Prethermalization, thermalization, and Fermi's golden rule in quantum many-body systems

TL;DR

This paper investigates the dynamics of prethermalization and thermalization in weakly perturbed nonintegrable quantum systems, revealing a two-step relaxation process and the role of Fermi's golden rule in slow thermalization.

Contribution

It demonstrates that slow thermalization follows a Fermi golden rule rate and that observables can be effectively described by projected ensembles during this process.

Findings

Fast prethermal dynamics followed by slow thermalization.

Thermalization rate scales as g^2, consistent with Fermi's golden rule.

Projected ensembles accurately describe observables during slow dynamics.

Abstract

We study the prethermalization and thermalization dynamics of local observables in weakly perturbed nonintegrable systems, with Hamiltonians of the form $\hat{H}_0+g\hat{V}$, where $\hat{H}_0$ is nonintegrable and $g\hat{V}$ is a perturbation. We explore the dynamics of far from equilibrium initial states in the thermodynamic limit using a numerical linked cluster expansion (NLCE), and in finite systems with periodic boundaries using exact diagonalization. We argue that generic observables exhibit a two-step relaxation process, with a fast prethermal dynamics followed by a slow thermalizing one, only if the perturbation breaks a conserved quantity of $\hat{H}_0$ and if the value of the conserved quantity in the initial state is $\mathcal{O}(1)$ different from the one after thermalization. We show that the slow thermalizing dynamics is characterized by a rate $\propto g^2$, which can be accurately determined using a Fermi golden rule (FGR) equation. We also show that during such a slow dynamics, observables can be described using projected diagonal and Gibbs ensembles, and we contrast their accuracy.