We review the study of the superfluid phase transition in a system of fermions whose interaction can be tuned continuously along the crossover from Bardeen–Cooper–Schrieffer (BCS) superconducting phase to a Bose–Einstein condensate (BEC), also in the presence of a spin–orbit coupling. Below a critical temperature the system is characterized by an order parameter. Generally a mean field approximation cannot reproduce the correct behavior of the critical temperature Tc over the whole crossover. We analyze the crucial role of quantum fluctuations beyond the mean-field approach useful to find Tc along the crossover in the presence of a spin–orbit coupling, within a path integral approach. A formal and detailed derivation for the set of equations useful to derive Tc is performed in the presence of Rashba, Dresselhaus and Zeeman couplings. In particular in the case of only Rashba coupling, for which the spin–orbit effects are more relevant, the two-body bound state exists for any value of the interaction, namely in the full crossover. As a result the effective masses of the emerging bosonic excitations are finite also in the BCS regime.
In order to study the pairing symmetry in non-centrosymmetric superconductors, we solve the linearized Eliashberg's equation on the two-dimensional extended Hubbard model in the presence of the Rashba-type spin-orbit coupling (RSOC) within the random phase approximation. In the presence of the RSOC, three types of pairing symmetries appear in the phase diagram with respect to the on-site Coulomb repulsion U and off-site one V. Each of pairing symmetries is admixture of spin-singlet and -triplet ones. On the basis of analytical study, it is found that the admixture of spin-singlet and -triplet components depends on not only the predominant pairing symmetry but also dispersion relation and pairing interaction.
We derive general conditions for the emergence of singlet Feshbach molecules in the presence of artificial Zeeman fields for arbitrary mixtures of Rashba and Dresselhaus spin-orbit coupling in two or three dimensions. We focus on the formation of two-particle bound states resulting from interactions between ultracold
spin-1/2 fermions, under the assumption that interactions are short ranged and occur only in the
s-wave channel. In this case, we calculate explicitly binding energies of Feshbach molecules and bound-state energy thresholds and analyze their dependence on spin-orbit couplings, Zeeman fields, interactions, and center-of-mass momentum, paying particular attention to the experimentally relevant case of spin-orbit couplings with equal Rashba and Dresselhaus (ERD) amplitudes.