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arXiv:1311.7510 · Physical Review A

Reexamination of the variational Bose-Hubbard model

Jan Major, Mateusz Łącki, Jakub Zakrzewski

Published Apr 30, 2014• 8 citations
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Abstract

For strongly interacting bosons in optical lattices, the standard description using the Bose-Hubbard model becomes questionable. The role of excited bands becomes important. In such a situation, we compare results of simulations using the multiband Bose-Hubbard model with a recent proposition based on a time-dependent variational approach. It is shown that the latter, in its original formulation, uses a too small variational space, often leading to spurious effects. Possible expansion of the variational approach is discussed.

DOI
10.1103/physreva.89.043626

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For strongly interacting bosons in optical lattices, the standard description using the Bose-Hubbard model becomes questionable. The role of excited bands becomes important. In such a situation, we compare results of simulations using the multiband Bose-Hubbard model with a recent proposition based on a time-dependent variational approach. It is shown that the latter, in its original formulation, uses a too small variational space, often leading to spurious effects. Possible expansion of the variational approach is discussed.
Cold atoms in optical lattices allow for accurate studies of many body dynamics. Rapid time-dependent modifications of optical lattice potentials may result in significant excitations in atomic systems. The dynamics in such a case is frequently quite incompletely described by standard applications of tight-binding models (such as, e.g., Bose-Hubbard model or its extensions) that typically neglect the effect of the dynamics on the transformation between the real space and the tight-binding basis. We illustrate the importance of a proper quantum mechanical description using a multiband extended Bose-Hubbard model with time-dependent Wannier functions. We apply it to situations directly related to experiments.
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Originally, the Hubbard model was derived for describing the behavior of strongly correlated electrons in solids. However, for over a decade now, variations of it have also routinely been implemented with ultracold atoms in optical lattices, allowing their study in a clean, essentially defect-free environment. Here, we review some of the vast literature on this subject, with a focus on more recent non-standard forms of the Hubbard model. After giving an introduction to standard (fermionic and bosonic) Hubbard models, we discuss briefly common models for mixtures, as well as the so-called extended Bose-Hubbard models, that include interactions between neighboring sites, next-neighbor sites, and so on. The main part of the review discusses the importance of additional terms appearing when refining the tight-binding approximation for the original physical Hamiltonian. Even when restricting the models to the lowest Bloch band is justified, the standard approach neglects the density-induced tunneling (which has the same origin as the usual on-site interaction). The importance of these contributions is discussed for both contact and dipolar interactions. For sufficiently strong interactions, the effects related to higher Bloch bands also become important even for deep optical lattices. Different approaches that aim at incorporating these effects, mainly via dressing the basis, Wannier functions with interactions, leading to effective, density-dependent Hubbard-type models, are reviewed. We discuss also examples of Hubbard-like models that explicitly involve higher p orbitals, as well as models that dynamically couple spin and orbital degrees of freedom. Finally, we review mean-field nonlinear Schrödinger models of the Salerno type that share with the non-standard Hubbard models nonlinear coupling between the adjacent sites. In that part, discrete solitons are the main subject of consideration. We conclude by listing some open problems, to be addressed in the future.
We determine the ground-state properties of a gas of interacting bosonic atoms in a one-dimensional optical lattice. The system is modeled by the Bose-Hubbard Hamiltonian. We show how to apply the time-evolving block decimation method to systems with periodic boundary conditions and employ it as a reference to find the ground state of the Bose-Hubbard model. Results are compared with recently proposed approximate methods, such as Hartree-Fock-Bogoliubov (HFB) theories generalized for strong interactions and the variational Bijl-Dingle-Jastrow method. We find that all HFB methods do not bring any improvement to the Bogoliubov theory and therefore provide correct results only in the weakly interacting limit, where the system is deeply in the superfluid regime. On the other hand, the variational Bijl-Dingle-Jastrow method is applicable for much stronger interactions but is essentially limited to the superfluid regime as it reproduces the superfluid--Mott-insulator transition only qualitatively.
We discuss the superfluid to Mott insulator transition of an atomic Bose gas in an optical lattice with high filling factors. We show that in this multiband situation, the long-wavelength physics is described by a single-band Bose-Hubbard model. We determine the many-body renormalization of the tunneling and interaction parameters in the effective Bose-Hubbard Hamiltonian, and consider the resulting model at nonzero temperatures. We show that, in particular, for a one- or two-dimensional optical lattice, the Mott-insulator phase is more difficult to realize than anticipated previously.