Statistical complexity and the road to equilibrium in many-body chaotic quantum systems

TL;DR

This paper investigates the process of equilibration in many-body quantum systems with chaotic Hamiltonians, revealing a hierarchy of diagonal ensembles and introducing statistical complexity to analyze the stability and uniqueness of the road to equilibrium.

Contribution

It provides an information-theoretic proof of the second law during equilibration and introduces the entropy-complexity plane to study the dynamics of quantum chaos.

Findings

Diagonal ensembles obey an emergent second law during equilibration

The road to equilibrium forms a hierarchy in time of diagonal ensembles

Entropy-complexity trajectories are stable under perturbations in chaotic systems

Abstract

In this work we revisit the problem of equilibration in isolated many-body interacting quantum systems. We pay particular attention to quantum chaotic Hamiltonians, and rather than focusing on the properties of the asymptotic states and how they adhere to the predictions of the Eigenstate Thermalization Hypothesis, we focus on the equilibration process itself, i.e., \emph{the road to equilibrium}. Along the road to equilibrium the diagonal ensembles obey an emergent form of the second law of thermodynamics and we provide an information theoretic proof of this fact. With this proof at hand we show that the road to equilibrium is nothing but a hierarchy in time of diagonal ensembles. Furthermore, introducing the notions of statistical complexity and the entropy-complexity plane, we investigate the uniqueness of the road to equilibrium in a generic many-body system by comparing its trajectories in the entropy-complexity plane to those generated by a random Hamiltonian. Finally by treating the random Hamiltonian as a perturbation we analyzed the stability of entropy-complexity trajectories associated with the road to equilibrium for a chaotic Hamiltonian and different types of initial states.