { j } (cid:104) a † j a i (cid:105) . On the other hand, no supersolidphase has been observed in [20]. In the present study of a 2D lattice, on the contrary,we do observe a supersolid behavior, but we do not find any indications for the existenceof a PSF phase. In the analysis of Hamiltonian (1), we employ the Stochastic Series Expansion(SSE) code, a QMC algorithm from the ALPS (Algorithms and Libraries for PhysicsSimulations) project [25]. We mainly rely on three observables. First, we study thedensity, ρ = (cid:104) n i (cid:105) , as a function of the chemical potential. Plateaus in the correspondinggraph as a function of chemical potential indicate insulating phases, such as MI or CDWphases. The employed variant of QMC works in the grand-canonical ensemble, i.e., atfixed chemical potential. Discontinuous jumps in the density as a function of chemicalpotential signify regions of phase separation (PS) in the canonical phase diagrams.Namely, when the filling is fixed to a value which is not stable at any chemical potential,the system acquires the required filling only in the mean, by forming domain wallsbetween two phases that are thermodynamically stable.To distinguish not only different insulating phases (MI, CDW I, and CDW II), butalso the superfluid (SF) and the supersolid phase (SS), we consider two other observables.These are the structure factor and the superfluid stiffness, which we analyze both as a ensity dependent tunneling in the extended Bose-Hubbard model S ( Q ) = (cid:42)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) N (cid:88) i =1 n i e ı Qr i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:43) /N . (2)Here, N denotes the number of lattice sites, and we focus on the wave vector Q = ( π, π ) ,which corresponds to a checkerboard modulation pattern. This observable has a peakwhen the particles are arranged in either of the CDW phases. This will help todistinguish the MI phase from the CDW phase, which cannot be done from the densitygraphs alone. For example, when a system is at unit filling, the structure factor is finitein the CDW II state, whereas it vanishes in a usual MI state.The other observable is the superfluid stiffness, which can be calculated from thewinding numbers of the QMC code. It is defined as ρ s = (cid:104) W (cid:105) β , (3)where W is the winding-number fluctuation of the world lines and β is the inversetemperature (in this study β = 20 ). This value shows what percent of the system isin a superfluid state. Taking superfluid stiffness and structure factor together, we canalso identify the SS phase. The SS phase occurs when both superfluid stiffness andstructure factor are non-zero. Note that, since PS regions do not correspond to stablegrand-canonical phases as computed in the SSE QMC code, we cannot assign any valuesof observables for them. This is not necessary, however, since PS regions are alreadyunambiguously identified by jumps in plots of density against chemical potential.From these three observables (density, structure factor, and superfluid stiffness)we are now able to distinguish the most prominent phases that we are looking for.These observables cannot, however, identify PSF phases, the signature of which is, asmentioned previously, a non-vanishing two-particle NN correlation function Φ i . In itscurrent version, the QMC code provided in the ALPS library is not able to calculatethese correlation functions. In order to extract this observable, the code would have tobe written with a two-headed worm, which could then be analyzed in a similar way as thesuperfluid stiffness, but with the difference that the winding numbers would representthe flowing of pairs instead of single particles [26]. Fortunately, one can identify adominant PSF order parameter using a different technique, namely by studying thedensity histograms of the QMC code. If these histograms show only even values ofparticles instead of a uniform distribution, this means that the bosons always pair up,indicating PSF behavior [26].
3. Ground-state phase diagrams
In this section, we present our QMC results for the ground-state phase diagram ofHamiltonian (1). We focus on a two-dimensional square lattice with linear system sizesranging from L = 8 to L = 16 (where N = L × L ). We present phase diagrams at twodifferent values of the on-site repulsion ( U = 20 and U = 5 ) for varying density and T ensity dependent tunneling in the extended Bose-Hubbard model V , since V = 10 | T | ). The two U values are chosen in sucha way that we can compare nearly hard-core like behavior, achieved at U = 20 , withsoft-core behavior, for U = 5 . Further, at U = 20 we can compare our data to knownresults of the usual EBHM, which was studied thoroughly in [8]. We compare phasediagrams obtained with and without density-dependent tunnelings. For simplicity andease of comparison to [8], we restrict our study to unit filling or less. Furthermore, for amore detailed evaluation of these phase diagrams, we study a few cuts at representativeparameter values. PSSSPSSF PS M I CD W I CD W II Ρ V (cid:144) a (cid:76) SSPSSF M I CD W II CD W I Ρ V (cid:144) b (cid:76) Figure 1.
