Abstract:
We analyze the vortex core states of two-species (mass imbalanced) superfluid fermion mixtures as a function of two-body binding energy in two dimensions. In particular we solve the Bogoliubov-de Gennes equations for a population balanced mixture of $^{6}$Li and $^{40}$K atoms at zero temperature. We find that the vortex core is mostly occupied by the light-mass ($^{6}$Li) fermions and that the core density of the heavy-mass ($^{40}$K) fermions is highly depleted. This is in contrast with the one-species (mass balanced) mixtures with balanced populations where an equal amount of density depletion is found at the vortex core for both pseudospin components.

Abstract:
We use the diagrammatic $T$-matrix approach to analyze the three-body scattering problem between two identical fermions and a third particle (which could be a different species of fermion or a boson). We calculate the s-wave dimer-atom scattering length for all mass ratios, and our results exactly match the results of Petrov. In particular, we list the exact dimer-atom scattering lengths for all available two-species Fermi-Fermi and Bose-Fermi mixtures. In addition, unlike that of the equal-mass particles case where the three-body scattering $T$-matrix decays monotonically as a function of the outgoing momentum, we show that, after an initial rapid drop, this function changes sign and becomes negative at large momenta and then decays slowly to zero when the mass ratio of the fermions to the third particle is higher than a critical value (around 6.5). As the mass ratio gets higher, modulations of the $T$-matrix become more apparent with multiple sign changes, related to the "fall of a particle to the center" phenomenon and to the emergence of three-body Efimov bound states.

Abstract:
To analyze the ground-state phase diagram of Bose-Bose mixtures loaded into $d$-dimensional hypercubic optical lattices, we perform a strong-coupling power-series expansion in the kinetic energy term (plus a scaling analysis) for the two-species Bose-Hubbard model with onsite boson-boson interactions. We consider both repulsive and attractive interspecies interaction, and obtain an analytical expression for the phase boundary between the incompressible Mott insulator and the compressible superfluid phase up to third order in the hoppings. In particular, we find a re-entrant quantum phase transition from paired superfluid (superfluidity of composite bosons, i.e. Bose-Bose pairs) to Mott insulator and again to a paired superfluid in all one, two and three dimensions, when the interspecies interaction is sufficiently large and attractive. We hope that some of our results could be tested with ultracold atomic systems.

Abstract:
It has recently been shown that the spin-orbit coupling gives rise to topologically-nontrivial and thermodynamically-stable gapless superfluid phases when the pseudo-spin populations of an atomic Fermi gas is imbalanced, with the possibility of featuring Majorana zero-energy quasiparticles. In this paper, we consider a Rashba-type spin-orbit coupling, and use the Bogoliubov-de Gennes formalism to analyze a single vortex line along a finite cylinder with a periodic boundary condition. We show that the signatures for the appearance of core- and edge-bound states can be directly found in the density of single-particle states and particle-current density. In particular, we find that the pseudo-spin components counterflow near the edge of the cylinder, the strength of which increases with increasing spin-orbit coupling.

Abstract:
We study the interplay between the Hofstadter butterfly, strong interactions and Zeeman field within the mean-field Bogoliubov-de Gennes theory in real space, and explore the ground states of the attractive single-band Hofstadter-Hubbard Hamiltonian on a square lattice, including the exotic possibility of imbalanced vector potentials. We find that the cooperation between the vector potential and superfluid order breaks the spatial symmetry of the system, and flourish stripe-ordered Fulde-Ferrell-Larkin-Ovchinnikov (FFLO)-like superfluid and supersolid phases that can be distinguished and characterized according to their coexisting pair-density (PDW), charge-density (CDW) and spin-density (SDW) wave orders. We also discuss confined systems and comment on the likelihood of observing such stripe-ordered phases by loading neutral atomic Fermi gases on laser-induced optical lattices under laser-generated artificial gauge fields.

Abstract:
After deriving the isothermal Hellmann-Feynman theorem (IHFT) that is suitable for mixed states in a canonical ensemble, we use this theorem to obtain the isothermal magnetic-field sweep theorems for the free, average and trapping energies, and for the entropy, specific heat, pressure and atomic compressibility of strongly-correlated ultra-cold quantum gases. In particular, we apply the sweep theorems to two-component Fermi gases in the weakly-interacting BCS and BEC limits, showing that the temperature dependence of the contact parameter can be determined by the variation of either the entropy or specific heat with respect to the scattering length. We also use the IHFT to obtain the Virial theorem in a canonical ensemble, and discuss its implications for quantum gases.

Abstract:
We analyze the momentum distribution function and its artificial-gauge-field dependence for the Mott insulator phases of the Hofstadter-Bose-Hubbard model. By benchmarking the results of the random-phase approximation (RPA) approach against those of the strong-coupling expansion (SCE) for the Landau and symmetric gauges, we find pronounced corrections to the former results in two dimensions.

Abstract:
We use the Bogoliubov-de Gennes formalism to analyze harmonically trapped Fermi gases with Rashba-type spin-orbit coupling in two dimensions. We consider both population-balanced and -imbalanced Fermi gases throughout the BCS-BEC evolution, and study the effects of spin-orbit coupling on the spontaneously induced countercirculating mass currents and the associated intrinsic angular momentum. In particular, we find that even a small spin-orbit coupling destabilizes Fulde-Ferrel-Larkin-Ovchinnikov (FFLO)-type spatially modulated superfluid phases as well as the phase-separated states against the polarized superfluid phase. We also show that the continuum of quasiparticle and quasihole excitation spectrum can be connected by zero, one or two discrete branches of interface modes, depending on the number of interfaces between a topologically trivial phase (e.g. locally unpolarized/low-polarized superfluid or spin-polarized normal) and a topologically nontrivial one (e.g. locally high-polarized superfluid) that may be present in a trapped system.

Abstract:
The standard mean-field theory for the Mott insulator-superfluid phase transition is not sufficient to describe the Mott insulator-paired superfluid phase transition. Therefore, by restricting the two-species Bose-Hubbard Hamiltonian to the subspace of paired particles, and using perturbation theory, here we derive an analytic mean-field expression for the Mott insulator-paired superfluid transition boundary.