A Physical Theory of Two-stage Thermalization

TL;DR

This paper develops an entanglement membrane theory to explain the two-stage exponential decay of purity during thermalization in local quantum circuits, linking decay rates to domain wall free energy and magnon dynamics.

Contribution

It introduces an entanglement membrane framework that interprets two-stage thermalization and decay rates, revealing how circuit geometry influences these processes.

Findings

Decay rate r1 corresponds to domain wall free energy.

Circuit geometry can produce a phantom eigenvalue.

Measuring local correlations can reveal entanglement growth.

Abstract

One indication of thermalization time is subsystem entanglement reaching thermal values. Recent studies on local quantum circuits reveal two exponential stages with decay rates $r_1$ and $r_2$ of the purity before and after thermalization. We provide an entanglement membrane theory interpretation, with $r_1$ corresponding to the domain wall free energy. Circuit geometry can lead to $r_1 < r_2$, producing a ``phantom eigenvalue". Competition between the domain wall and magnon leads to $r_2 < r_1$ when the magnon prevails. However, when the domain wall wins, this mechanism provides a practical approach for measuring entanglement growth through local correlation functions.