Fermionic superfluidity in atomic Fermi gases across a Feshbach resonance is normally described by the atom-molecule theory, which treats the closed channel as a noninteracting point boson. In this work we present a theoretical description of the resonant superfluidity in analogy to the two-band superconductors. We employ the underlying two-channel scattering model of Feshbach resonance where the closed channel is treated as a composite boson with binding energy ${\ensuremath{\varepsilon}}_{0}$ and the resonance is triggered by the microscopic interchannel coupling
U12. The binding energy ${\ensuremath{\varepsilon}}_{0}$ naturally serves as an energy scale of the system, which has been sent to infinity in the atom-molecule theory. We show that the atom-molecule theory can be viewed as a leading-order low-energy effective theory of the underlying fermionic theory in the limit ${\ensuremath{\varepsilon}}_{0}\ensuremath{\rightarrow}\ensuremath{\infty}$ and ${U}_{12}\ensuremath{\rightarrow}0$, while keeping the phenomenological atom-molecule coupling finite. The resulting two-band description of the superfluid state is in analogy to the BCS theory of two-band superconductors. In the dilute limit ${\ensuremath{\varepsilon}}_{0}\ensuremath{\rightarrow}\ensuremath{\infty}$, the two-band description recovers precisely the atom-molecule theory. The two-band theory provides a natural approach to study the corrections because of a finite binding energy ${\ensuremath{\varepsilon}}_{0}$ in realistic experimental systems. For broad and moderate resonances, the correction is not important for current experimental densities. However, for extremely narrow resonance, we find that the correction becomes significant. The finite binding energy correction could be important for the stability of homogeneous polarized superfluid against phase separation in imbalanced Fermi gases across a narrow Feshbach resonance.
We use a finite-temperature effective field theory recently developed for superfluid Fermi gases to investigate the properties of dark solitons in these superfluids. Our approach provides an analytic solution for the dip in the order parameter and the phase profile across the soliton, which can be compared with results obtained in the framework of the Bogoliubov--de Gennes equations. We present results in the whole range of the BCS-BEC crossover, for arbitrary temperatures and taking into account Gaussian fluctuations about the saddle point. The obtained analytic solutions yield an exact energy-momentum relation for a dark soliton showing that the soliton in a Fermi gas behaves like a classical particle even at nonzero temperatures. The spatial profile of the pair field and for the parameters of state for the soliton are analytically studied.
The standard Ginzburg-Landau theory is valid only near the critical temperature, where the order parameter is small. As a consequence it is not straightforward to develop a Ginzburg-Landau framework for two-bandgap superconductors such as magnesium diboride, where two order parameters exist with different intrinsic critical temperatures. We derive an extended Ginzburg-Landau functional, capable of describing both single-band and two-bandgap superfluids at all temperatures below the critical temperature. In the limit of small temperatures, we retrieve the ground state energy obtained from the microscopic path-integral theory for Fermi superfluids, and for temperatures near the critical temperature, we obtain the standard Ginzburg-Landau expressions. The present formalism is applied to reveal hidden criticality and the presence of two healing length scales in two-bandgap superfluids. The developed method is focused on strong-coupling superfluid systems, in particular, the ultracold atomic Fermi gases in the BCS-BEC crossover regime.