The phase diagram in the ρ − V parameter space without density-dependenttunneling term, T = 0 , for (a) U = 20 and (b) U = 5 . The energy unit is t = 1 . Panela) reporoduces the results of [8]. The model contains various phases. The red solidline indicates the charge density wave (CDW I) at half filling; other phases present arethe superfluid (SF), supersolid (SS), and at unit filling either Mott insulator (MI) oranother charge density wave (CDW II); PS denotes phase-separated regions. Whenthe on-site interaction becomes weaker, as shown in panel b), the SS phase becomeslarger and PS regions disappear at filling larger than / . We begin our analysis with phase diagrams of the regular EBHM, illustrated in Fig. 1.This provides an overview of the behavior of the considered systems under a morecommon Hamiltonian, which does not have a density-dependent term T . We consider thecase of strong repulsion U = 20 , discussed previously in [8], as well as softer interactingbosons with U = 5 , where up to 4 bosons are allowed per site. U = 20 ) For ease of comparison,and for later reference, Fig. 1a reproduces the phase diagram of U = 20 that has beenthoroughly investigated in [8]. It is well known that for ρ < there exist only twodistinct regions, the SF phase and a PS region. For sufficiently low values of V , the ensity dependent tunneling in the extended Bose-Hubbard model V , which in thepresent case lies around V = 2 . . A system in a checkerboard phase (CDW I) can bedoped by holes or particles. When it is doped with holes, these create domain walls andcause the system to phase separate, preventing the appearance of a SS phase. In thecase of hardcore bosons, this behavior would be mirrored for ρ > , due to particle–holesymmetry. In the case of soft-core bosons, such particle–hole symmetry can break down.At sufficiently low V , a region of PS appears, and the system does present a hardcore-like behavior, but as the NN repulsion is increased this PS region disappears. Sincenow the particles can occupy either an empty or occupied site, it is no longer necessaryfor the domain walls to form and the system can move into a SS state. Moreover, ata certain value of V , upon increasing ρ the SS phase is followed by a region of PS,instead of going into a SF phase and then becoming a MI. At unit filling, this PS regionthen changes to the CDW II phase, which is characterized by a checkerboard patternconsisting of an alternation of doubly-filled sites and empty ones. (cid:230) (cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230) (cid:230)(cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) Μ (cid:144) U Ρ a (cid:76) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224)(cid:224)(cid:224)(cid:224)(cid:224) (cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224) (cid:224)(cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224)(cid:224)(cid:224)(cid:230) (cid:230) (cid:230) (cid:230) (cid:230)(cid:230)(cid:230)(cid:230)(cid:230) (cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230) (cid:230)(cid:230)(cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230)(cid:230)(cid:230) Ρ S (cid:72) Π , Π (cid:76) Ρ s b (cid:76) Figure 2.
