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arXiv:1305.6937 · Physical Review E

Effects of the interplay between initial state and Hamiltonian on the thermalization of isolated quantum many-body systems

Eduardo Jonathan Torres-Herrera, Léa Ferreira dos Santos

Published Oct 11, 2013• 49 citations
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Abstract

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.

DOI
10.1103/physreve.88.042121

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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 H\ifmmode^\elseˆ\fiI{\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{H}}_{I}H\ifmmode^\elseˆ\fi​I​ and it evolves according to a different final Hamiltonian H\ifmmode^\elseˆ\fiF{\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{H}}_{F}H\ifmmode^\elseˆ\fi​F​. If the initial state has a chaotic structure with respect to H\ifmmode^\elseˆ\fiF{\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{H}}_{F}H\ifmmode^\elseˆ\fi​F​, i.e., if it fills the energy shell ergodically, thermalization is certain to occur. This happens when H\ifmmode^\elseˆ\fiI{\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{H}}_{I}H\ifmmode^\elseˆ\fi​I​ is a full random matrix, because its states projected onto H\ifmmode^\elseˆ\fiF{\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{H}}_{F}H\ifmmode^\elseˆ\fi​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 H\ifmmode^\elseˆ\fiF{\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{H}}_{F}H\ifmmode^\elseˆ\fi​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.
Using the ergodicity principle for the expectation values of several types of observables, we investigate the thermalization process in isolated fermionic systems.These are described by the two-body random ensemble, which is a paradigmatic model to study quantum chaos and especially the dynamical transition from integrability to chaos.By means of exact diagonalizations we analyze the relevance of the eigenstate thermalization hypothesis as well as the influence of other factors, such as the energy and structure of the initial state, or the dimension of the Hilbert space.We also obtain analytical expressions linking the degree of thermalization for a given observable with the so-called number of principal components for transition strengths originating at a given energy, with the dimensions of the whole Hilbert space and microcanonical energy shell, and with the correlations generated by the observable.As the strength of the residual interaction is increased, an order-to-chaos transition takes place, and we show that the onset of Wigner spectral fluctuations, which is the standard signature of chaos, is not sufficient to guarantee thermalization in finite systems.When all the signatures of chaos are fulfilled, including the quasicomplete delocalization of eigenfunctions, the eigenstate thermalization hypothesis is the mechanism responsible for the thermalization of certain types of observables, such as (linear combinations of) occupancies and strength function operators.Our results also suggest that fully chaotic systems will thermalize relative to most observables in the thermodynamic limit.
We study sudden quantum quenches in which the initial states are selected to be either eigenstates of an integrable Hamiltonian that is nonmappable to a noninteracting one or a nonintegrable Hamiltonian, while the Hamiltonian after the quench is always integrable and mappable to a noninteracting one. By studying weighted energy densities and entropies, we show that quenches starting from nonintegrable (chaotic) eigenstates lead to an ``ergodic'' sampling of the eigenstates of the final Hamiltonian, while those starting from the integrable eigenstates do not (or at least it is not apparent for the system sizes accessible to us). This goes in parallel with the fact that the distribution of conserved quantities in the initial states is thermal in the nonintegrable cases and nonthermal in the integrable ones, and means that, in general, thermalization occurs in integrable systems when the quench starts form an eigenstate of a nonintegrable Hamiltonian (away from the edges of the spectrum), while it fails (or requires larger system sizes than those studied here to become apparent) for quenches starting at integrable points. We test those conclusions by studying the momentum distribution function of hard-core bosons after a quench.
We study the onset of chaos and statistical relaxation in two isolated dynamical quantum systems of interacting spins 1/2, one of which is integrable and the other chaotic. Our approach to identifying the emergence of chaos is based on the level of delocalization of the eigenstates with respect to the energy shell, the latter being determined by the interaction strength between particles or quasiparticles. We also discuss how the onset of chaos may be anticipated by a careful analysis of the Hamiltonian matrices, even before diagonalization. We find that despite differences between the two models, their relaxation processes following a quench are very similar and can be described analytically with a theory previously developed for systems with two-body random interactions. Our results imply that global features of statistical relaxation depend on the degree of spread of the eigenstates within the energy shell and may happen to both integrable and nonintegrable systems.
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 study how the proximity to an integrable point or to localization as one approaches the atomic limit, as well as the mixing of symmetries in the chaotic domain, may affect the onset of thermalization in finite one-dimensional systems. We consider systems of hard-core bosons at half-filling with nearest-neighbor hopping and interaction, and next-nearest-neighbor interaction. The latter breaks integrability and induces a ground-state superfluid to insulator transition. By full exact diagonalization, we study chaos indicators and few-body observables. We show that when different symmetry sectors are mixed, chaos indicators associated with the eigenvectors, contrary to those related to the eigenvalues, capture the onset of chaos. The results for the complexity of the eigenvectors and for the expectation values of few-body observables confirm the validity of the eigenstate thermalization hypothesis in the chaotic regime, and therefore the occurrence of thermalization. We also study the properties of the off-diagonal matrix elements of few-body observables in relation to the transition from integrability to chaos and from chaos to localization.
Significant attention has been devoted to the problem of thermalization of observables in isolated quantum setups by individual eigenstates. Here, we address this issue from an open quantum system perspective, examining an isolated setup where a small system (specifically, a single fermionic level) is coupled to a macroscopic fermionic bath. We argue that in such a model, despite its full integrability, the system observables exhibit thermalization when the system-bath setup resides in a typical eigenstate of its Hamiltonian, a phenomenon known as weak eigenstate thermalization. This thermalization occurs unless it is suppressed by localization due to strong coupling. We further show that, following the quench of the system Hamiltonian, the system occupancy typically relaxes to the thermal value corresponding to the new Hamiltonian. Finally, we demonstrate that system thermalization also arises when the system is coupled to a bath that has been initialized in a typical eigenstate of its Hamiltonian. Our findings suggest that nonintegrability is not the sole driver of thermalization, highlighting the need for complementary approaches to fully understand the emergence of statistical mechanics.