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We determine the phase diagram of a quasi-one-dimensional conductor (weakly coupled chains system with an open Fermi surface) in a magnetic field. The usual Ginzburg-Landau regime is followed, when the field is increased, by a cascade of superconducting phases separated by first order transitions, which ends with a strong reentrance of the superconducting phase. These new phases show a novel kind of symmetry of laminar type. The Zeeman splitting does not completely suppress the reentrance in very strong field, the ground state being in this case a Larkin-Ovchinnikov-Fulde-Ferrell state.

We analyze the phase diagram of uniform superfluidity for two-species fermion mixtures from the Bardeen-Cooper-Schrieffer to Bose-Einstein condensation (BEC) limit as a function of the scattering parameter and population imbalance. We find at zero temperature that the phase diagram of population imbalance versus scattering parameter is asymmetric for unequal masses, having a larger stability region for uniform superfluidity when the lighter fermions are in excess. In addition, we find topological quantum phase transitions associated with the disappearance or appearance of momentum space regions of zero quasiparticle energies. Lastly, near the critical temperature, we derive the Ginzburg-Landau equation and show that it describes a dilute mixture of composite bosons and unpaired fermions in the BEC limit.

We discuss ultracold Fermi gases in two dimensions, which could be realized in a strongly confining one-dimensional optical lattice. We obtain the temperature versus effective interaction phase diagram for an s-wave superfluid and show that, below a certain critical temperature Tc, spontaneous vortex-antivortex pairs appear for all coupling strengths. In addition, we show that the evolution from weak-to-strong coupling is smooth, and that the system forms a square vortex-antivortex lattice at a lower critical temperature TM.

We discuss the possibility of a quantum phase transition in ultra-cold spin-polarized Fermi gases which exhibit a p-wave Feshbach resonance. We show that when fermionic atoms form a condensate that can be externally tuned between the BCS and BEC limits, the zero temperature compressibility and the spin susceptibility of the fermionic gas are non-analytic functions of the two-body bound state energy. This non-analyticity is due to a massive rearrangement of the momentum distribution in the ground state of the system. Furthermore, we show that the low temperature superfluid density is also nonanalytic, and exhibits a dramatic change in behavior when the critical value of the bound state energy is crossed.

We study phase transitions and hysteresis in a system of dipolar bosons loaded into triangular optical lattices at zero temperature. We find that the quantum melting transition from supersolid to superfluid phase is first order, in contrast with the previous report. We also find that due to strong quantum fluctuations the supersolid (or solid)-superfluid transition can exhibit an anomalous hysteretic behavior, in which the curve of density versus chemical potential does not form a standard loop structure. Furthermore, we show that the transition occurs unidirectionally along the anomalous hysteresis curve.

We discuss the evolution from BCS to BEC superfluids in the presence of spin-orbit coupling, and show that this evolution is just a crossover in the balanced case. The dependence of several thermodynamic properties, such as the chemical potential, order parameter, pressure, entropy, isothermal compressibility and spin susceptibility tensor on the spin-orbit coupling and interaction parameter at low temperatures are analyzed. We studied both the case of equal Rashba and Dresselhaus (ERD) and the Rashba-only (RO) spin-orbit coupling. Comparisons between the two cases reveal several striking differences in the corresponding thermodynamic quantities. Finally we propose measuring the spin susceptibility as a means to detect the spin-orbit coupling effect.