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arXiv:2310.03053 · Journal of Physics A Mathematical and Theoretical

Thermalization of closed chaotic many-body quantum systems

Hans A. Weidenmüller

Published Mar 27, 2024• 3 citations
arXivHTMLPDF

Abstract

Abstract We investigate thermalization of a closed chaotic many-body quantum system by combining the Hartree–Fock approach with the Bohigas–Giannoni–Schmit conjecture. The conjecture implies that locally, the residual interaction causes the statistics of eigenvalues and eigenfunctions of the full Hamiltonian to agree with random-matrix predictions. The agreement is confined to an interval Δ (the correlation width). The results are used to calculate Tr ( A ρ ( t ) ) . Here ρ ( t ) is the time-dependent density matrix of the system, and A represents an observable. In the semiclassical regime, the average ⟨ Tr ( A ρ ( t ) ) ⟩ decays on the time scale ℏ / Δ toward an asymptotic value. If the energy spread of the system is of order Δ, that value is given by Tr ( A ρ eq ) where ρ eq is the density matrix of statistical equilibrium. The correlation width Δ is the central parameter of our approach. We argue that Δ occurs generically in chaotic quantum systems and plays the same central role.

DOI
10.1088/1751-8121/ad389c

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Abstract We investigate thermalization of a closed chaotic many-body quantum system by combining the Hartree–Fock approach with the Bohigas–Giannoni–Schmit conjecture. The conjecture implies that locally, the residual interaction causes the statistics of eigenvalues and eigenfunctions of the full Hamiltonian to agree with random-matrix predictions. The agreement is confined to an interval Δ (the correlation width). The results are used to calculate Tr ( A ρ ( t ) ) . Here ρ ( t ) is the time-dependent density matrix of the system, and A represents an observable. In the semiclassical regime, the average ⟨ Tr ( A ρ ( t ) ) ⟩ decays on the time scale ℏ / Δ toward an asymptotic value. If the energy spread of the system is of order Δ, that value is given by Tr ( A ρ eq ) where ρ eq is the density matrix of statistical equilibrium. The correlation width Δ is the central parameter of our approach. We argue that Δ occurs generically in chaotic quantum systems and plays the same central role.
Abstract Thermalization of a closed chaotic quantum system is commonly addressed in terms of the ‘eigenstate thermalization hypothesis’ (ETH). An alternative approach uses the Bohigas–Giannoni–Schmit (BGS) conjecture. The comparison shows that the two approaches differ significantly. In contrast to ETH, BGS fully uses the statistical properies of the chaotic Hamiltonian. In both approaches, the criterion for thermalization is similar. BGS goes beyond ETH in predicting quantitatively the time dependence of thermalization.
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.
We explore the role of the initial state on the onset of thermalization in isolated quantum many-body systems after a quench. The initial state is an eigenstate of an initial Hamiltonian ${\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{H}}_{I}$ and it evolves according to a different final Hamiltonian ${\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{H}}_{F}$. If the initial state has a chaotic structure with respect to ${\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{H}}_{F}$, i.e., if it fills the energy shell ergodically, thermalization is certain to occur. This happens when ${\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{H}}_{I}$ is a full random matrix, because its states projected onto ${\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{H}}_{F}$ are fully delocalized. The results for the observables then agree with those obtained with thermal states at infinite temperature. However, finite real systems with few-body interactions, as the ones considered here, are deprived of fully extended eigenstates, even when described by a nonintegrable Hamiltonian. We examine how the initial state delocalizes as it gets closer to the middle of the spectrum of ${\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{H}}_{F}$, causing the observables to approach thermal averages, be the models integrable or chaotic. Our numerical studies are based on initial states with energies that cover the entire lower half of the spectrum of one-dimensional Heisenberg spin-1/2 systems.
There are two types of quantum chaos: eigenbasis chaos and spectral chaos. The first type controls the early-time physics, e.g. the thermal relaxation and the sensitivity of the system to initial conditions. It can be traced back to the Eigenstate Thermalization Hypothesis (ETH), a statistical hypothesis about the eigenvectors of the Hamiltonian. The second type concerns very late-time physics, e.g. the ramp of the Spectral Form Factor. It can be traced back to Random Matrix Universality (RMU), a statistical hypothesis about the eigenvalues of the Hamiltonian. The Bohigas-Giannoni-Schmit (BGS) conjecture asserts a direct relationship between the two types of chaos for quantum systems with a chaotic semiclassical limit. The BGS conjecture is challenged by the Poissonian Hamiltonian ensembles, which can be used to model any quantum system displaying RMU. In this paper, we start by analyzing further aspects of such ensembles. On general and numerical grounds, we argue that these ensembles can have chaotic semiclassical limits. We then study the Poissonian ensemble associated with the Sachdev-Ye-Kitaev (SYK) model. While the distribution of couplings peaks around the original SYK model, the Poissonian ensemble is not kkk-local. This suggests that the link between ETH and RMU requires of physical kkk-locality as an assumption. We test this hypothesis by modifying the couplings of the SYK Hamiltonian via the Metropolis algorithm, rewarding directions in the space of couplings that do not display RMU. The numerics converge to a kkk-local Hamiltonian with eigenbasis chaos but without spectral chaos. We finally comment on ways out and corollaries of our results.
In this paper, we employ a semiperturbative theory to study the statistical structural properties of energy eigenfunctions (EFs) in many-body quantum chaotic systems consisting of a central system coupled to an environment. Under certain assumptions, we derive both the average shape and the statistical fluctuations of EFs on the basis formed by the direct product of the energy eigenbases of the system and the environment. Furthermore, we apply our results to two fundamental questions: (i) the properties of the reduced density matrix of the central system in an eigenstate, and (ii) the structure of the off-diagonal smooth function within the framework of the eigenstate thermalization hypothesis. Numerical results are also presented in support of our main findings.
Abstract In this paper, we employ a semiperturbative theory to study the statistical structural properties of energy eigenfunctions (EFs) in many-body quantum chaotic systems consisting of a central system coupled to an environment. Under certain assumptions, we derive both the average shape and the statistical fluctuations of EFs on the basis formed by the direct product of the energy eigenbases of the system and the environment. Furthermore, we apply our results to two fundamental questions: (i) the properties of the reduced density matrix of the central system in an eigenstate, and (ii) the structure of the off-diagonal smooth function within the framework of the eigenstate thermalization hypothesis. Numerical results are also presented in support of our main findings.