Identification of different phases as exemplified for U = 20 and V = 3 . based on the density in (a) as well as the structure factor (blue circles) and thesuperfluid stiffness (red squares) shown versus density in (b). Plateaus in ρ as afunction of µ indicate incompressible crystal phases. Jumps denote phase separation(the densities that are jumped over do not correspond to thermodynamically stablephases). Moreover, a finite superfluid stiffness characterizes SF phases and a finitestructure factor CDW order. When both are finite, the system is SS. Figure 2 shows, for a fixed V = 3 , the observables described in Section 2.1 that weused to determine the various phases. The boson density as a function of the chemicalpotential displays clear plateaus, corresponding to gapped insulating phases (Fig. 2a).As mentioned above, jumps in Fig. 2a correspond to PS regions in Fig. 1. The structurefactor and the superfluid stiffness are shown in Fig. 2b. For low chemical potential(density) the system is in a SF state with non-zero superfluid stiffness and vanishingstructure factor. At ρ ≈ . , the system phase separates, and there are no values forthese observables. At half filling, when the system moves to CDW I phase, the structurefactor becomes finite. There is a small region, roughly around . < ρ < . , where thesystem is in a SS phase – here both the superfluid stiffness and the structure factor arenon-zero. This phase is followed by a second region of PS that extends up to ρ ≈ . . ensity dependent tunneling in the extended Bose-Hubbard model U = 5 ) We now consider U = 5 ,a case of weaker repulsion that has not been studied earlier. For the moment, westill retain T = 0 . The phase diagram Fig. 1b seems a bit simpler than for U = 20 .Importantly, the particle-doped side now has to deal with much "softer" bosons allowingfor multiple occupancy on any given site. The hole-doped side is much less affected sincethe on-site repulsion has a lesser influence on lower densities. For weak NN repulsion V ,the system stays SF across the entire range of densities from empty to unit filling andthen goes into the MI state. At V ≈ . up to V ≈ . , the system goes directly from aSF phase into a SS phase, which ends at a CDW II phase at unit filling. At larger V , aPS region appears. The biggest difference between U = 20 and U = 5 cases appears forhigher values of V , where the PS region at the particle-doped side disappears and theSS phase occupies the entire region between the CDW I at half filling and the CWD IIat unit filling. (cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230) Μ (cid:144) U Ρ a (cid:76) (cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230) Μ (cid:144) U N N b (cid:76) (cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230) Μ (cid:144) U Ρ c (cid:76) (cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224)(cid:224)(cid:224)(cid:224)(cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) Ρ S (cid:72) Π , Π (cid:76) d (cid:76) (cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224) (cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230) (cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) Ρ N N e (cid:76) (cid:224)(cid:224)(cid:224)(cid:224)(cid:224) (cid:224) (cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224) (cid:224)(cid:224)(cid:224)(cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224)(cid:224)(cid:224) (cid:224) (cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224) (cid:224) (cid:224) (cid:224) (cid:224)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230) (cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230) (cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230)(cid:230)(cid:230) (cid:230) (cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230) (cid:230) (cid:230) (cid:230) (cid:230) Ρ Ρ s f (cid:76) Figure 3.
Top row: density versus chemical potential for U = 5 and different valuesof V : V = 3 . , V = 4 . and V = 6 . (left to right). The bottom row shows thecorresponding structure factor (blue circles) and the superfluid stiffness (red squares). The different transitions are revealed by a slices through the phase diagram at fixed V (exemplified for a few values in Fig. 3). At V = 3 . , the density is strictly increasingacross the entire range of µ (Fig. 3a). Notice, however, a change of the slope around µ/U = 4 , corresponding to ρ = 0 . . As seen in Fig. 3d, the structure factor starts to risein a similar parameter range, namely around ρ = 0 . . At the same time, the superfluidstiffness only has a