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arXiv:2403.05197

Generic ETH: Eigenstate Thermalization beyond the Microcanonical

Elena Cáceres, Stefan Eccles, Jason Pollack, Sarah Racz

Published Mar 8, 2024• 0 citations
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Abstract

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.'

DOI
10.48550/arxiv.2403.05197

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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.'
How does an isolated quantum system evolve toward thermal equilibrium from a far-from-equilibrium initial state-for instance, one prepared by a global quantum quench? This question has attracted significant interest in recent years. The Eigenstate Thermalization Hypothesis (ETH) provides a compelling framework for addressing it, with far-reaching implications from high-energy physics to condensed matter. This primer offers a pedagogical introduction to quantum equilibrium and ETH for a broad physics audience, particularly high-energy physicists seeking a comprehensive understanding of these topics. We assume only basic knowledge of quantum mechanics and statistical mechanics.
The eigenstate thermalization hypothesis (ETH), which asserts that every eigenstate of a many-body quantum system is indistinguishable from a thermal ensemble, plays a pivotal role in understanding thermalization of isolated quantum systems. Yet, no evidence has been obtained as to whether the ETH holds for all few-body operators in a chaotic system; such few-body operators include key quantities in statistical mechanics, such as the total magnetization, the momentum distributions, and their low-order thermal and quantum fluctuations. Here, we formulate a conjecture that for a generic nonintegrable system the ETH holds simultaneously for all mmm-body operators with m<α∗Nm < α_{\ast} Nm<α∗​N in the thermodynamic limit for some nonzero constant α∗>0α_{\ast} > 0α∗​>0. We first show the existence of such nontrivial constants for idealized (pseudo) random-matrix descriptions of many-body eigenstates. We then verify the conjecture for generic spin, Bose, and Fermi systems with local and few-body interactions by large-scale numerical calculations. Our results imply that generic systems satisfy the ETH simultaneously for all few-body operators, including their thermal and quantum fluctuations.
A very fundamental problem in quantum statistical mechanics involves whether—and how—an isolated quantum system will reach thermal equilibrium after waiting a long time. In quantum systems that do thermalize, the long-time expectation value of any “reasonable” operator will match its predicted value in the canonical ensemble. The Eigenstate Thermalization Hypothesis (ETH) posits that this thermalization occurs at the level of each individual energy eigenstate; in fact, any single eigenstate in a microcanonical energy window will predict the expectation values of such operators exactly. In the first part of this dissertation, we identify, for a generic model system, precisely which operators satisfy ETH, as well as limits to the information contained in a single eigenstate. Remarkably, our results strongly suggest that a single eigenstate can contain information about energy densities—and therefore temperatures—far away from the energy density of the eigenstate.Additionally, we study the possible breakdown of quantum thermalization in a model of itinerant electrons on a one-dimensional chain, with both spin and charge degrees of freedom. This model exhibits peculiar properties in the entanglement entropy, the apparent scaling of which is modified from a “volume law” to an “area law” after performing a partial, site-wise measurement on the system. These properties and others suggest that this model realizes a new, non-thermal phase of matter, known as a Quantum Disentangled Liquid (QDL). The putative existence of this phase has striking implications for the foundations of quantum statistical mechanics.
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.
Identifying the general mechanics behind the equilibration of a complex isolated quantum system towards a state described by only a few parameters has been the focus of attention in non-equilibrium thermodynamics. And several experimentally unproven conjectures are proposed for the statistical description of quantum (non-)integrable models. The plausible eigenstate thermalization hypothesis (ETH), which suggests that each energy eigenstate itself is thermal, plays a crucial role in understanding the quantum thermalization in non-integrable systems; it is commonly believed that it does not exist in integrable systems. Nevertheless, integrable systems can still relax to the generalized Gibbs ensemble. From a microscopic perspective, understanding the origin of this generalized thermalization that occurs in an isolated integrable system is a fundamental open question lacking experimental investigations. Herein, we experimentally investigated the spin subsystem relaxation in an isolated spin-orbit coupling quantum system. By applying the quantum state engineering technique, we initialized the system with various distribution widths in the mutual eigenbasis of the conserved quantities. Then, we compared the steady state of the spin subsystem reached in a long-time coherent dynamics to the prediction of a generalized version of ETH and the underlying mechanism of the generalized thermalization is experimentally verified for the first time. Our results facilitate understanding the origin of quantum statistical mechanics.