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Lyapunov spectrum scaling for many-body dynamics close to integrability

Merab Malishava, Sergej Flach A

Published Sep 3, 2021• 0 citations
OpenAlex

Abstract

We use unitary map evolution to address the thermalization of many-body dynamical systems close to an integrable limit. Integrable limits are classified as either long-range or short-range, depending on the connectivity between actions. The connectivity type is imposed by the nonintegrable perturbation. We compute the Lyapunov spectra and analyze their scaling properties. Long-range limits result in a single parameter scaling of the Lyapunov spectrum, with the largest Lyapunov exponent being the only control parameter. Short-range networks have an additional second scaling parameter which describes the exponential suppression of all relative to the largest Lyapunov exponent.

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Thermalization universality-class transition induced by Anderson localization

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We study the disorder-induced crossover between the two recently discovered thermalization slowing-down universality classes, characterized by long- and short-range coupling, in classical unitary-circuit maps close to integrability. We compute Lyapunov spectra, which display qualitatively distinct features depending on whether the proximity to the integrable limit is short or long range. For sufficiently small nonlinearity, translationally invariant systems fall into the long-range class. Adding disorder to such a system triggers a transition to the short-range class, implying a breaking of this invariance, and in the very limit of vanishing nonlinearity Anderson localization emerges. The crossover from the long- to the short-range class is attained by tuning the localization length ξ from ξ≈N to ξ≪N , where N is the system size. As a consequence, the Lyapunov spectrum becomes exponentially suppressed, depending on the extent to which its translational invariance is destroyed. We expect that this disorder-induced crossover will lead to prethermalized phases and, following quantization, to many-body localization. Published by the American Physical Society 2024

arXiv:2308.08921•Physical Review Research•Mar 25, 2024•9 citations
We use unitary map evolution to address the thermalization of many-body dynamical systems close to an integrable limit. Integrable limits are classified as either long-range or short-range, depending on the connectivity between actions. The connectivity type is imposed by the nonintegrable perturbation. We compute the Lyapunov spectra and analyze their scaling properties. Long-range limits result in a single parameter scaling of the Lyapunov spectrum, with the largest Lyapunov exponent being the only control parameter. Short-range networks have an additional second scaling parameter which describes the exponential suppression of all relative to the largest Lyapunov exponent.
We propose a novel framework to characterize the thermalization of many-body dynamical systems close to integrable limits using the scaling properties of the full Lyapunov spectrum. We use a classical unitary map model to investigate macroscopic weakly nonintegrable dynamics beyond the limits set by the KAM regime. We perform our analysis in two fundamentally distinct long-range and short-range integrable limits which stem from the type of nonintegrable perturbations. Long-range limits result in a single parameter scaling of the Lyapunov spectrum, with the inverse largest Lyapunov exponent being the only diverging control parameter and the rescaled spectrum approaching an analytical function. Short-range limits result in a dramatic slowing down of thermalization which manifests through the rescaled Lyapunov spectrum approaching a non-analytic function. An additional diverging length scale controls the exponential suppression of all Lyapunov exponents relative to the largest one.
We devote our studies to the subject of weakly nonintegrable dynamics of systems with a macroscopic number of degrees of freedom. Our main points of interest are the relations between the timescales of thermalization and the timescales of chaotization; the choice of appropriate observables and the structure of equations coupling them; identifying the classes of weakly nonintegrable dynamics and developing tools to diagnose the properties specific to such classes. We discuss the traditional in the field methods for thermalization timescale computation and employ them to study the scaling the timescale with the proximity to the integrable limit. We then elaborate on a novel framework based on the full Lyapunov spectra computation for large systems as a powerful tool for the characterization of weak nonintegrability. In particular, the Lyapunov spectrum scaling offers a quantitative description allowing us to infer the structure of the underlying network of observables. Proximity to integrable limit is associated with the rapid growth of thermalization timescales and, thus, potential numerical challenges. We solve these challenges by performing numerical tests using computationally efficient model - unitary maps. The great advantage of unitary maps for numerical applications is time-discrete error-free evolution. We use these advantages to perform large timescale and system size computations in extreme proximity to the integrable limit. To demonstrate the scope of obtained results we report on the application of the developed framework to Hamiltonian systems.
We investigate prethermalization by studying the statistical properties of the time-dependent largest Lyapunov exponent Λ(t)Λ(t)Λ(t) for unitary-circuit maps upon approaching integrability. We follow the evolution of trajectories for different initial conditions and compute the mean μ(t)μ(t)μ(t) and standard deviation σ(t)σ(t)σ(t) of Λ(t)Λ(t)Λ(t). Thermalization implies a temporal decay σ∼t−1/2σ\sim t^{-1/2}σ∼t−1/2 at a converged finite value of μμμ. We report prethermalization plateaus that persist for long times where both μμμ and σσσ appear to have converged to finite values, seemingly implying differing saturated Lyapunov exponent values for different trajectories. The lifetime of such plateaus furnishes a novel time scale characterizing the thermalization dynamics of many-body systems close to integrability. We also find that the plateaus converge to their respective thermal values for long enough times.
We investigate prethermalization by studying the statistical properties of the time-dependent largest Lyapunov exponent Λ(t) for unitary-circuit maps upon approaching integrability. We follow the evolution of trajectories for different initial conditions and compute the mean μ(t) and standard deviation σ(t) of Λ(t). Thermalization implies a temporal decay σ∼t−1/2 at a converged finite value of μ. We report prethermalization plateaus that persist for long times where both μ and σ appear to have converged to finite values, seemingly implying differing saturated Lyapunov exponent values for different trajectories. The lifetime of such plateaus furnishes a novel time scale characterizing the thermalization dynamics of many-body systems close to integrability. We also find that the plateaus converge to their respective thermal values for long enough times.
We investigate prethermalization by studying the statistical properties of the time-dependent largest Lyapunov exponent Λ(t)\Lambda(t)Λ(t) for unitary-circuit maps upon approaching integrability. We follow the evolution of trajectories for different initial conditions and compute the mean μ(t)\mu(t)μ(t) and standard deviation σ(t)\sigma(t)σ(t) of Λ(t)\Lambda(t)Λ(t). Thermalization implies a temporal decay σ∼t−1/2\sigma \sim t^{-1/2}σ∼t−1/2 at a converged finite value of μ\muμ. We report prethermalization plateaus that persist for long times where both μ\muμ and σ\sigmaσ appear to have converged to finite values, seemingly implying differing saturated Lyapunov exponent values for different trajectories. The lifetime of such plateaus furnishes a novel time scale characterizing the thermalization dynamics of many-body systems close to integrability. We also find that the plateaus converge to their respective thermal values for long enough times.
We study the disorder-induced crossover between the two recently discovered thermalization slowing-down universality classes, characterized by long- and short-range coupling, in classical unitary-circuit maps close to integrability. We compute Lyapunov spectra, which display qualitatively distinct features depending on whether the proximity to the integrable limit is short or long range. For sufficiently small nonlinearity, translationally invariant systems fall into the long-range class. Adding disorder to such a system triggers a transition to the short-range class, implying a breaking of this invariance, and in the very limit of vanishing nonlinearity Anderson localization emerges. The crossover from the long- to the short-range class is attained by tuning the localization length ξ from ξ≈N to ξ≪N , where N is the system size. As a consequence, the Lyapunov spectrum becomes exponentially suppressed, depending on the extent to which its translational invariance is destroyed. We expect that this disorder-induced crossover will lead to prethermalized phases and, following quantization, to many-body localization. Published by the American Physical Society 2024