peak at ρ = 0 . , but remains finite for all values of µ considered.Therefore, the increase of the structure factor is a clear sign of a second-order transition ensity dependent tunneling in the extended Bose-Hubbard model ρ = 1 ,the unit-filling phase will be a CDW II and not a MI.The next slice is taken at V = 4 . , where the state changes from SF to PS to SSwithout ever settling into the CDW I phase at half filling. In the density graph, Fig. 3b,we can see a small jump that bypasses ρ = . This explains why the CDW I phase doesnot appear at this value of V . The SF at small ρ is identified by a non-zero superfluidfraction and vanishing structure factor (Fig. 3e). This phase is followed by the PS regionfrom ρ ≈ . to ρ ≈ . . At higher densities, a SS state appears as characterized bynon-zero structure factor and superfluid stiffness. Finally, the system settles into theCDW II state at unit filling.The last slice at V = 6 . is similar to the previous one at V = 4 . with one majordifference, the appearance of the CDW I phase at half filling. As before, we can see ajump (this time slightly larger) in the density, Fig. 3c, but now it is followed by a plateauthat signifies the CDW I phase. In Fig. 3f, we see again the three distinct phases, SFup to ρ ≈ . , then a region of PS up to ρ = 0 . , and from half filling to unit fillingthere is the SS phase, once again ending in the CDW II state. As we have seen in the previous section, the phase diagram of the EBHM at vanishing T displays a large variety of phases: MI, CDW, SF, and SS. Additionally, there arevarious regions of phase separation, some of which (the ones at filling larger than 1/2)disappear with decreasing on-site repulsion of the bosons. In this section, we study howthis phase diagram of the usual EBHM is changed by the density-dependent hopping. The first case we study is U = 20 when the two tunneling amplitudes t and T have the same sign. Comparison of Fig. 4with Fig. 2 shows that in the presence of density-dependent tunneling the PS regionat low V values has disappeared and there is no PS region between the SS and CDWII phases. One can explain this behavior by the increase in the total hopping due tothe additional tunneling term T . Thus, the on-site repulsion U behaves as if effectivelyrescaled to smaller values. Similar arguments explain the shift of the point where the ρ = plateau first appears and, therefore, the CDW I phase moves from V ≈ . (with T = 0 , Fig. 2) to V ≈ . (Fig. 4). As a consequence, the phase diagram at U = 20 with t and T of the same sign looks very similar to the one at U = 5 with vanishing T .The behavior in the U = 20 phase diagram becomes more interesting when the twotunneling terms compete due to opposite signs, T < . The phase diagram is presentedin Fig. 4b. The CDW I phase now starts at a lower value of | T | than in the previouslydiscussed case. Similarly, the region of PS at the lower values of | T | (and thus V ) nowbecomes much larger. This shows that the system has a hardcore behavior for a largerrange of parameters. Additionally, the SS region diminishes and finally disappears as V gets larger. These findings can be explained through the competition between t and T , ensity dependent tunneling in the extended Bose-Hubbard model M I CD W II CD W I SF PS SS Ρ T a (cid:76) SF PS SS PS CD W I CD W II M I Ρ T b (cid:76) Figure 4.
The phase diagrams for U = 20 at finite T (with V = 10 | T | , and t = 1 the unit of energy). (a) If t and T are of the same sign, the relative importance ofinteractions decreases, leading to the disappearance of PS phases at greater than halffilling. Compared to the T = 0 cases presented in Fig. 1, this phase diagram resemblesmore the case U = 5 than U = 20 . (b) If T and t compete due to opposite sign, therelative importance of interactions is enhanced, increasing the PS regions. In fact thetwo separate regions of PS in Fig. 1b increase to the point of overlapping. which decreases the effective, overall tunneling strength. This decrease can alternativelybe seen as an effective relative increase of the interaction parameters U and V . As aresult, the hard-core behavior of the system becomes more pronounced, and the PSregions become more important. (cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230) (cid:45) Μ (cid:144) U Ρ a (cid:76) 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(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230) (cid:230)(cid:230) (cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230) Ρ Ρ s f (cid:76) Figure 5.
