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 Mott phases and the superfluid-insulator transition of two-component ultracold bosons on a square optical lattice in the presence of a non-Abelian synthetic gauge field, which renders a SU(2)-hopping matrix for the bosons. Using a resummed hopping expansion, we calculate the excitation spectra in the Mott insulating phases and demonstrate that the superfluid-insulator phase boundary displays a nonmonotonic dependence on the gauge-field strength. We also compute the momentum distribution of the bosons in the presence of the non-Abelian field and show that they develop peaks at nonzero momenta as the superfluid-insulator transition point is approached from the Mott side. Finally, we study the superfluid phases near the transition and discuss the induced spatial pattern of the superfluid density due to the presence of the non-Abelian gauge potential. PHYSICAL REVIEW A 84, 053632 (2011) FIG. 4. (Color online) Momentum distributions of the bosons at J ≈ 0.97J c . Bright regions correspond to high densities. The upper and lower rows correspond to = 0 and = 1/2, respectively. The non-Abelian-field strengths α are, from left to right in each panel, 0, 0.7, 0.8, and π/2.

Exact diagonalization (ED) techniques are a powerful method for studying many-body problems. Here, we apply this method to systems of few bosons in an optical lattice, and use it to demonstrate the emergence of interesting quantum phenomena such as fragmentation and coherence. Starting with a standard Bose-Hubbard Hamiltonian, we first revise the characterisation of the superfluid to Mott insulator (MI) transitions. We then consider an inhomogeneous lattice, where one potential minimum is made much deeper than the others. The MI phase due to repulsive on-site interactions then competes with the trapping of all atoms in the deep potential. Finally, we turn our attention to attractively interacting systems, and discuss the appearance of strongly correlated phases and the onset of localisation for a slightly biased lattice. The article is intended to serve as a tutorial for ED of Bose-Hubbard models.