Subsystem Evolution Speed as Indicator of Relaxation

TL;DR

This paper introduces a new method to detect relaxation in quantum systems by measuring the evolution speed of subsystems, which diminishes as the system relaxes, aligning with eigenstate thermalization predictions.

Contribution

It proposes a novel indicator based on subsystem evolution speed to assess relaxation directly from the state, applicable across various quantum models.

Findings

Subsystem evolution speed decreases with system size in relaxing systems.

The method accurately distinguishes between localized and thermalizing phases.

Results align with eigenstate thermalization hypothesis predictions.

Abstract

In studying the time evolution of isolated many-body quantum systems, a key focus is determining whether the system undergoes relaxation and reaches a steady state at a given point in time. Traditional approaches often rely on specific local operators or a detailed understanding of the stationary state. In this letter, we introduce an alternative method that assesses relaxation directly from the time-dependent state by focusing on the evolution speed of the subsystem. The proposed indicator evaluates the rate of change in the reduced density matrix of the subsystem over time. We demonstrate that in systems reaching relaxation, as the overall system size increases, the evolution speed of sufficiently small yet still finite-sized subsystems notably diminishes. This leads to small fluctuations in the expectation values of operators, which is also consistent with the predictions made by the eigenstate thermalization hypothesis. We apply this approach across various models, including the chaotic Ising chain, XXZ chains with and without many-body localization, and the transverse field Ising chain. Our results confirm the robustness and accuracy of subsystem evolution speed as a reliable indicator for relaxation.