The inhomogeneous extended Bose‐Hubbard model: A mean‐field theory

Abstract

Dedicated to Ulrich Eckern on the occasion of his 60th birthday.We develop an inhomogeneous mean-field theory for the extended Bose-Hubbard model with a quadratic, confining potential. In the absence of this potential, our mean-field theory yields the phase diagram of the homogeneous extended Bose-Hubbard model. This phase diagram shows a superfluid (SF) phase and lobes of Mott-insulator (MI), density-wave (DW), and supersolid (SS) phases in the plane of the chemical potential μ and on-site repulsion U ; we present phase diagrams for representative values of V , the repulsive energy for bosons on nearest-neighbor sites. We demonstrate that, when the confining potential is present, superfluid and density-wave order parameters are nonuniform; in particular, we obtain, for a few representative values of parameters, spherical shells of SF, MI, DW, and SS phases. We explore the implications of our study for experiments on cold-atom dipolar condensates in optical lattices in a confining potential.Original Paper is different on the two sublattices of the hypercubic lattices we consider, and a super-solid (SS) phase (see, e.g., [15,18,19]).Before we present the details of our study, we summarize our principal results. We first develop a mean-field theory for the homogeneous, extended Bose-Hubbard model by developing on the work of our group on Bose-Hubbard models for the spinless and spin-1 cases [7,12]; this yields the SF, MI, DW, and SS phases and the transitions between them, which have been studied by a Gutzwiller-type approximation [15] that is akin to, but not the same as, our mean-field theory. We then develop an inhomogeneous mean-field theory for the inhomogeneous extended, Bose-Hubbard model by generalizing our inhomogeneous mean-field theory for the Bose-Hubbard model [14]. In particular, when we use a quadratic confining potential in three dimensions (3D), our theory yields inhomogeneous phases with spherical shells of SF, MI, DW, and SS states. The precise way in which these phases alternate depends on the parameters of the model; we study a few illustrative cases explicitly for which we present order-parameter profiles and their Fourier transforms. We also discuss the experimental implications of our work.The remaining part of this paper is organized as follows. In Sect. 2 we introduce the inhomogeneous extended Bose-Hubbard model and then develop an inhomogeneous mean-field theory for it. In Sect. 3 we present the results of our mean-field theory. Section 4 contains concluding remarks; here we give a brief comparison of our work with earlier studies and we explore the experimental implications of our study.

Spin‐1 Bose Hubbard Model with Nearest Neighbour Extended Interaction

“…[61,65]. We have also checked the spin eigenvalues and the order parameter profiles in presence of one dimensional trapping potential without zV/U 0 in both the AF and ferromagnetic cases are in agreement with the results obtained in ref.…”

“…To decouple both the hopping and the density-density interaction terms, we use the mean field approximation as given by, [20,61] …”

Supersolid phases of dipolar fermions in a two-dimensional-lattice bilayer array

“…6 exhibit a small region where both phases spatially overlap. Other studies [39,40] with different systems in the presence of a harmonic trap have shown coexistence of phases without spatial overlapping, for instance, in the extended Bose-Hubbard model the Mott insulator and superfluid phases tend to form rings and disks. In 2D those studies agree quantitatively with Quantum Monte Carlo calculations and more Finally, when the interlayer spacing is large enough, the intralayer repulsive interaction dominates over the interlayer attraction and the superfluid order parameter almost vanishes.…”

Phase Diagram of Correlated Bosons in a Tripartite Lattice with Harmonic Confinement

“…(iv) i and j both even : no neighbours To find the ground state properties of the system, first we have computed the matrix elements of the Hamiltonian in Eq. (3) in the occupation number basis |n truncated at some number n max (n max = 8 used here) [9,10] and diagonalize it to obtain the self consistent values of ψ i and ρ i . It is equivalent to the minimization of free energy of the system.…”

“…[9]. In presence of trapping potential, it was extended for the inhomogeneous case for a bipartite lattice [10]. In a previous work, we have considered a tripartite lattice and solved extended Bose Hubbard model (EBHM) via mean field theory in the homogeneous case and obtained a rich phase diagram, in which the ground state of the model exhibits SF, different density ordered (DW), supersolid (SS) and MI phases as function of the chemical potential [11].…”