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 study the extended Bose-Hubbard model describing an ultracold gas of dipolar molecules in an optical lattice, taking into account all on-site and nearest-neighbor interactions, including occupation-dependent tunneling and pair tunneling terms. Using exact diagonalization and the multiscale entanglement renormalization ansatz, we show that these terms can destroy insulating phases and lead to novel quantum phases. These considerable changes of the phase diagram have to be taken into account in upcoming experiments with dipolar molecules. PACS numbers: 37.10Jk,67.85.Hj,75.40.Cx Trapping and manipulating ultracold gases in optical lattices has allowed the realization of many-body physics in a controlled environment. For atoms interacting via contact interaction, a quantum phase transition from a superfluid (SF) to a Mott insulator (MI) has been predicted and observed [1]. In the simplest case, these systems can be theoretically described by the Bose-Hubbard (BH) model, which has two parameters: a tunneling J and an on-site interaction U [2, 3]. A natural extension of the Bose-Hubbard model comes from including longrange interactions between particles. Experiments on ultracold polar molecules have renewed interest in extended Bose-Hubbard models which can model such systems in optical lattices [4][5][6][7]. Because of the strong electric dipole moment of polar molecules, long-range interactions play a crucial role in the collective behavior of the system, leading to the appearance of states with long-range order, like various structured insulating states, supersolids, Wigner crystals, pair-supersolids, etc. [9][10][11][12][13][14][15].In this Letter, we study the ground-state of dipolar molecules in a 2D square optical lattice with a harmonic trapping along the polarization direction of the dipoles. We derive a modified BH model which includes additional occupation-dependent nearest-neighbor (NN) hopping processes arising from long-range dipolar interactions in the lowest Bloch band. Usually, interactioninduced hopping terms are neglected when discussing dipolar bosonic molecules. In this Letter, we show that these terms considerably change the physics of dipolar soft-core bosons. Soft-core bosons in square and onedimensional lattices have been discussed in the literature within the extended Hubbard model, focusing on the presence of stable supersolidity [17,18]. In the usual case with only NN interaction, at sufficient dipolar strength, the ground states at half-and unit-filling are checkerboard (CB) insulating states. Using exact diagonalization (ED) and multiscale entanglement renormalization ansatz (MERA), we solve the one-dimensional extended Hubbard model including the novel occupationdependent NN hopping processes. We find that with increasing dipolar interaction, the system enters from the CB phases to a novel state which has a one-particle superfluid (SF) and pair-superfluid (PSF) properties. Particularly we find a region where both of them coexists with the SF order parameter has alternating sign at ...

We study spin-1/2 fermions, interacting via a two-body contact potential, in a one-dimensional harmonic trap. Applying exact diagonalization, we investigate their behavior at finite interaction strength, and discuss the role of the ground-state degeneracy which occurs for sufficiently strong repulsive interaction. Even low temperature or a completely depolarizing channel may then dramatically influence the system's behavior. We calculate level occupation numbers as signatures of thermalization, and we discuss the mechanisms to break the degeneracy. PACS numbers: 67.85.-d, 67.85.Lm

We study the Hartree ground state of a dipolar condensate of atoms or molecules in an threedimensional anisotropic geometry and at T = 0. We determine the stability of the condensate as a function of the aspect ratios of the trap frequencies and of the dipolar strength. We find numerically a rich phase space structure characterized by various structures of the ground-state density profile.

We study the ground-state properties of ultracold bosons in an optical lattice in the regime of strong interactions. The system is described by a nonstandard Bose-Hubbard model with both occupation-dependent tunneling and on-site interaction. We find that for sufficiently strong coupling the system features a phasetransition from a Mott insulator with one particle per site to a superfluid of spatially extended particle pairs living on top of the Mott background -instead of the usual transition to a superfluid of single particles/holes. Increasing the interaction further, a superfluid of particle pairs localized on a single site (rather than being extended) on top of the Mott background appears. This happens at the same interaction strength where the Mott-insulator phase with 2 particles per site is destroyed completely by particlehole fluctuations for arbitrarily small tunneling. In another regime, characterized by weak interaction, but high occupation numbers, we observe a dynamical instability in the superfluid excitation spectrum. The new ground state is a superfluid, forming a 2D slab, localized along one spatial direction that is spontaneously chosen.

Attractive ultracold fermions trapped in a one-dimensional periodically shaken optical lattice are considered. For an appropriate resonant shaking, a dimerized structure emerges for which the system realizes paradigmatic physics described by the Rice-Mele model. The emergent nature of the system together with density fluctuations or controlled modifications of lattice filling allow for the creation of defects. Those defects lead to topologically protected localized modes carrying the fractional particle number. Their possible experimental signatures are discussed.

Fundamental forces of Nature are described by field theories, also known as gauge theories, based on a local gauge invariance. The simplest of them is quantum electrodynamics (QED), which is an example of an Abelian gauge theory. Such theories describe the dynamics of massless photons and their coupling to matter. However, in two spatial dimension (2D) they are known to exhibit gapped phases at low temperature. In the realm of quantum spin systems, it remains a subject of considerable debate if their low energy physics can be described by emergent gauge degrees of freedom. Here we present a class of simple two-dimensional models that admit a low energy description in terms of an Abelian gauge theory. We find rich phase diagrams for these models comprising exotic deconfined phases and gapless phases - a rare example for 2D Abelian gauge theories. The counter-intuitive presence of gapless phases in 2D results from the emergence of additional symmetry in the models. Moreover, we propose schemes to realize our model with current experiments using ultracold bosonic atoms in optical lattices.Comment: Accepted versio

We investigate the coupling of a nanomechanical oscillator in the quantum regime with molecular (electric) dipoles. We find theoretically that the cantilever can produce single-mode squeezing of the center-of-mass motion of an isolated trapped molecule and two-mode squeezing of the phonons of an array of molecules. This work opens up the possibility of manipulating dipolar crystals, which have been recently proposed as quantum memory, and more generally, is indicative of the promise of nanoscale cantilevers for the quantum detection and control of atomic and molecular systems.