Absence of thermalization in a Fermi liquid

We study a weak interaction quench in a three-dimensional Fermi gas. We first show that, under some general assumptions on time-dependent perturbation theory, the perturbative expansion of the long-wavelength structure factor S(q) is not compatible with the hypothesis that steady-state averages correspond to thermal ones. In particular, S(q) does develop an analytical component similar to const + O(q(2)) at q -> 0, as implied by thermalization, but, in contrast, it maintains a nonanalytic part similar to vertical bar q vertical bar characteristic of a Fermi liquid at zero-temperature. In real space, this nonanalyticity corresponds to persisting power-law decaying density-density correlations, whereas thermalization would predict only an exponential decay. We next consider the case of a dilute gas, where one can obtain nonperturbative results in the interaction strength but at lowest order in the density. We find that in the steady state the momentum distribution jump at the Fermi surface remains finite, though smaller than in equilibrium, up to second order in k(F) f(0), where f(0) is the scattering length of two particles in the vacuum. Both results question the emergence of a finite length scale in the quench dynamics as expected by thermalization.