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 Thermalisation Hypothesis by Deutsch (Phys Rev A 43:2046–2049, 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 and Yau (Commun Math Phys 350:231–278, 2017) and Bourgade et al. (Commun Pure Appl Math 73:1526–1596, 2020).
Highlights
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
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 and Yau (Commun Math Phys 350:231–278, 2017) and Bourgade et al (Commun Pure Appl Math 73:1526–1596, 2020)
A fundamental phenomenon of such systems is Quantum Ergodicity (QE), stating that the eigenvectors tend to become uniformly distributed in the phase space
Summary
We use the spectral decomposition of G to estimate traces of products of many G s and A s by the lower degree term G AG A This requires to extract sufficiently many Λ-factors in the cumulant expansion, which we achieve by a subtle Feynman graph analysis to estimate all high moments of | W G AG A |. For the proof of (2) we manage to extract the asymptotic orthogonality effect between the eigenvectors ui and their complex conjugates ui optimally, resulting in the bound GGt 1, gaining a full power of η over e.g. GG∗ ∼ 1/η After this introduction and presenting the main results, we prove the local laws involving two resolvents in Sect. We use the convention that ξ > 0 denotes an arbitrary small constant which is independent of N
Sign-in/Register to view highlights & summary Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
