From quantum chaos and eigenstate thermalization to statistical mechanics and thermodynamics

By Luca D'Alessio, Yariv Kafri, A. Polkovnikov, Marcos Rigol. 2015

2,356 citations recorded by the source corpus.

This review gives a pedagogical introduction to the eigenstate thermalization hypothesis (ETH), its basis, and its implications to statistical mechanics and thermodynamics. In the first part, ETH is introduced as a natural extension of ideas from quantum chaos and random matrix theory (RMT). To this end, we present a brief overview of classical and quantum chaos, as well as RMT and some of its most important predictions. The latter include the statistics of energy levels, eigenstate components, and matrix elements of observables. Building on these, we introduce the ETH and show that it allows one to describe thermalization in isolated chaotic systems without invoking the notion of an external bath. We examine numerical evidence of eigenstate thermalization from studies of many-body lattice systems. We also introduce the concept of a quench as a means of taking isolated systems out of equilibrium, and discuss results of numerical experiments on quantum quenches. The second part of the review explores the implications of quantum chaos and ETH to thermodynamics. Basic thermodynamic relations are derived, including the second law of thermodynamics, the fundamental thermodynamic relation, fluctuation theorems, the fluctuation–dissipation relation, and the Einstein and Onsager relations. In particular, it is shown that quantum chaos allows one to prove these relations for individual Hamiltonian eigenstates and thus extend them to arbitrary stationary statistical ensembles. In some cases, it is possible to extend their regimes of applicability beyond the standard thermal equilibrium domain. We then show how one can use these relations to obtain nontrivial universal energy distributions in continuously driven systems. At the end of the review, we briefly discuss the relaxation dynamics and description after relaxation of integrable quantum systems, for which ETH is violated. We present results from numerical experiments and analytical studies of quantum quenches at integrability. We introduce the concept of the generalized Gibbs ensemble and discuss its connection with ideas of prethermalization in weakly interacting systems.

Subfields: Atomic and Molecular Physics, and Optics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/cba6ea25e55a1f53a5eb8b7c8bd16ca1855707a9

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Atomic and Molecular Physics, and Optics

1. A. Nahum
2. V. Khemani
3. Pasquale Calabrese
4. J. Cardy
5. Matthew Fisher
6. M. Fagotti
7. B. Doyon
8. F. Essler

In Statistical and Nonlinear Physics

1. Pasquale Calabrese
2. J. Cardy
3. B. Doyon
4. V. Khemani
5. L. Vidmar
6. M. Fagotti
7. Juan P. Garrahan
8. F. Essler

Citations in the corpus

• Generalized Gibbs ensemble in integrable lattice models (2016, background): “A system is said to exhibit eigenstate thermalization if eigenstates with close energies have similar expectation values of observables (with deviations that decrease exponentially with increasing system size [112]).”
• Aspects of non-equilibrium in classical and quantum systems: Slow relaxation and glasses, dynamical large deviations, quantum non-ergodicity, and open quantum dynamics (2017, background): “lly constrained models, Refs. [8, 9]; for large deviations, Refs. [10, 11, 12]. In the case of quantum systems, comprehensive recent reviews on thermalisation and many-body localisation include Refs. [13, 14, 15]; and on open quantum systems, Refs. [16, 17, 18, 19]. The selection of topics, many of ”
• Quench dynamics and relaxation in isolated integrable quantum spin chains (2016, background): “In the literature a dierent definition of relaxation after quantum quenches in finite systems is widely used [118, 121].”
• Random statistics of OPE coefficients and Euclidean wormholes (2020, background): “[1, 2]) to the fact that typical states thermalize (see [3, 4] for reviews).”
• Ergodicity Breaking Arising from Hilbert Space Fragmentation in Dipole-Conserving Hamiltonians (2019, background): “While no proof of ETH exists, there are many cases where it has been shown numerically that indeed all eigenstates satisfy this hypothesis [7, 11, 12].”