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Does thermalization occur in an isolated system after a global quantum quench

Giulio Biroli, Corinna Kollath, Andreas Martin Läuchli

Published Jul 21, 2009• 5 citations
OpenAlex

Abstract

We consider the question of thermalization for isolated quantum systems after a sudden parameter change, a so-called quantum quench. In particular we investigate the pre-requisites for thermalization focusing on the statistical properties of the time-averaged density matrix and of the expectation values of observables in the final eigenstates. We find that eigenstates, which are rare compared to the typical ones sampled by the micro-canonical distribution, are responsible for the absence of thermalization of some infinite integrable models and play an important role for some finite size non-integrable systems, such as the Bose-Hubbard model. We stress the importance of finite size effects for the thermalization of isolated quantum systems and propose a condition, relying on the Kullback-Leibler entropy, to obtain thermalization.

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We consider the question of thermalization for isolated quantum systems after a sudden parameter change, a so-called quantum quench. In particular, we investigate the prerequisites for thermalization, focusing on the statistical properties of the time-averaged density matrix and of the expectation values of observables in the final eigenstates. We find that eigenstates, which are rare compared to the typical ones sampled by the microcanonical distribution, are responsible for the absence of thermalization of some infinite integrable models and play an important role for some nonintegrable systems of finite size, such as the Bose-Hubbard model. We stress the importance of finite size effects for the thermalization of isolated quantum systems and discuss two scenarios for thermalization.

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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.

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We analyze the thermalization properties and the validity of the eigenstate thermalization hypothesis in a generic class of quantum Hamiltonians where the quench parameter explicitly breaks a ${Z}_{2}$ symmetry. Natural realizations of such systems are given by random matrices expressed in a block form where the terms responsible for the quench dynamics are the off-diagonal blocks. Our analysis examines both dense and sparse random matrix realizations of the Hamiltonians and the observables. Sparse random matrices may be associated with local quantum Hamiltonians and they show a different spread of the observables on the energy eigenstates with respect to the dense ones. In particular, the numerical data seem to support the existence of rare states, i.e., states where the observables take expectation values that are different compared to the typical ones sampled by the microcanonical distribution. In the case of sparse random matrices, we also extract the finite-size behavior of two different time scales associated with the thermalization process.

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Understanding how isolated quantum systems thermalize has recently gathered renewed interest almost 100 years after the first work by von Neumann, thanks to the experimental realizations of such systems. Experimental and numerical pieces of evidence imply that nonintegrability of the system plays an important role in thermalization. Nonintegrable systems that conserve energy alone are expected to be effectively described by the (micro)canonical ensemble due to the so-called eigenstate thermalization hypothesis (ETH) in the thermodynamic limit. In contrast, it is expected that stationary states in integrable systems are described not by the canonical ensemble but by the generalized Gibbs ensemble (GGE) due to the existence of many nontrivial conserved quantities. In this thesis, we study thermalization and its mechanism in nonintegrable systems from two perspectives. We first study how well the ETH and its finite-size corrections can be predicted by random matrix theory (RMT). Next, we present our study on the emergence of the GGE in a nonintegrable system with an extensive number of local symmetries.

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Eigenstate Gibbs ensemble in integrable quantum systems

Sourav Nandy, Arnab Sen, Arnab Das +1 more

The eigenstate thermalization hypothesis conjectures that for a thermodynamically large system in one of its energy eigenstates, the reduced density matrix describing any finite subsystem is determined solely by a set of relevant conserved quantities. In a chaotic quantum system, only the energy is expected to play that role and hence eigenstates appear locally thermal. Integrable systems, on the other hand, possess an extensive number of such conserved quantities and therefore the reduced density matrix requires specification of all the corresponding parameters (generalized Gibbs ensemble). However, here we show by unbiased statistical sampling of the individual eigenstates with a given finite energy density that the local description of an overwhelming majority of these states of even such an integrable system is actually Gibbs-like, i.e., requires only the energy density of the eigenstate. Rare eigenstates that cannot be represented by the Gibbs ensemble can also be sampled efficiently by our method and their local properties are then shown to be described by appropriately truncated generalized Gibbs ensembles. We further show that the presence of these rare eigenstates differentiates the model from the chaotic case and leads to the system being described by a generalized Gibbs ensemble at long time under a unitary dynamics following a sudden quench, even when the initial state is a typical (Gibbs-like) eigenstate of the prequench Hamiltonian.

