We derive upper and lower bounds on the critical temperature
Tc and the energy gap $\ensuremath{\Xi}$ (at zero temperature) for the BCS gap equation, describing spin-
21 fermions interacting via a local two-body interaction potential $\ensuremath{\lambda}V(x)$. At weak coupling $\ensuremath{\lambda}⪡1$ and under appropriate assumptions on
V(x), our bounds show that ${T}_{c}\ensuremath{\sim}A\phantom{\rule{0.2em}{0ex}}\mathrm{exp}(\ensuremath{-}B∕\ensuremath{\lambda})$ and $\ensuremath{\Xi}\ensuremath{\sim}C\phantom{\rule{0.2em}{0ex}}\mathrm{exp}(\ensuremath{-}B∕\ensuremath{\lambda})$ for some explicit coefficients
A,
B, and
C depending on the interaction
V(x) and the chemical potential $\ensuremath{\mu}$. The ratio
A∕C turns out to be a universal constant, independent of both
V(x) and $\ensuremath{\mu}$. Our analysis is valid for any $\ensuremath{\mu}$; for small $\ensuremath{\mu}$, or low density, our formulas reduce to well-known expressions involving the scattering length of
V(x).
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.
We investigate the
1/N expansion proposed recently as a strategy to include quantum fluctuation effects in the nonrelativistic, attractive Fermi gas at and near unitarity. We extend the previous results by calculating the next-to-leading order corrections to the critical temperature along the whole crossover from Bardeen-Cooper-Schrieffer (BCS) superconductivity to Bose-Einstein condensation. We demonstrate explicitly that the extrapolation from the mean-field approximation, based on the
1/N expansion, provides a useful approximation scheme only on the BCS side of the crossover. We then apply the technique to the study of strongly interacting relativistic many-fermion systems. Having in mind the application to color superconductivity in cold dense quark matter, we develop, within a simple model, a formalism suitable to compare the effects of order parameter fluctuations in phases with different pairing patterns. Our main conclusion is that the relative correction to the critical temperature is to a good accuracy proportional to the mean-field ratio of the critical temperature and the chemical potential. As a consequence, it is significant even rather deep in the BCS regime, where phenomenologically interesting values of the quark--quark coupling are expected. Possible impact on the phase diagram of color-superconducting quark matter is discussed.