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

Recent theoretical and experimental progress on studying one-dimensional systems of bosonic, fermionic, and Bose-Fermi mixtures of a few ultracold atoms confined in traps is reviewed in the broad context of mesoscopic quantum physics. We pay special attention to limiting cases of very strong or very weak interactions and transitions between them. For bosonic mixtures, we describe the developments in systems of three and four atoms as well as different extensions to larger numbers of particles. We also briefly review progress in the case of spinor Bose gases of a few atoms. For fermionic mixtures, we discuss a special role of spin and present a detailed discussion of the two-and three-atom cases. We discuss the advantages and disadvantages of different computation methods applied to systems with intermediate interactions. In the case of very strong repulsion, close to the infinite limit, we discuss approaches based on effective spin chain descriptions. We also report on recent studies on higherspin mixtures and inter-component attractive forces. For both statistics, we pay particular attention to impurity problems and mass imbalance cases. Finally, we describe the recent advances on trapped Bose-Fermi mixtures, which allow for a theoretical combination of previous concepts, well illustrating the importance of quantum statistics and inter-particle interactions. Lastly, we report on fundamental questions related to the subject which we believe will inspire further theoretical developments and experimental verification. :1903.12189v3 [cond-mat.quant-gas] CONTENTS arXiv

-Recently developed techniques allow for simultaneous measurements of the positions of all ultra cold atoms in a trap with high resolution. Each such single shot experiment detects one element of the quantum ensemble formed by the cloud of atoms. Repeated single shot measurements can be used to determine all correlations between particle positions as opposed to standard measurements that determine particle density or two-particle correlations only. In this paper we discuss the possible outcomes of such single shot measurements in case of cloud of ultra-cold noninteracting Fermi atoms. We show that the Pauli exclusion principle alone leads to correlations between particle positions that originate from unexpected spatial structures formed by the atoms.Introduction. -Tremendous progress in experimental techniques of preparing, manipulating and probing ultra-cold gases have opened new possibilities of optical methods of monitoring atomic systems. Atomic fluorescence microscopes with resolution in the range of hundreds of nanometers became accessible [1][2][3][4][5][6][7]. The microscopes allow for observation of both boson and fermion atoms with resolution comparable to the optical wavelength. Single shot pictures of such systems correspond to a single realization of the N -body probability density as opposed to a one-particle probability distribution. Difference between the two is tremendous, they differ by N body correlations. The seminal work of [8] shows how interference fringes, visible in a simultaneous single shot picture of N atoms, arise in the course of measurement. No fringes are observed in a single particle detection instead. In a similar way the solitons emerge in a process of detection of N -particles prepared in a type II excited state of a 1D system of bosons interacting via short-range potential described by the Lieb-Linger model [9]. Single shot time-dependent simulations of many-body dynamics showing appearance of fluctuating vortices and center-ofmass fluctuations of attractive BEC have been reported recently [10].

We study a strongly attractive system of a few spin-1/2 fermions confined in a one-dimensional harmonic trap, interacting via two-body contact potential. Performing exact diagonalization of the Hamiltonian we analyze the ground state and the thermal state of the system in terms of one-and two-particle reduced density matrices. We show how for strong attraction the correlated pairs emerge in the system. We find that the fraction of correlated pairs depends on temperature and we show that this dependence has universal properties analogous to the gap function known from the theory of superconductivity. In contrast to the standard approach based on the variational ansatz and/or perturbation theory, our predictions are exact and are valid also in a strong attraction limit. Our findings contribute to the understanding of strongly correlated few-body systems and can be verified in current experiments on ultra-cold atoms.The Model. -In this Letter we study a few fermions of mass m, confined in a one-dimensional harmonic trap of frequency Ω, mutually interacting via short-range deltap-1 arXiv:1406.0400v2 [cond-mat.quant-gas] 30 Jan 2015

In this Brief Report the extended Bose-Hubbard model with local two-and three-body interactions is studied by the exact diagonalization approach. The shapes of the first two insulating lobes are discussed and the values of the critical tunneling for which the insulating phase loses stability for repulsive and attractive three-body interactions are predicted.

A system of two species of fermions of different mass confined in a one-dimensional harmonic trap is studied with an exact diagonalization approach. It is shown that a mass difference between fermionic species induces a separation in the density of the lighter flavour independently of the number of particles. The mechanism behind the emergent separation is explained phenomenologically and confirmed by direct studies of the ground state of the system. Finally, it is shown that the separation driven by a mass difference, in contrast to the separation induced by a difference of populations, is robust to the interactions with a thermal environment.