Jeudi 8 décembre 2022
This is a colloquium in Physics and Mathematics, organised biannually in Strasbourg by a group of scientists from IRMA, ISIS and IPCMS.
This fourth session is devoted to Eigenstate Thermalization Hypothesis.
Organizers : N. Anantharaman, R. Côte, S. Klevtsov, Y. Le Floch, C. Tauber, M. Vogel, X. Zeng (IRMA) — G. Pupillo, J. Schachenmayer (ISIS) — P.-A. Hervieux, R. Jalabert, G. Manfredi, G. Weick, D. Weinmann (IPCMS)
- László Erdös (IST, Austria)
- Rodolfo Jalabert (IPCMS, Strasbourg)
- Silvia Pappalardi (ENS, Paris)
Location : Salle de conférences IRMA
For practical and other questions please contact the organizers.
Jeudi 8 décembre 2022
Silvia Pappalardi, ENS, Paris
Eigenstate thermalization hypothesis and free probabilityThe general form of the Eigenstate Thermalization Hypothesis (ETH), describing all the relevant correlations of matrix elements, may be understood on the basis of a `typicality' argument of invariance with respect to local rotations involving nearby energy levels. In this talk, I will discuss the close relation between ETH and Free Probability theory, as applied to a thermal ensemble or an energy shell. This mathematical framework allows one to express in an unambiguous way high order connected correlation functions (here identified as free cumulants) in terms of standard correlation functions. This perspective naturally incorporates the consistency property that local functions of ETH operators also satisfy ETH. The present results open a direct connection between quantum thermalization and the mathematical structure of Free Probability, thus offering the basis for insightful analogies and new developments. S. Pappalardi, L. Foini and J. Kurchan - Phys. Rev. Lett 129, 18-603 (2022).
Déjeuner au 32
Rodolfo Jalabert, IPCMS, Strasbourg
OTOC, ETH, MBL, other acronyms, and their rich interconnectionsQuantum Chaos has originally emerged as the field which studies how the properties of classical chaotic systems arise in their quantum counterparts. The growing interest on quantum many-body systems, with no obvious classical meaning, has led to consider time-dependent quantities that can help to characterize and redefine Quantum Chaos. Among them, the Out of Time Ordered Correlator (OTOC) is particularly useful, in view of its universal properties and its connection with other Quantum Chaos indicators, like the level statistics, the Loschmidt echo (LE), the Many-Body Localization (MBL), and the eigenstate thermalization hypothesis (ETH). Semiclassical approaches and numerical calculations of the OTOC are presented in order to understand the signatures of chaos in short and long-time scales.
László Erdös, IST, Austria
Rank-uniform local law and quantum unique ergodicity for Wigner matricesWe consider quadratic forms of any deterministic observable on the eigenvectors of an N by N Wigner matrix and show that it has a Gaussian fluctuation in the large N limit. Viewing the Wigner matrix as a chaotic quantum Hamiltonian, the convergence of this quadratic form to its mean value is the analogue of the celebrated quantum unique ergodicity and our result reveals even its fluctuation. The proof is a combination of Dyson Brownian motion for eigenvectors and a new general local law for Wigner matrices that optimally handles observables of arbitrary rank.