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Checking the validity of the eigenstate thermalization hypothesis by applying operator exponentials to pure states

Robin Steinigeweg, Hendrik Niemeyer, Christian Gogolin, Jochen Gemmer B

Published Nov 1, 2013• 0 citations
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Extended Eigenstate Thermalization and the role of FZZT branes in the Schwarzian theory

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In this paper we provide a universal description of the behavior of the basic operators of the Schwarzian theory in pure states. When the pure states are energy eigenstates, expectation values of non-extensive operators are thermal. On the other hand, in coherent pure states, these same operators can exhibit ergodic or non-ergodic behavior, which is characterized by elliptic, parabolic or hyperbolic monodromy of an auxiliary equation; or equivalently, which coadjoint Virasoro orbit the state lies on. These results allow us to establish an extended version of the eigenstate thermalization hypothesis (ETH) in theories with a Schwarzian sector. We also elucidate the role of FZZT-type boundary conditions in the Schwarzian theory, shedding light on the physics of microstates associated with ZZ branes and FZZT branes in low dimensional holography.

•Jan 1, 2020•22 citations
We extend the notion of the eigenstate thermalization hypothesis (ETH) to open quantum systems governed by the Gorini-Kossakowski-Lindblad-Sudarshan (GKLS) master equation. We present evidence that the eigenstates of nonequilibrium steady-state (NESS) density matrices obey a generalization of ETH in boundary-driven systems when the bulk Hamiltonian is nonintegrable, just as eigenstates of Gibbs density matrices are conjectured to do in equilibrium. This generalized ETH, which we call NESS ETH, can be used to obtain representative pure states that reproduce the expectation values of few-body operators in the NESS. The density matrices of these representative pure states can be further interpreted as weak solutions of the GKLS master equation. Additionally, we explore the validity and breakdown of NESS-ETH in the presence of symmetries, integrability, and many-body localization in the bulk Hamiltonian.
Understanding the evolution towards thermal equilibrium of an isolated quantum system is at the foundation of statistical mechanics and a subject of interest in such diverse areas as cold atom physics or the quantum mechanics of black holes. Since a pure state can never evolve into a thermal density matrix, the Eigenstate Thermalization Hypothesis (ETH) has been put forward by Deutsch and Srednicki as a way to explain this apparent thermalization, similarly to what the ergodic theorem does in classical mechanics. In this paper this hypothesis is tested numerically. First, it is observed that thermalization happens in a subspace of states (the Krylov subspace) with dimension much smaller than that of the total Hilbert space. We check numerically the validity of ETH in such a subspace, for a system of hard core bosons on a two-dimensional lattice. We then discuss how well the eigenstates of the Hamiltonian projected on the Krylov subspace represent the true eigenstates. This discussion is aided by bringing the projected Hamiltonian to the tridiagonal form and interpreting it as an Anderson localization problem for a finite one-dimensional chain. We also consider thermalization of a subsystem and argue that generation of a large entanglement entropy can lead to a thermal density matrix for the subsystem well before the whole system thermalizes. Finally, we comment on possible implications of ETH in quantum gravity.
In this paper we provide a universal description of the behavior of the basic operators of the Schwarzian theory in pure states. When the pure states are energy eigenstates, expectation values of non-extensive operators are thermal. On the other hand, in coherent pure states, these same operators can exhibit ergodic or non-ergodic behavior, which is characterized by elliptic, parabolic or hyperbolic monodromy of an auxiliary equation; or equivalently, which coadjoint Virasoro orbit the state lies on. These results allow us to establish an extended version of the eigenstate thermalization hypothesis (ETH) in theories with a Schwarzian sector. We also elucidate the role of FZZT-type boundary conditions in the Schwarzian theory, shedding light on the physics of microstates associated with ZZ branes and FZZT branes in low dimensional holography.