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Low Temperature Physics

Observation of prethermalization in weakly nonintegrable unitary maps

Xiaodong Zhang, Gabriel M. Lando, Barbara Dietz, Sergej Flach A

Published Jun 1, 2025• 1 citation
Publisher

Abstract

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.

DOI
10.1063/10.0036801

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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)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)\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 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 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 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 the crossover of the entanglement entropy toward its thermal value in nearly integrable systems. We employ equations-of-motion techniques to study the entanglement dynamics in a lattice model of weakly interacting spinless fermions after a quantum quench. For weak enough interactions we observe a two-step relaxation of the entanglement entropies of finite subsystems. Initially, the entropies follow a nearly integrable evolution, approaching the value predicted by the generalized Gibbs ensemble (GGE) of the unperturbed model. Then, they start a slow drift toward the thermal stationary value described by a standard Gibbs ensemble (GE). While the initial relaxation to the GGE is independent of the interaction, the slow drift from GGE to GE values happens on timescales proportional to the inverse interaction squared. For asymptotically large times and subsystem sizes the dynamics of the entropies can be predicted using a modified quasiparticle picture that keeps track of the evolution of the fermionic occupations caused by the integrability breaking. This picture gives a quantitative description of the results as long as the integrability-breaking timescale is much larger than the one associated with the (quasi)saturation to the GGE. In the opposite limit, the quasiparticle picture still provides the correct late-time behavior, but it underestimates the initial slope of the entanglement entropy.