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New Journal of Physics

Eigenstate thermalization: Deutsch’s approach and beyond

Peter Reimann

Published May 27, 2015• 93 citations
Publisher

Abstract

The eigenstate thermalization hypothesis postulates that the energy eigenstates of an isolated manybody system are thermal, i.e., each of them already yields practically the same expectation values as the microcanonical ensemble at the same energy.Here, we review, compare, and extend some recent approaches to corroborate this hypothesis and discuss the implications for the system's equilibration and thermalization.

DOI
10.1088/1367-2630/17/5/055025

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The eigenstate thermalization hypothesis postulates that the energy eigenstates of an isolated manybody system are thermal, i.e., each of them already yields practically the same expectation values as the microcanonical ensemble at the same energy.Here, we review, compare, and extend some recent approaches to corroborate this hypothesis and discuss the implications for the system's equilibration and thermalization.
We consider the set of all initial states within a microcanonical energy shell of an isolated many-body quantum system, which exhibit an arbitrary but fixed nonequilibrium expectation value for some given observable A. On the condition that this set is not too small, it is shown by means of a dynamical typicality approach that most such initial states exhibit thermalization if and only if A satisfies the so-called weak eigenstate thermalization hypothesis (wETH). Here, thermalization means that the expectation value of A spends most of its time close to the microcanonical value after initial transients have died out. The wETH means that, within the energy shell, most eigenstates of the pertinent system Hamiltonian exhibit very similar expectation values of A.
The Eigenstate Thermalization Hypothesis (ETH) has played a key role in recent advances in the high energy and condensed matter communities. It explains how an isolated quantum system in a far-from-equilibrium initial state can evolve to a state that is indistinguishable from thermal equilibrium, with observables relaxing to almost time-independent results that can be described using traditional statistical mechanics ensembles. In this work we probe the limits of ETH, pushing it outside its prototypical applications in several directions. We design a qutrit lattice system with conserved quasilocal charge, in which we verify a form of generalized eigenstate thermalization. We also observe signatures of thermalization in states well outside microcanonical windows of both charge and energy, which we dub `generic ETH.'
The eigenstate thermalization hypothesis (ETH) states that for nonintegrable systems, each energy eigenstate accurately gives microcanonical expectation values for a class of observables. In this paper, we explore the ETH in terms of the time-energy uncertainty and an intrinsic thermal nature shared by the majority of quasi eigenstates of operationally defined 'time operator': First, we show that the energy eigenstates are superposition of uncountably many quasi eigenstates of suitably defined 'time operator'. Majority of such quasi eigenstates are thermal for thermodynamic isolated quantum many-body systems and approximately orthogonal in terms of extremely short relaxation time of the fidelity. In this manner, our scenario provides a theoretical explanation of ETH.