Equilibration, thermalisation, and the emergence of statistical mechanics in closed quantum systems

Abstract

We review selected advances in the theoretical understanding of complex quantum many-body systems with regard to emergent notions of quantum statistical mechanics. We cover topics such as equilibration and thermalisation in pure state statistical mechanics, the eigenstate thermalisation hypothesis, the equivalence of ensembles, non-equilibration dynamics following global and local quenches as well as ramps. We also address initial state independence, absence of thermalisation, and many-body localisation. We elucidate the role played by key concepts for these phenomena, such as Lieb-Robinson bounds, entanglement growth, typicality arguments, quantum maximum entropy principles and the generalised Gibbs ensembles, and quantum (non-)integrability. We put emphasis on rigorous approaches and present the most important results in a unified language.

Black holes as random particles: entanglement dynamics in infinite range and matrix models

“…It is a gaussian system of coupled fermionic oscillators in which the mass matrix belongs to the Gaussian Orthogonal Ensemble of random matrices. 15 This system was described in [4], as an analytical example of the Eigenstate Thermalization Hypothesis [15][16][17]. The advantages of choosing such a model are the following:…”

“…We thank an anonymous referee for pointing this aspect to us. 4 See [16] for an earlier discovery of ETH and [17] for an extensive review of the subject of quantum thermalization.…”

“…It was then more rigorously (and abstractly) defined by the 'Page's test' [35], as the time by which subsystems are maximally entangled with the environment, see [5,28,36]. The logic behind this last definition is well known in the context of quantum thermalization, see the recent review [17]. If we choose a generic subsystem A smaller than half of the system, with reduced density matrix ρ A , the non-equilibrium unitary process will drive this reduced state towards the thermal distribution at temperature β.…”

Equilibration via Gaussification in Fermionic Lattice Systems

“…The argument gives rise to rigorously proven instances of a convergence to a generalized Gibbs ensemble. Our results allow to develop an intuition of equilibration that is expected to be more generally valid and relates to current experiments of cold atoms in optical lattices.Despite the great complexity of quantum many-body systems out-of-equilibrium, local expectation values in such systems show the remarkable tendency to equilibrate to stationary values that do not depend on the microscopic details of the initial state, but rather can be described with few parameters using thermal states or generalized Gibbs ensembles [1][2][3]. Such behavior has been successfully studied in many settings theoretically and experimentally, most notably in instances of quantum simulations in optical lattices [2,4,5].…”

Many-body localization phase in a spin-driven chiral multiferroic chain

“…For integrable systems this issue has long been in the focus of research with many interesting findings (cf., for instance, Refs. [20][21][22][23][24][25][26][27]). Generically, integrable systems with a large number of degrees of freedom turn chaotic for a weak perturbation.…”