Eigenstate Thermalization Hypothesis for Wigner Matrices

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].

arXiv comment39 pages. Added remark on choice of renormalisation

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• Quantitative CLT for linear eigenvalue statistics of Wigner matrices
• Equipartition principle for Wigner matrices
• Eigenvectors of the square grid plus GUE
• Benchmarking Quantum Simulators