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arXiv:1805.07085 · Physical review. E

Dynamical typicality of isolated many-body quantum systems

Peter Reimann

Published Jun 18, 2018• 49 citations
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Abstract

Dynamical typicality refers to the property that two pure states, which initially exhibit (almost) the same expectation value for some given observable A, are very likely to exhibit also very similar expectation values when evolving in time according to the pertinent Schrödinger equation. We unify and generalize a variety of previous findings of this type for sufficiently high-dimensional quantum mechanical model systems. Particular emphasis is put on the necessary and sufficient conditions, which the initial expectation value and the spectrum of A have to fulfill.

DOI
10.1103/physreve.97.062129

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Dynamical typicality refers to the property that two pure states, which initially exhibit (almost) the same expectation value for some given observable A, are very likely to exhibit also very similar expectation values when evolving in time according to the pertinent Schrödinger equation. We unify and generalize a variety of previous findings of this type for sufficiently high-dimensional quantum mechanical model systems. Particular emphasis is put on the necessary and sufficient conditions, which the initial expectation value and the spectrum of A have to fulfill.
We show that the vast majority of all pure states featuring a common expectation value of some generic observable at a given time will yield very similar expectation values of the same observable at any later time. This is meant to apply to Schrödinger type dynamics in high dimensional Hilbert spaces. As a consequence individual dynamics of expectation values are then typically well described by the ensemble average. Our approach is based on the Hilbert space average method. We support the analytical investigations with numerics obtained by exact diagonalization of the full time-dependent Schrödinger equation for some pertinent, abstract Hamiltonian model. Furthermore, we discuss the implications for the applicability of projection operator methods with respect to initial states, as well as for irreversibility in general.
Abstract Typicality is a well-established and very general property of quantum many-body systems, referring to the phenomenon that the expectation values of any given observable are practically indistinguishable for the overwhelming majority of all pure states (normalized vectors) in a sufficiently high-dimensional Hilbert (sub-)space. Here, we provide very simple and general arguments that analogous typicality properties of pure states (phase space points) in classical many-body systems are still expected to hold true for macroscopic observables, but not any more for microscopic (few-body) observables.
We consider the set of all initial states within a microcanonical energy shell of an isolated many-body quantum system, which exhibit an arbitrary but fixed nonequilibrium expectation value for some given observable A. On the condition that this set is not too small, it is shown by means of a dynamical typicality approach that most such initial states exhibit thermalization if and only if A satisfies the so-called weak eigenstate thermalization hypothesis (wETH). Here, thermalization means that the expectation value of A spends most of its time close to the microcanonical value after initial transients have died out. The wETH means that, within the energy shell, most eigenstates of the pertinent system Hamiltonian exhibit very similar expectation values of A.
Investigation on foundational aspects of quantum statistical mechanics recently entered a renaissance period due to novel intuitions from quantum information theory and to increasing attention on the dynamical aspects of single quantum systems. In the present contribution a simple but effective theoretical framework is introduced to clarify the connections between a purely mechanical description and the thermodynamic characterization of the equilibrium state of an isolated quantum system. A salient feature of our approach is the very transparent distinction between the statistical aspects and the dynamical aspects in the description of isolated quantum systems. Like in the classical statistical mechanics, the equilibrium distribution of any property is identified on the basis of the time evolution of the considered system. As a consequence equilibrium properties of quantum system appear to depend on the details of the initial state due to the abundance of constants of the motion in the Schrodinger dynamics. On the other hand the study of the probability distributions of some functions, such as the entropy or the equilibrium state of a subsystem, in statistical ensembles of pure states reveals the crucial role of typicality as the bridge between macroscopic thermodynamics and microscopic quantum dynamics. We shall consider two particular ensembles: the random pure state ensemble and the fixed expectation energy ensemble. The relation between the introduced ensembles, the properties of a given isolated system, and the standard quantum statistical description are discussed throughout the presentation. Finally we point out the conditions which should be satisfied by an ensemble in order to get meaningful thermodynamical characterization of an isolated quantum system.
Characterizing the time evolution of generic quantum many-body systems is a fundamental challenge, as representing the exact state requires exponentially scaling computational resources. While hydrodynamics and statistical mechanics successfully simplify this task by predicting the expectation values of local observables, these macroscopic frameworks provide no information about nonlinear characteristics of the quantum state. In this work, we demonstrate that for systems exhibiting a timescale separation, with dynamics governed by the transport of conserved charges, this lost information can be systematically recovered. By constraining the maximum-entropy Scrooge ensemble solely by the system's slow modes, we accurately reconstruct complex, nonlinear quantum properties of the global time-evolved state, including the half-chain entanglement entropy and the participation entropy in the computational basis. Our results generalize the paradigm of canonical typicality, revealing that a hydrodynamically bottlenecked system behaves as a typical pure state within a dynamically restricted submanifold of the Hilbert space - a phenomenon we term global restricted typicality.
An analytical prediction is established of how an isolated many-body quantum system relaxes towards its thermal longtime limit under the action of a time-independent perturbation, but still remaining sufficiently close to a reference case whose temporal relaxation is known. This is achieved within the conceptual framework of a typicality approach by showing and exploiting that the time-dependent expectation values behave very similarly for most members of a suitably chosen ensemble of perturbations. The predictions are validated by comparison with various numerical and experimental results from the literature.