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arXiv:1802.08265 · Physical review. E

Exponentially fast dynamics of chaotic many-body systems

F. Borgonovi, Felix M. Izrailev, Léa Ferreira dos Santos

Published Jan 2, 2019• 35 citations
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Abstract

We demonstrate analytically and numerically that in isolated quantum systems of many interacting particles, the number of many-body states participating in the evolution after a quench increases exponentially in time, provided the eigenstates are delocalized in the energy shell. The rate of the exponential growth is defined by the width Γ of the local density of states and is associated with the Kolmogorov-Sinai entropy for systems with a well-defined classical limit. In a finite system, the exponential growth eventually saturates due to the finite volume of the energy shell. We estimate the timescale for the saturation and show that it is much larger than ℏ/Γ. Numerical data obtained for a two-body random interaction model of bosons and for a dynamical model of interacting spin-1/2 particles show excellent agreement with the analytical predictions.

DOI
10.1103/physreve.99.010101

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arXiv:1305.6937•Physical Review E•Oct 11, 2013•49 citations
We demonstrate analytically and numerically that in isolated quantum systems of many interacting particles, the number of many-body states participating in the evolution after a quench increases exponentially in time, provided the eigenstates are delocalized in the energy shell. The rate of the exponential growth is defined by the width Γ of the local density of states and is associated with the Kolmogorov-Sinai entropy for systems with a well-defined classical limit. In a finite system, the exponential growth eventually saturates due to the finite volume of the energy shell. We estimate the timescale for the saturation and show that it is much larger than ℏ/Γ. Numerical data obtained for a two-body random interaction model of bosons and for a dynamical model of interacting spin-1/2 particles show excellent agreement with the analytical predictions.
We demonstrate analytically and numerically that in isolated quantum systems of many interacting particles, the number of states participating in the evolution after a quench increases exponentially in time, provided the eigenstates are delocalized in the energy shell. The rate of the exponential growth is defined by the width Γ\GammaΓ of the local density of states (LDOS) and is associated with the Kolmogorov-Sinai entropy for systems with a well defined classical limit. In a finite system, the exponential growth eventually saturates due to the finite volume of the energy shell. We estimate the time scale for the saturation and show that it is much larger than 1/Γ1/\Gamma1/Γ. Numerical data obtained for a two-body random interaction model of bosons and for a dynamical model of interacting spin-1/2 particles show excellent agreement with the analytical predictions.
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.
We analyze time evolution of spread complexity (SC) in an isolated interacting quantum many-body system when it is subjected to a sudden quench. Characteristics features of the time evolution of the SC after the quench are analyzed for different timescales, both in integrable and chaotic models. For a short time after the quench, the SC shows universal quadratic growth, irrespective of the initial state or the nature of the Hamiltonian, with the timescale of this growth being determined by the local density of states. The characteristics of the SC in the next phase depend on the nature of the system, and we show that, depending on whether the survival probability of an initial state is Gaussian or exponential, the SC can continue to grow quadratically, or it can show linear growth. To understand the behavior of the SC at late times, we consider sudden quenches in two models, a full random matrix in the Gaussian orthogonal ensemble, and a spin-1/21/21/2 system with disorder. We observe that, for the full random matrix model and the chaotic phase of the spin-1/21/21/2 system, the SC shows linear growth at early times and saturation at late times. The full random matrix case shows a peak in the intermediate-time region, whereas this feature is less prominent in the spin-1/21/21/2 system, as we explain.
We study isolated quantum systems with two-body interactions after a quench. In these systems, the energy shell is a Gaussian of width $\ensuremath{\sigma}$, and it gives the maximum possible spreading of the energy distribution of the initial states. When the distribution achieves this shape, the fidelity decay can be Gaussian until saturation. This establishes a lower bound for the fidelity decay in realistic systems. An ultimate bound for systems with many-body interactions is also derived based on the analysis of full random matrices. We find excellent agreement between numerical and analytical results. We also provide the conditions under which the short-time dynamics of few-body observables is controlled by $\ensuremath{\sigma}$. The analyses are developed for systems, initial states, and observables accessible to experiments.
Numerically, we study the time fluctuations of few-body observables after relaxation in isolated dynamical quantum systems of interacting particles. Our results suggest that they decay exponentially with system size in both regimes, integrable and chaotic. The integrable systems considered are solvable with the Bethe ansatz and have a highly nondegenerate spectrum. This is in contrast with integrable Hamiltonians mappable to noninteracting ones. We show that the coefficient of the exponential decay depends on the level of delocalization of the initial state with respect to the energy shell.
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.