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arXiv:1807.07577 · Physical review. B./Physical review. B

Thouless and relaxation time scales in many-body quantum systems

Mauro Schiulaz, Eduardo Jonathan Torres-Herrera, Léa Ferreira dos Santos

Published May 28, 2019• 142 citations
arXivHTMLPDF

Abstract

A major open question in studies of nonequilibrium quantum dynamics is the identification of the time scales involved in the relaxation process of isolated quantum systems that have many interacting particles. We demonstrate that long time scales can be analytically found by analyzing dynamical manifestations of spectral correlations. Using this approach, we show that the Thouless time ${t}_{\text{Th}}$ and the relaxation time ${t}_{\text{R}}$ increase exponentially with system size. We define ${t}_{\text{Th}}$ as the time at which the spread of the initial state in the many-body Hilbert space is complete and verify that it agrees with the inverse of the Thouless energy. ${t}_{\text{Th}}$ marks the point beyond which the dynamics acquire universal features, while relaxation happens later when the evolution reaches a stationary state. In chaotic systems, ${t}_{\text{Th}}\ensuremath{\ll}{t}_{\text{R}}$, while for systems approaching a many-body localized phase, ${t}_{\text{Th}}\ensuremath{\rightarrow}{t}_{\text{R}}$. Our analytical results for ${t}_{\text{Th}}$ and ${t}_{\text{R}}$ are obtained for the survival probability, which is a global quantity. We show numerically that the same time scales appear also in the evolution of the spin autocorrelation function, which is an experimental local observable. Our studies are carried out for realistic many-body quantum models. The results are compared with those for random matrices.

DOI
10.1103/physrevb.99.174313

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We provide an overview of our numerical and analytical studies of isolated interacting quantum systems that are abruptly quenched out of equilibrium instantaneously. We describe the relaxation process to a new equilibrium and obtain lower bounds for the relaxation time of full random matrices and realistic systems with two-body interactions. We show that the size of the time fluctuations after relaxation decays exponentially with system size for systems without too many degeneracies. We also discuss the conditions for thermalization and demonstrate that it can happen after local and global quenches in space. The analyses are developed for systems, initial states, and few-body observables accessible to experiments with optical lattices.
Abstract We consider time-dependent relaxation of observables in quantum systems of chaotic and regular type. Using statistical arguments and exact numerical solutions we show that the spread of the initial wave function in the Hilbert space and the main characteristics of evolution of observables have certain generic features. The study compares examples of regular dynamics, a completely chaotic case of the Gaussian orthogonal ensemble, a bosonic system with random interactions, and a fully realistic case of the time evolution of various initial non-stationary states in the nuclear shell model. In the case of the Gaussian orthogonal ensemble we show that the survival probability obtained analytically also fully defines the relaxation timescale of observables. This is not the case in general. Using the realistic nuclear shell model and the quadrupole moment as an observable we demonstrate that the relaxation time is significantly longer than defined by the survival probability of the initial state. The full analysis does not show the presence of an analog of the Lyapunov exponent characteristic for examples of classical chaos.
Phenomena in quantum chaotic systems such as spectral fluctuations are known to be described remarkably well by the random matrix theory (RMT) in small energy intervals and over large time scales. However, in exchange for analytical tractability, RMT has been constructed to leave out an important element of realistic physical systems { the notion of locality. Since quantum information propagates at a finite speed in a spatially-extended quantum chaotic system, there exists time scales before which RMT is effective. How do we describe the non-equilibrium quantum dynamics at intermediate timescales which are much longer than the local relaxation time but too short for RMT to apply? What are the signatures of many-body quantum chaotic systems beyond the description of RMT? In this thesis, we address these questions by analysing minimal models that incorporate a notion of locality and at the same time retain some analytical tractability of RMT. More specifically, we study Floquet quantum circuits that are random in space but periodic in time. These circuits consist of local and random quantum gates that act on a q-state `spin' at each site. After a brief review of classical chaos, quantum chaos, and quantum circuits in Chapter 1, we present three main results. In Chapter 2, we formulate a diagrammatical approach for computing ensemble averages of observables over random unitary circuits, and derive exact results on spectral statistics and dynamical observables for a Floquet circuit in the large-q limit. In Chapter 3, we further investigate the spectral statistics in spatiallyextended many-body quantum chaotic systems by computing the spectral form factor K(t) analytically and numerically. We show that K(t) follows the RMT description at times longer than the Thouless time tTh and identify the scaling of tTh at the large-q limit of a Floquet circuit. In Chapter 4, we discuss eigenstate correlations for spatially-extended many-body quantum chaotic systems, in terms of the statistical properties of matrix elements of local observables. While the eigenstate thermalization hypothesis (ETH) is known to give an excellent description of these quantities, the phenomenon of scrambling and the butterfly effect imply structure beyond ETH. We determine analytically and verify numerically the universal form of this structure at long distances and small eigenvalue separations for Floquet systems.
Understanding the timescales associated with relaxation to equilibrium in closed quantum many-body systems is one of the central focuses in the study of their nonequilibrium dynamics. At late times, these relaxation processes exhibit universal behavior, emerging from the inherent randomness of chaotic Hamiltonians. In this work, we investigate a disordered spin- 1 / 2 XX ladder—an experimentally realizable model known for its diffusive dynamics—to explore the connection between transport properties and spectral measures derived solely from the energy levels of the system via these relaxation timescales. We begin by analyzing the spectral form factor, which yields the time when the system begins to follow the random matrix theory (RMT) statistics, known as the RMT time. We then determine the Thouless times—the average times for a local excitation to diffuse across the entire finite system—through the linear-response theory for both spin and energy transport. Our numerical results confirm that the RMT time scales quadratically with system size and upper bounds the Thouless times. Interestingly, we also find that, unlike other nonintegrable models, spin diffusion proceeds faster than energy diffusion.
Isolated many-body quantum systems quenched far from equilibrium can eventually equilibrate, but it is not yet clear how long they take to do so. To answer this question, we use exact numerical methods and analyze the entire evolution, from perturbation to equilibration, of a paradigmatic disordered many-body quantum system in the chaotic regime. We investigate how the equilibration time depends on the system size and observables. We show that if dynamical manifestations of spectral correlations in the form of the correlation hole (``ramp'') are taken into account, the time for equilibration scales exponentially with system size, while if they are neglected, the scaling is better described by a power law with system size, although with an exponent larger than what is expected for diffusive transport.
As recently manifested [ 1 ], the quench dynamics of isolated quantum systems consisting of a finite number of particles, is characterized by an exponential spreading of wave packets in the many-body Hilbert space. This happens when the inter-particle interaction is strong enough, thus resulting in a chaotic structure of the many-body eigenstates considered in the non-interacting basis. The semi-analytical approach used here, allows one to estimate the rate of the exponential growth as well as the relaxation time, after which the equilibration (thermalization) emerges. The key ingredient parameter in the description of this process is the width Γ of the local density of states (LDoS) defined by the initially excited state, the number of particles and the interaction strength. In this paper we show that apart from the meaning of Γ as the decay rate of survival probability, the width of the LDoS is directly related to the diagonal entropy and the latter can be linked to the thermodynamic entropy of a system equilibrium state emerging after the complete relaxation. The analytical expression relating the two entropies is derived phenomenologically and numerically confirmed in a model of bosons with random two-body interaction, as well as in a deterministic model which becomes completely integrable in the continuous limit.