Results are presented for the time evolution of fermions initially in a nonzero temperature normal phase, following the switch on of an attractive interaction. The dynamics are studied in the disordered phase close to the critical point, where the superfluid fluctuations are large. The analysis is conducted within a two-particle irreducible, large
N approximation. The system is considered from the perspective of critical quenches where it is shown that the fluctuations follow universal model A dynamics. A signature of this universality is found in a singular correction to the fermion lifetime, given by a scaling form ${t}^{(3\ensuremath{-}d)/2}{S}_{d}({\ensuremath{\varepsilon}}^{2}t)$, where
d is the spatial dimension,
t is the time since the quench, and $\ensuremath{\varepsilon}$ is the fermion energy. The singular behavior of the spectral density is interpreted as arising due to incoherent Andreev reflections off superfluid fluctuations.
We present a comprehensive analysis of quantum fluctuation effects in the superfluid ground state of an attractively interacting Fermi system, employing the attractive Hubbard model as a prototype. The superfluid order parameter and fluctuations thereof are implemented by a bosonic Hubbard-Stratonovich field, which splits into two components corresponding to longitudinal and transverse (Goldstone) fluctuations. Physical properties of the system are computed from a set of approximate flow equations obtained by truncating the exact functional renormalization group flow of the coupled fermion-boson action. The equations capture the influence of fluctuations on nonuniversal quantities such as the fermionic gap, as well as the universal infrared asymptotics present in every fermionic superfluid. We solve the flow equations numerically in two dimensions and compute the asymptotic behavior analytically in two and three dimensions. The fermionic gap $\ensuremath{\Delta}$ is reduced significantly compared to the mean-field gap, and the bosonic order parameter $\ensuremath{\alpha}$, which is equivalent to $\ensuremath{\Delta}$ in mean-field theory, is suppressed to values below $\ensuremath{\Delta}$ by fluctuations. The fermion-boson vertex is only slightly renormalized. In the infrared regime, transverse order-parameter fluctuations associated with the Goldstone mode lead to a strong renormalization of longitudinal fluctuations: the longitudinal mass and the bosonic self-interaction vanish linearly as a function of the scale in two dimensions, and logarithmically in three dimensions, in agreement with the exact behavior of an interacting Bose gas.
We study the effect of disorder on the superfluidity of the hard-sphere Bose gas in one dimension. This system is equivalent to the spin-(1/2 XY model with a random transverse field, which in turn can be mapped onto a disordered spinless-fermion model. We show that the localization of all fermionic states implies an exponential decay in the spin-spin correlation function and hence the instability of the superfluid against any amount of disorder. We point out a fundamental difference in the characteristics of the Jordan-Wigner transformation between the pure and disordered systems. Generalization of our results beyond the present model and implications to disordered superconductivity are discussed.