Density graphs of U = 20 for T = − . , T = − . and T = − . (left toright). Structure factor (blue circles) and superfluid stiffness (red squares) graphs for U = 20 at T = 0 . , T = 0 . and T = 0 . (left to right) ensity dependent tunneling in the extended Bose-Hubbard model V ( T ),presented in Fig. 5. The first slice we present is for T = − . ( V = 3 . ). As seenin Fig. 5a, the plateau at half filling — a CDW I, as indicated by the finite structurefactor, Fig. 5d — is surrounded by discontinuities in the density, thus implying regionsof PS. These are surrounded by SF phases, with a MI appearing at unit filling.The next slice cuts through the phase diagram at T = − . ( V = 5 . ) and thistime shows also a region of the SS phase for densities just above half filling (Fig. 5e).This SS may also be observed in the density plot, Fig. 5b: Above half filling, there isa small interval of steady increase before a discontinuity occurs around ρ = 0 . . Afterthis PS region, there is a small region where the system becomes superfluid before onceagain phase separating. At unit filling, the system finally transitions into a CDW IIphase. Below half filling, another jump in the density indicates yet another PS.The final cut is taken at T = − . ( V = 6 . ). Again, at low densities the systemstarts in a SF phase and then jumps through a region of PS to reach the CDW I phase athalf filling. For higher densities, the system first enters a SS phase, and around ρ = 0 . a transition to PS occurs. This time, the system ends in the CDW II phase when unitfilling is reached. M I CD W II PSSF SS Ρ T a (cid:76) SF PS PSSS M I CD W II CD W I Ρ T b (cid:76) Figure 6.
Phase diagrams for U = 5 and finite T . (a) If T and t have the same sign, therelative strength of tunneling is strongly increased with respect to the interactions. Asa consequence, the CDW I phase has disappeared completely from this phase diagram.(b) When T and t are of opposite sign, the role of interactions is enhanced, leading toincreased PS regions and again the CDW I phase is present. U = 5 ) In the previous section, we saw that theadditional density-dependent tunneling term T can increase or decrease the effectiveimportance of the interactions U and V , depending whether it competes with or supportsthe single-particle tunneling t . In this section, we study this effect for weaker on-siteinteraction U = 5 . The corresponding phase diagrams are presented in Fig. 6. ensity dependent tunneling in the extended Bose-Hubbard model T diagram reveals that the CDW I phase, present for T = 0 , disappearscompletely (Fig. 6a). This means that at no point does there exist a plateau in thedensity graphs at ρ = . Instead, a discontinuity bypasses half filling altogether. Therest of the behavior is rather similar to the system without the density-dependent term.There are still only three phases below unit filling, i.e., the SF phase at low densitiesand T (and therefore at low V ), the PS region near half filling for larger T , and finallythe SS phase for still higher T and larger densities. As can be expected, when the SFphase persists through the entire range of densities, the system ends in a MI state atunit filling. Instead, when the system at fixed V passes through the SS state, the finalphase at unit filling is, as before, the CDW II phase.Consider now the phase diagram of a system with U = 5 when the tunneling termshave opposite signs (Fig. 6b). Here, contrary to the case of positive T , the CDW I existsat half filling. This indicates that the relative importance of the effective total tunnelingis suppressed for T < . Moreover, now a second region of PS appears above half filling.As a result, for T (cid:46) − . there is no stable phase with a density between the CDW Iand the CDW II. (cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230) (cid:45) (cid:45) Μ (cid:144) U Ρ a (cid:76) (cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230) (cid:45) (cid:45) Μ (cid:144) U N N b (cid:76) (cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230) (cid:45) (cid:45) Μ (cid:144) U Ρ c (cid:76) (cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224) (cid:224) (cid:224) (cid:224)(cid:224)(cid:224)(cid:224)(cid:224) (cid:224) (cid:224) (cid:224) (cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224) (cid:224)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230) (cid:230) (cid:230)(cid:230)(cid:230)(cid:230)(cid:230) (cid:230) (cid:230) (cid:230) (cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230) Ρ S (cid:72) Π , Π (cid:76) d (cid:76) (cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224) (cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224) (cid:224) (cid:224)(cid:224) (cid:224)(cid:224)(cid:224) (cid:224) (cid:224) (cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230) (cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230) (cid:230) (cid:230)(cid:230) (cid:230)(cid:230)(cid:230) (cid:230) (cid:230) (cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230) Ρ N N e (cid:76) (cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224) (cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224) (cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230) (cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230) (cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230) Ρ Ρ s f (cid:76) Figure 7.
