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arXiv:1811.03114 · Physical review. B./Physical review. B

Extension of the eigenstate thermalization hypothesis to nonequilibrium steady states

Sanjay Moudgalya, Trithep Devakul, Daniel P. Arovas, Shivaji L. Sondhi

Published Jul 9, 2019• 8 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physrevb.100.045112

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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.
We investigate steady states of macroscopic quantum systems under dissipation not obeying the detailed balance condition. We argue that the Gibbs state at an effective temperature gives a good description of the steady state provided that the system Hamiltonian obeys the eigenstate thermalization hypothesis (ETH) and the perturbation theory in the weak system-environment coupling is valid in the thermodynamic limit. We derive a criterion to guarantee the validity of the perturbation theory, which is satisfied in the thermodynamic limit for sufficiently weak dissipation when the Liouvillian is gapped for bulk-dissipated systems, while the perturbation theory breaks down in boundary-dissipated chaotic systems due to the presence of diffusive transports. We numerically confirm these theoretical predictions. This paper suggests a connection between steady states of macroscopic open quantum systems and the ETH.
We use the eigenstate thermalization hypothesis to derive a quantum master equation for a system that is weakly coupled to a chaotic finite-sized bath prepared in a pure state. We show that the emergence of Markovianity is controlled by the spectral function of the ETH and that local detailed balance emerges in the Markovian regime for a broad class of pure bath states. We numerically verify this result by comparing the master equation to dynamics computed using exact diagonalization of a chaotic Hamiltonian. We also compare the master equation to exact dynamics for an integrable bath and find that at finite size they strongly disagree. Our work puts forward eigenstate thermalization as a foundation for open quantum systems theory, thus extending it beyond ensemble bath preparations to chaotic many-body environments in generic pure states.
Deriving conditions under which a macroscopic system thermalizes directly from the underlying quantum many-body dynamics of its microscopic constituents is a long-standing challenge in theoretical physics. The well-known eigenstate thermalization hypothesis (ETH) is presumed to be a key mechanism, but has defied rigorous verification for generic systems thus far. A weaker variant (weak ETH), by contrast, is provably true for a large variety of systems, including even many integrable models, but its implications with respect to the problem of thermalization are still largely unexplored. Here we analytically demonstrate that systems satisfying the weak ETH exhibit thermalization for two very natural classes of far-from-equilibrium initial conditions: the overwhelming majority of all pure states with a preset nonequilibrium expectation value of some given local observable, and the Gibbs states of a Hamiltonian which subsequently is subject to a quantum quench in the form of a sudden change of some local system properties.
We ask whether the eigenstate thermalization hypothesis (ETH) is valid in a strong sense: in the limit of an infinite system, every eigenstate is thermal. We examine expectation values of few-body operators in highly excited many-body eigenstates and search for "outliers," the eigenstates that deviate the most from ETH. We use exact diagonalization of two one-dimensional nonintegrable models: a quantum Ising chain with transverse and longitudinal fields, and hard-core bosons at half-filling with nearest- and next-nearest-neighbor hopping and interaction. We show that even the most extreme outliers appear to obey ETH as the system size increases and thus provide numerical evidences that support ETH in this strong sense. Finally, periodically driving the Ising Hamiltonian, we show that the eigenstates of the corresponding Floquet operator obey ETH even more closely. We attribute this better thermalization to removing the constraint of conservation of the total energy.
Integrable systems do not obey the strong eigenstate thermalization hypothesis (ETH), which has been proposed as a mechanism of thermalization in isolated quantum systems. It has been suggested that an integrable system reaches a steady state described by a generalized Gibbs ensemble (GGE) instead of thermal equilibrium. We prove that a generalized version of the strong ETH holds for noninteracting integrable systems with translation invariance. Our generalized ETH states that any pair of energy eigenstates with similar values of local conserved quantities looks similar with respect to local observables, such as local correlations. This result tells us that an integrable system relaxes to a GGE for any initial state that has subextensive fluctuations of macroscopic local conserved quantities. Contrary to the previous derivations of the GGE, it is not necessary to assume the cluster decomposition property for an initial state.
The Lindblad description of an open quantum system gives rise to two types of integrability, since the nonequilibrium steady state can be integrable independently of the Liouvillian. Taking boundary-driven and dephasing spin chains as a representative example, we discriminate Liouvillian and steady-state chaos by combining level spacing statistics and an extension of the eigenstate thermalization hypothesis to open quantum systems. Moreover, we analyze the structure of the steady states by expanding it in the basis of Pauli strings and comparing the weight of strings of different lengths. We show that the natural expectation that integrable steady states are "simple" (i.e., built from few-body local operators) does not hold: the steady states of both chaotic and integrable models have relevant contributions coming from Pauli strings of all possible lengths, including long-range and many-body interactions. Nevertheless, we show that one can effectively use the operator-size distribution to distinguish chaotic and integrable steady states.