arXiv:1605.09225•Physical review. B./Physical review. B•Dec 21, 2016•45 citations
We consider the question of thermalization for isolated quantum systems after a sudden parameter change, a so-called quantum quench. In particular we investigate the pre-requisites for thermalization focusing on the statistical properties of the time-averaged density matrix and of the expectation values of observables in the final eigenstates. We find that eigenstates, which are rare compared to the typical ones sampled by the micro-canonical distribution, are responsible for the absence of thermalization of some infinite integrable models and play an important role for some finite size non-integrable systems, such as the Bose-Hubbard model. We stress the importance of finite size effects for the thermalization of isolated quantum systems and propose a condition, relying on the Kullback-Leibler entropy, to obtain thermalization.
We consider the question of thermalization for isolated quantum systems after a sudden parameter change, a so-called quantum quench. In particular, we investigate the prerequisites for thermalization, focusing on the statistical properties of the time-averaged density matrix and of the expectation values of observables in the final eigenstates. We find that eigenstates, which are rare compared to the typical ones sampled by the microcanonical distribution, are responsible for the absence of thermalization of some infinite integrable models and play an important role for some nonintegrable systems of finite size, such as the Bose-Hubbard model. We stress the importance of finite size effects for the thermalization of isolated quantum systems and discuss two scenarios for thermalization.
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.
Abstract The observable long-time behavior of an isolated many-body system after a quantum quench is considered, i.e. an eigenstate (or an equilibrium ensemble) of some pre-quench Hamiltonian H serves as initial condition which then evolves in time according to some post-quench Hamiltonian H p . Absence of thermalization is analytically demonstrated for a large class of quite common pre- and post-quench spin Hamiltonians. The main requirement is that the pre-quench Hamiltonian must exhibit a Z 2 (spin-flip) symmetry, which would be spontaneously broken in the thermodynamic limit, though we actually focus on finite (but large) systems. On the other hand, the post-quench Hamiltonian must violate the Z 2 symmetry, but for the rest may be non-integrable and may obey the eigenstate thermalization hypothesis for (sums of) few-body observables.
We analyze the thermalization properties and the validity of the eigenstate thermalization hypothesis in a generic class of quantum Hamiltonians where the quench parameter explicitly breaks a Z2{Z}_{2}Z2​ symmetry. Natural realizations of such systems are given by random matrices expressed in a block form where the terms responsible for the quench dynamics are the off-diagonal blocks. Our analysis examines both dense and sparse random matrix realizations of the Hamiltonians and the observables. Sparse random matrices may be associated with local quantum Hamiltonians and they show a different spread of the observables on the energy eigenstates with respect to the dense ones. In particular, the numerical data seem to support the existence of rare states, i.e., states where the observables take expectation values that are different compared to the typical ones sampled by the microcanonical distribution. In the case of sparse random matrices, we also extract the finite-size behavior of two different time scales associated with the thermalization process.
Understanding how isolated quantum systems thermalize has recently gathered renewed interest almost 100 years after the first work by von Neumann, thanks to the experimental realizations of such systems. Experimental and numerical pieces of evidence imply that nonintegrability of the system plays an important role in thermalization. Nonintegrable systems that conserve energy alone are expected to be effectively described by the (micro)canonical ensemble due to the so-called eigenstate thermalization hypothesis (ETH) in the thermodynamic limit. In contrast, it is expected that stationary states in integrable systems are described not by the canonical ensemble but by the generalized Gibbs ensemble (GGE) due to the existence of many nontrivial conserved quantities. In this thesis, we study thermalization and its mechanism in nonintegrable systems from two perspectives. We first study how well the ETH and its finite-size corrections can be predicted by random matrix theory (RMT). Next, we present our study on the emergence of the GGE in a nonintegrable system with an extensive number of local symmetries.
The eigenstate thermalization hypothesis conjectures that for a thermodynamically large system in one of its energy eigenstates, the reduced density matrix describing any finite subsystem is determined solely by a set of relevant conserved quantities. In a chaotic quantum system, only the energy is expected to play that role and hence eigenstates appear locally thermal. Integrable systems, on the other hand, possess an extensive number of such conserved quantities and therefore the reduced density matrix requires specification of all the corresponding parameters (generalized Gibbs ensemble). However, here we show by unbiased statistical sampling of the individual eigenstates with a given finite energy density that the local description of an overwhelming majority of these states of even such an integrable system is actually Gibbs-like, i.e., requires only the energy density of the eigenstate. Rare eigenstates that cannot be represented by the Gibbs ensemble can also be sampled efficiently by our method and their local properties are then shown to be described by appropriately truncated generalized Gibbs ensembles. We further show that the presence of these rare eigenstates differentiates the model from the chaotic case and leads to the system being described by a generalized Gibbs ensemble at long time under a unitary dynamics following a sudden quench, even when the initial state is a typical (Gibbs-like) eigenstate of the prequench Hamiltonian.