Prethermalisation and Thermalisation in the Entanglement Dynamics

TL;DR

This paper studies how entanglement entropy in nearly integrable quantum systems relaxes from a generalized Gibbs ensemble to a thermal state, revealing a two-step process with a slow drift governed by weak interactions.

Contribution

It introduces a modified quasiparticle picture to quantitatively describe entanglement dynamics during the crossover from GGE to GE in weakly interacting systems.

Findings

Initial entanglement follows integrable evolution and approaches GGE.

A slow drift from GGE to GE occurs on timescales proportional to the inverse square of the interaction.

The quasiparticle picture accurately predicts late-time behavior and the initial slope of entanglement entropy.

Abstract

We investigate the crossover of the entanglement entropy towards its thermal value in nearly integrable systems. We employ equation of motion techniques to study the entanglement dynamics in a lattice model of weakly interacting spinless fermions after a quantum quench. For weak enough interactions we observe a two-step relaxation of the entanglement entropies of finite subsystems. Initially the entropies follow a nearly integrable evolution, approaching the value predicted by the Generalized Gibbs Ensemble (GGE) of the unperturbed model. Then, they start a slow drift towards the thermal stationary value described by a standard Gibbs Ensemble (GE). While the initial relaxation to the GGE is independent of the interaction, the slow drift from GGE to GE values happens on time scales proportional to the inverse interaction squared. For asymptotically large times and subsystem sizes the dynamics of the entropies can be predicted using a modified quasiparticle picture that keeps track of the evolution of the fermionic occupations caused by the integrability breaking. This picture gives a quantitative description of the results as long as the integrability-breaking timescale is much larger than the one associated with the (quasi) saturation to the GGE. In the opposite limit the quasiparticle picture still provides the correct late-time behaviour, but it underestimates the initial slope of the entanglement entropy.