Top row: Density graphs for U = 5 and T = − . , T = − . and T = − . (left to right). The bottom row shows the structure factor (blue circles) and thesuperfluid stiffness (red squares) for the same parameters These observations about the phase diagram are supported by an analysis of cuts ata few chosen values of T (and thus V ), see Fig. 7. At T = − . ( V = 3 . ), one observes asmooth density increase all the way until unit filling, where a plateau appears (Fig. 7a).The structure factor starts increasing near half filling, indicating the transition from theSF to the SS phase (Fig. 7d); at unit filling, the system lands in the CDW II phase.A cut at the slightly higher absolute value T = − . ( V = 4 . ) reveals a plateau athalf filling (CDW I) and a second one at unit filling (CDW II). Comparing the densityplot (Fig. 7b) with those of superfluid stiffness and structure factor (Fig. 7e), we see that ensity dependent tunneling in the extended Bose-Hubbard model T and inter-siterepulsion V , namely T = − . ( V = 8 . ). Below half filling, the density graduallyincreases up to the value of ρ ≈ . and then jumps to the CDW I phase (Fig. 7c).After this phase, the density behaves step-like, jumping directly into the CDW II phaseat ρ = 1 . This behavior is seen clearly in the data presented in Fig. 7f, where the SFphase for low densities is followed by two distinct regions of PS. These regions are onlyinterrupted by the CDW I phase at half filling and the CDW II phase at full filling.As these results show, for the lower on-site interaction U = 5 , the density-dependentterm T does not change much the overall behavior of the phase diagram if it has thesame sign as the single-particle tunneling t . Instead, if the two tunneling terms haveopposite sign, a large part of the SS phase disappears into a phase-separated region, dueto the increased relative importance of the interaction terms.
4. Conclusion
In summary, we have studied the extended Bose–Hubbard model on a square lattice withadditional terms coming from density-dependent tunnelings. Taking these terms intoaccount is relevant for experiments on ultracold dipolar molecules in optical lattices.The competition between the density-dependent tunneling, a standard single-particlehopping, finite on-site repulsion, and nearest-neighbor repulsion gives rise to a richphase diagram of the system.Specifically, as has been found previously [8], at large on-site repulsion and withoutdensity-dependent tunneling, there are Mott-insulator, charge-density wave, superfluid,and supersolid phases, as well as phase-separated parameter regions. Depending on theparameter strengths, this phase diagram undergoes considerable deformations. If eitherwe reduce on-site repulsion or introduce density-dependent tunnelings that have thesame sign as the single-particle hopping, some of the phase-separated regions disappear.Remarkably, if we introduce both of these effects simultaneously, additionally the chargedensity wave at half filling disappears. In this case of same-sign tunnelings, both hoppingprocesses act constructively producing an effective larger tunneling, or respectively,weaker interactions.We have also studied the phase diagram when the density-dependent tunnelingand single-particle hopping compete due to the their signs being opposite. Due to thiscompetition, the relative importance of interaction terms is enhanced. In this case, themost striking effect is the disappearance of the supersolid into a phase-separated region.This occurs on the particle-doped side of the half-filling charge density wave, and atstrong V . Contrary to similar models in one dimension [20], we find no indications forpair-superfluid behavior for all considered parameter values. ensity dependent tunneling in the extended Bose-Hubbard model Acknowledgements
This work was supported by the International PhD Projects Programme of theFoundation for Polish Science within the European Regional Development Fund of theEuropean Union, agreement no. MPD/2009/6. We acknowledge financial support fromSpanish Government Grant TOQATA (FIS2008-01236) and Consolider Ingenio EU IPSIQS, ERC Advanced Grant QUAGATUA, CatalunyaCaixa, Alexander von HumboldtFoundation and Hamburg Theory Award. M.M. and J.Z. thank Lluis Torner, SusanaHorváth and all ICFO personnel for hospitality. O.D. and J.Z. acknowledge supportfrom Polish National Center for Science project No. DEC-2012/04/A/ST2/00088. P.H.acknowledges support from the Austrian Science Fund (SFB F40 FOQUS) and the MarieCurie Initial Training Network COHERENCE.
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