© 2026 Aqora Quantum S.A.S.

Terms Privacy Legal Notice

arXiv:2211.01887

Thermalization of Classical Weakly Nonintegrable Many-Body Systems

Merab Malishava

Published Nov 3, 2022• 0 citations

Abstract

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.

DOI: 10.48550/arxiv.2211.01887

Related papers

Lyapunov Spectrum Scaling for Classical Many-Body Dynamics Close to Integrability

Merab Malishava, Sergej Flach A

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.

arXiv:2109.01361•Physical Review Letters•Mar 31, 2022•23 citations

Lyapunov spectrum scaling for many-body dynamics close to integrability

Merab Malishava, Sergej Flach A

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.

•Sep 3, 2021•0 citations

Thermalization slowing down of weakly nonintegrable quantum spin dynamics

Budhaditya Bhattacharjee, Alexei Andreanov, Sergej Flach A

We study thermalization slowing down of a quantum many-body spin system upon approach to two distinct integrability limits. Motivated by previous studies of classical systems, we identify two thermalization timescales: one quantum Lyapunov timescale is extracted by quantifying operator growth in time on an appropriately defined basis, while another ergodization timescale is related to the statistics of fluctuations of the time-evolved operator around its mean value based on the eigenstate thermalization hypothesis. Using a paradigmatic quantum Ising chain, we find that both timescales diverge upon approach to integrability. We investigate the relative strength of the divergence in the two limits and find that, despite significant qualitative differences in the mechanism of integrability breaking, the timescales diverge in a similar fashion. This allows us to establish a universality of integrability breaking in quantum spin dynamics.

arXiv:2405.00786•Physical Review Research•May 14, 2025•2 citations

Observation of prethermalization in weakly nonintegrable unitary maps

Xiaodong Zhang, Gabriel M. Lando, Barbara Dietz +1 more

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 $σ\sim t^{-1/2}$ 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.

arXiv:2501.07956•Jan 14, 2025•0 citations

Observation of prethermalization in weakly nonintegrable unitary maps

Xiaodong Zhang, Gabriel M. Lando, Barbara Dietz +1 more

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.

Low Temperature Physics•Jun 1, 2025•1 citation

Observation of Prethermalization in Weakly Nonintegrable Unitary Maps

Xiaodong Zhang, Gabriel M. Lando, Barbara Dietz +1 more

We investigate prethermalization by studying the statistical properties of the time-dependent largest Lyapunov exponent \(\Lambda(t)\) for unitary-circuit maps upon approaching integrability. We follow the evolution of trajectories for different initial conditions and compute the mean \(\mu(t)\) and standard deviation \(\sigma(t)\) of \(\Lambda(t)\). Thermalization implies a temporal decay \(\sigma \sim t^{-1/2}\) at a converged finite value of \(\mu\). We report prethermalization plateaus that persist for long times where both \(\mu\) and \(\sigma\) 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.

Qeios•Feb 7, 2025•0 citations

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 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 study thermalization slowing down of a quantum many-body spin system upon approach to two distinct integrability limits. Motivated by previous studies of classical systems, we identify two thermalization timescales: one quantum Lyapunov timescale is extracted by quantifying operator growth in time on an appropriately defined basis, while another ergodization timescale is related to the statistics of fluctuations of the time-evolved operator around its mean value based on the eigenstate thermalization hypothesis. Using a paradigmatic quantum Ising chain, we find that both timescales diverge upon approach to integrability. We investigate the relative strength of the divergence in the two limits and find that, despite significant qualitative differences in the mechanism of integrability breaking, the timescales diverge in a similar fashion. This allows us to establish a universality of integrability breaking in quantum spin dynamics.

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\sim 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) 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

\Lambda(t)

for unitary-circuit maps upon approaching integrability. We follow the evolution of trajectories for different initial conditions and compute the mean

\mu(t)

and standard deviation

\sigma(t)

of

\Lambda(t)

. Thermalization implies a temporal decay

σ

at a converged finite value of

\mu

. We report prethermalization plateaus that persist for long times where both

\mu

and

\sigma

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.