Theory of Eigenstate Thermalisation

TL;DR

This paper derives the eigenstate thermalization hypothesis (ETH) as a rigorous theory using random matrix theory, showing that large, non-integrable quantum systems naturally exhibit thermal behavior without relying on ergodicity or entropy concepts.

Contribution

It provides a theoretical derivation of ETH from first principles, connecting eigenvalue distributions and Dyson Brownian motion to quantum thermalization.

Findings

Eigenvalue distribution of interacting systems is Gaussian

ETH can be derived from random matrix theory

Thermodynamic equilibrium arises from quantum mechanics and non-integrability

Abstract

If we prepare an isolated, interacting quantum system in an eigenstate and perturb a local observable at an initial time, its expectation value will relax towards a thermal expectation value, even though the time evolution of the system is deterministic. The eigenstate thermalization hypothesis (ETH) of Deutsch and Srednicki suggests that this is possible because each eigenstate of the full quantum system acts as a thermal bath to its subsystems, such that the reduced density matrices of the subsystems resemble thermal density matrices. Here, we use the observation that the eigenvalue distribution of interacting quantum systems is a Gaussian under very general circumstances, and Dyson Brownian motion random matrix theory, to derive the ETH and thereby elevate it from hypothesis to theory. Our analysis provides a derivation of statistical mechanics which neither requires the concepts of ergodicity or typicality, nor that of entropy. Thermodynamic equilibrium follows solely from the applicability of quantum mechanics to large systems and the absence of integrability.