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Exponentially fast dynamics in the Fock space of chaotic many-body systems

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

Published Feb 22, 2018• 4 citations
OpenAlex

Abstract

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 $\Gamma$ 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 $1/\Gamma$. 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.

Related papers

Exponentially fast dynamics of chaotic many-body systems

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

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.

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Process of equilibration in many-body isolated systems: diagonal versus thermodynamic entropy

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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.

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Quench dynamics of isolated many-body quantum systems

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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.

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Spread complexity evolution in quenched interacting quantum systems

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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-$1/2$ system with disorder. We observe that, for the full random matrix model and the chaotic phase of the spin-$1/2$ 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-$1/2$ system, as we explain.

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Thouless and relaxation time scales in many-body quantum systems

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arXiv:1807.07577•Physical review. B./Physical review. B•May 28, 2019•142 citations

Role of the Initial State in the Nonequilibrium Quantum Dynamics of Many-Body Systems

Eduardo Jonathan Torres-Herrera, Léa Ferreira dos Santos

We study isolated many-body quantum systems after a quench and show that their dynamics depends on the interplay between the initial state and the Hamiltonian dictating the evolution. The systems are in the nonperturbative regime. The relaxation process is controlled by the width of the energy distribution of the initial state and may be very similar for both chaotic and integrable Hamiltonians. We derive an analytical expression for the fidelity decay, which displays excellent agreement with our numerical results. This decay is Gaussian and may persist until saturation. Analytical expressions that describe very well the initial evolution of the Shannon entropy and of few-body observables are also provided. The analyses are developed for deterministic one-dimensional systems and initial states of interest to current experiments with cold atoms in optical lattices.

•Oct 18, 2013•2 citations
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.
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.
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 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.
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.
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 tTh{t}_{\text{Th}}tTh​ and the relaxation time tR{t}_{\text{R}}tR​ increase exponentially with system size. We define tTh{t}_{\text{Th}}tTh​ 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. tTh{t}_{\text{Th}}tTh​ 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 tTh{t}_{\text{Th}}tTh​ and tR{t}_{\text{R}}tR​ 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.
We study isolated many-body quantum systems after a quench and show that their dynamics depends on the interplay between the initial state and the Hamiltonian dictating the evolution. The systems are in the nonperturbative regime. The relaxation process is controlled by the width of the energy distribution of the initial state and may be very similar for both chaotic and integrable Hamiltonians. We derive an analytical expression for the fidelity decay, which displays excellent agreement with our numerical results. This decay is Gaussian and may persist until saturation. Analytical expressions that describe very well the initial evolution of the Shannon entropy and of few-body observables are also provided. The analyses are developed for deterministic one-dimensional systems and initial states of interest to current experiments with cold atoms in optical lattices.