Eigenstate Thermalization Hypothesis for Wigner Matrices

TL;DR

This paper rigorously proves the Eigenstate Thermalization Hypothesis for Wigner matrices, showing that deterministic matrices become approximately the identity in the eigenbasis of large random matrices, confirming a key aspect of quantum chaos.

Contribution

It establishes the strong form of Quantum Unique Ergodicity with optimal convergence rates for all eigenvectors of Wigner matrices, extending previous probabilistic results.

Findings

Verification of ETH for Wigner matrices

Optimal convergence rate for eigenvector ergodicity

Generalization of previous QUE results

Abstract

We prove that any deterministic matrix is approximately the identity in the eigenbasis of a large random Wigner matrix with very high probability and with an optimal error inversely proportional to the square root of the dimension. Our theorem thus rigorously verifies the Eigenstate Thermalization Hypothesis by Deutsch [Deutsch 1991] for the simplest chaotic quantum system, the Wigner ensemble. In mathematical terms, we prove the strong form of Quantum Unique Ergodicity (QUE) with an optimal convergence rate for all eigenvectors simultaneously, generalizing previous probabilistic QUE results in [Bourgade, Yau 2017] and [Bourgade, Yau, Yin 2020].