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arXiv:2206.05310 · Physical Review Letters

Non-Abelian Eigenstate Thermalization Hypothesis

Chaitanya Murthy, Arman Babakhani, Fernando Iniguez, Mark Srednicki, Nicole Yunger Halpern

Published Apr 6, 2023• 48 citations
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Abstract

The eigenstate thermalization hypothesis (ETH) explains why nonintegrable quantum many-body systems thermalize internally if the Hamiltonian lacks symmetries. If the Hamiltonian conserves one quantity ("charge"), the ETH implies thermalization within a charge sector-in a microcanonical subspace. But quantum systems can have charges that fail to commute with each other and so share no eigenbasis; microcanonical subspaces may not exist. Furthermore, the Hamiltonian will have degeneracies, so the ETH need not imply thermalization. We adapt the ETH to noncommuting charges by positing a non-Abelian ETH and invoking the approximate microcanonical subspace introduced in quantum thermodynamics. Illustrating with SU(2) symmetry, we apply the non-Abelian ETH in calculating local operators' time-averaged and thermal expectation values. In many cases, we prove, the time average thermalizes. However, we find cases in which, under a physically reasonable assumption, the time average converges to the thermal average unusually slowly as a function of the global-system size. This work extends the ETH, a cornerstone of many-body physics, to noncommuting charges, recently a subject of intense activity in quantum thermodynamics.

DOI
10.1103/physrevlett.130.140402

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The eigenstate thermalization hypothesis (ETH) explains why nonintegrable quantum many-body systems thermalize internally if the Hamiltonian lacks symmetries. If the Hamiltonian conserves one quantity ("charge"), the ETH implies thermalization within a charge sector-in a microcanonical subspace. But quantum systems can have charges that fail to commute with each other and so share no eigenbasis; microcanonical subspaces may not exist. Furthermore, the Hamiltonian will have degeneracies, so the ETH need not imply thermalization. We adapt the ETH to noncommuting charges by positing a non-Abelian ETH and invoking the approximate microcanonical subspace introduced in quantum thermodynamics. Illustrating with SU(2) symmetry, we apply the non-Abelian ETH in calculating local operators' time-averaged and thermal expectation values. In many cases, we prove, the time average thermalizes. However, we find cases in which, under a physically reasonable assumption, the time average converges to the thermal average unusually slowly as a function of the global-system size. This work extends the ETH, a cornerstone of many-body physics, to noncommuting charges, recently a subject of intense activity in quantum thermodynamics.
The eigenstate thermalization hypothesis (ETH) explains how generic quantum many-body systems thermalize internally. It implies that local operators' time-averaged expectation values approximately equal their thermal expectation values, regardless of microscopic details. The ETH's range of applicability therefore impacts theory and experiments. Murthy et al. [Phys. Rev. Lett. 130, 140402 (2023)0031-900710.1103/PhysRevLett.130.140402] recently showed that non-Abelian symmetries conflict with the ETH. Such symmetries have excited interest in quantum thermodynamics lately, as they are equivalent to conserved quantities that fail to commute with each other and noncommutation is a quintessentially quantum phenomenon. Murthy et al. proposed a non-Abelian ETH, which we support numerically. The numerics model a one-dimensional next-nearest-neighbor Heisenberg chain of 18 qubits. We represent local operators with matrices relative to an energy eigenbasis. The matrices bear out seven predictions of the non-Abelian ETH. We also prove analytically that the non-Abelian ETH exhibits a self-consistency property. The proof relies on a thermodynamic-entropy definition different from that of Murthy et al. This work initiates the observation and application of the non-Abelian ETH.
The eigenstate thermalization hypothesis (ETH) posits that energy eigenstates encode local properties of the microcanonical ensemble. Motivated by recent interest in the physics of non-commuting conserved charges and the non-Abelian ETH, we study chaotic eigenstates in the presence of symmetries described by general compact Lie groups, such as SU(2). By applying non-Abelian symmetry resolution, we develop a non-Abelian microcanonical entropy and relate this entropy to the entanglement entropy of chaotic eigenstates. We find that microcanonical entropy is closely related to the symmetry-resolved entanglement entropy, which differs from conventional entanglement entropy by a universal logarithmic correction. Our results depend on the global Casimir charge, e.g. total spin. At finite charge density, we find a logarithmic enhancement to conventional entanglement entropy. At zero density, we find no such correction to entanglement entropy, but a logarithmic reduction to microcanonical entropy and symmetry-resolved entanglement entropy. We discuss the implications of our approach for non-Abelian eigenstate thermalization.
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.
Studying noncommuting conserved quantities, or ‘charges,’ has revealed a conceptual puzzle: noncommuting charges hinder thermalization in some ways yet promote it in others. While many quantum systems thermalize according to the Eigenstate Thermalization Hypothesis (ETH), systems with ‘dynamical symmetries’ violate the ETH and exhibit non-stationary dynamics, preventing them from equilibrating, much less thermalizing. We demonstrate that each pair of dynamical symmetries corresponds to a specific charge. We find that introducing new charges that do not commute with existing ones disrupts these symmetries, thereby eliminating non-stationary dynamics and facilitating thermalization. We illustrate this behavior across various models, including the Hubbard model and Heisenberg spin chains. Our findings demonstrate that noncommuting charges can enhance thermalization by reducing the number of local observables that thermalize according to the ETH. Noncommuting charges pose a puzzle: they hinder thermalization in some ways yet promote it in others. Here, the authors demonstrate how noncommuting charges enhance thermalization by reducing the number of local observables that thermalize.
Quantum simulators have recently enabled experimental observations of the internal thermalization of quantum many-body systems.Often, the global energy and particle number are conserved and the system is prepared with a well-defined particle number-in a microcanonical subspace.However, quantum evolution can also conserve quantities, or charges, that fail to commute with each other.Noncommuting charges have recently emerged as a subfield at the intersection of quantum thermodynamics and quantum information.Until now, this subfield has remained theoretical.We initiate the experimental testing of its predictions, with a trapped-ion simulator.We prepare 6-21 spins in an approximate microcanonical subspace, a generalization of the microcanonical subspace for accommodating noncommuting charges, which cannot necessarily have well-defined nontrivial values simultaneously.We simulate a Heisenberg evolution using laser-induced entangling interactions and collective spin rotations.The noncommuting charges are the three spin components.We find that small subsystems equilibrate to near a recently predicted non-Abelian thermal state.This work bridges quantum many-body simulators to the quantum thermodynamics of noncommuting charges, the predictions of which can now be tested.