Equilibration times in clean and noisy systems

Abstract

We study the equilibration dynamics of closed finite quantum systems and address the question of the time needed for the system to equilibrate. In particular, we focus on the scaling of the equilibration time ${T}_{\mathrm{eq}}$ with the system size $L$. For clean systems, we give general arguments predicting ${T}_{\mathrm{eq}}=O({L}^{0})$ for clustering initial states, while for small quenches around a critical point we find ${T}_{\mathrm{eq}}=O({L}^{\ensuremath{\zeta}})$ where $\ensuremath{\zeta}$ is the dynamical critical exponent. We then analyze noisy systems where exponentially large time scales are known to exist. Specifically, we consider the tight-binding model with diagonal impurities and give numerical evidence that in this case ${T}_{\mathrm{eq}}\ensuremath{\sim}B{e}^{C{L}^{\ensuremath{\psi}}}$ where $B,C,\phantom{\rule{0.16em}{0ex}}\ensuremath{\psi}$ are observable-dependent constants. Finally, we consider another noisy system whose evolution dynamics is randomly sampled from a circular unitary ensemble. Here, we are able to prove analytically that ${T}_{\mathrm{eq}}=O(1)$, thus showing that noise alone is not sufficient for slow equilibration dynamics.