From quantum chaos and eigenstate thermalization to statistical mechanics and thermodynamics

Abstract

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. To this end, we present a brief overview of classical and quantum chaos, as well as random matrix theory 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.

Eigenstate thermalization hypothesis in quantum dimer models

“…Predictions for off-diagonal elements are also contained in Eq. (1), but an alternative perspective on this question involves the time dependence of observables after a quantum quench [2]. Our results here suggest that interesting behavior should occur at large V (around V /t 5), where ETH begins to break down, and we indeed observe signatures of slow relaxation and metastability at such parameters [36].…”

“…Considerable attention has recently been devoted to the question of whether and how an isolated quantum many-body system thermalizes [1][2][3]. At the center of the topic is the eigenstate thermalization hypothesis (ETH), which states that each energy eigenstate of a generic many-body Hamiltonian is indistinguishable from a microcanonical ensemble with the same energy [4] (see Sec.…”

Quench-induced entanglement and relaxation dynamics in Luttinger liquids

“…Here, very different fundamental aspects can be mentioned, including photo-induced biological processes [1, 2], formation of strongly-correlated bound-states [3,4], quantum phase transitions [5][6][7], relaxation and equilibration dynamics [8][9][10]. Among all, the recent technological developments in atomic and condensed matter physics offer a very promising platform to investigate these important issues.…”

Multipoint entanglement in disordered systems

“…One prominent result is the (LiebRobinson) linear in-time growth of entanglement in ballistic and diffusive systems 12 . By contrast, there exist localized interacting many-body systems, known as "manybody localized" phases [13][14][15][16][17][18][19][20][21][22] , whose subsystems' entanglement grows only logarithmically in time 16,[23][24][25][26] .…”