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 $S\left(\mathbf{q}\right)$ is not compatible with the hypothesis that steady-state averages correspond to thermal ones. In particular, $S\left(\mathbf{q}\right)$ does develop an analytical component $\sim \mathrm{const}+O\left(q^2\right)$ at $\mathbf{q}\to \mathbf{0}$, as implied by thermalization, but, in contrast, it maintains a nonanalytic part $\sim \left|\mathbf{q}\right|$ 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.

Figure 1

Scattering processes that refer to Eq. (B16b). The circle represents the Fermi sphere, open dots holes and solid ones particles. The initial particle-hole pairs in $|n\rangle$ are encircled and have opposite spins. The scattering process, i.e., the motion of particles/holes indicated by the arrows, leads to a new state $|m\rangle$.

Figure 2

Same as Fig. 1 but for initial pairs that have the same spin.

Figure 3

Scattering processes that refer to Eq. (B16c) and are produced by $n_{\mathbf{q}\sigma }{n}_{-\mathbf{q}-\sigma }$. The sign of the process is indicated and the initial particle-hole pairs in $|n\rangle$ are encircled.

Figure 4

Scattering processes that refer to Eq. (B16c) and are produced by $n_{\mathbf{q}\sigma }{n}_{-\mathbf{q}\sigma }$. The sign of the process is indicated and the initial particle-hole pairs in $|n\rangle$ are encircled.

Figure 5

Scattering processes that refer to Eq. (B16c) and are produced by $n_{\mathbf{q}\sigma }{n}_{-\mathbf{q}\sigma }$ in the case in which $|n\rangle =|m\rangle$. The sign of the process is indicated and the initial particle-hole pairs in $|n\rangle$ are encircled.