Non-standard Hubbard models in optical lattices: a review

Abstract

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.

Causality and quantum criticality in long-range lattice models

“…The powerful tools of field theory used in this paper can be employed to study a range of interesting, and experimentally relevant, long-range interacting systems [16,[53][54][55][56], such as a huge variety of spin-1/2 [57][58][59][60], spin-1 [41,61,62], and higher-spin [63,64] models, generalized Hubbard [63,65,66] and t-J models [58,59], and spin-boson problems [67], among many others, in one or more spatial dimensions. In general, these models exhibit new universal behavior not captured by standard long-range interacting classical models, since the quantum-to-classical mapping generates classical models with long-range interactions in all spatial directions except the one corresponding to the imaginary time dimension of the quantum model [39].…”

Vortices and vortex lattices in quantum ferrofluids

“…The relatively recent realization of dipolar cold atoms and molecules now provides an alternative venue for investigating supersolid phases. Apart from the strongly correlated dipolar systems [70,[72][73][74][75][76] cited in section 2, extended Bose-Hubbard lattice models are also thought to support supersolidity [287][288][289][290][291][292][293][294][295][296][297]. The common ingredient in both sets of investigations is the presence of long-range interactions.…”

Quantum Simulation of a 2D Quasicrystal with Cold Atoms

“…By varying the relative strength of the hopping and onsite interaction terms, it is possible, for example, to induce a superfluid to Mott insulator transition in a bosonic system [13]. Extensions of the Hubbard model to further near neighbors are reviewed in [14].…”