Universal Dynamical Response to Slow Driving in Chaotic Systems

Technical Summary: Universal Dynamical Response to Slow Driving in Chaotic Systems

Problem Statement
Traditional definitions of chaos diverge between classical and quantum mechanics. Classical chaos is characterized by trajectory instability and positive Lyapunov exponents, while quantum chaos is typically identified via level-spacing statistics (Wigner-Dyson) or the Eigenstate Thermalization Hypothesis (ETH). These distinct metrics fail to provide a unified framework that explains the emergence of classical chaos from quantum dynamics or the nature of dynamical instabilities in quantum systems near integrable limits. The authors argue for a unified perspective grounded not in trajectories or spectral statistics, but in the stability of a system's stationary state under slow, cyclic driving.

Methodology
The authors propose characterizing chaos through the breakdown of adiabaticity. They introduce a protocol where a system, initially in a stationary state $\rho_{-\infty} = P(H_0)$, is subjected to a cyclic perturbation $H(t) = H_0 + \lambda(t)V$ with $\lambda(\pm\infty) = 0$. The protocol is parametrized by a characteristic time scale $T = 1/\mu$ and an average speed $\bar{v}$.

To quantify the system's susceptibility to the drive speed, the authors define the "speed-Fisher information," denoted as $I_{\bar{v}}(\mu)$. This is derived from the fidelity $F(\bar{v}, \mu)$ between the initial and final states:
$$I_{\bar{v}}(\mu) = -2 \frac{\partial^2 F(\bar{v}, \mu)}{\partial \bar{v}^2} \bigg|_{\bar{v}=0}$$
This quantity measures the leading-order loss of fidelity due to non-zero drive speed in the quasi-static limit ($\mu \to 0$).

Theoretical analysis links $I_{\bar{v}}(\mu)$ to a "score-weighted symmetric autocorrelation function" $D_V(t)$ and its Fourier transform, the spectral function $\tilde{D}_V(\omega)$. The speed-Fisher information is expressed as:
$$I_{\bar{v}}(\mu) = \frac{1}{\mu} \int_{-\infty}^{\infty} \frac{d\omega}{2\pi} \tilde{D}_V(\mu\omega) |\tilde{g}(\omega)|^2$$
where $\tilde{g}(\omega)$ is the Fourier transform of the protocol's time derivative. The authors demonstrate that $I_{\bar{v}}(\mu)$ corresponds to a thermodynamic drag, specifically the irreversible production of "modular energy" (or heating in Gibbs ensembles), quantified by the relative entropy between final and initial states.

Key Contributions and Results

1. Unified Framework: The paper establishes that the divergence of the speed-Fisher information in the quasi-static limit serves as a universal signature of chaos, applicable to classical Hamiltonian systems, quantum Hamiltonian systems, and non-Hamiltonian flows.
2. Spectral Weight Criterion: The response is controlled by the low-frequency spectral weight of the perturbation.
   • Regular/Integrable Systems: Spectral functions vanish at low frequencies (typically as $\omega^\alpha$ with $\alpha > 1$). Consequently, $I_{\bar{v}}(\mu)$ remains finite or vanishes as $\mu \to 0$, indicating adiabatic stability.
   • Chaotic Thermalizing Systems: Generic perturbations exhibit non-zero low-frequency spectral weight (a plateau). This leads to a robust divergence $I_{\bar{v}}(\mu) \sim 1/\mu$, signaling a breakdown of adiabaticity.
   • Marginal/Non-thermalizing Chaos: Systems with singular low-frequency spectra (e.g., critical or sub-Ohmic) exhibit intermediate or stronger divergences depending on the exponent $\alpha$.
3. Thermodynamic Interpretation: The speed-Fisher information is identified as a thermodynamic drag. For a Gibbs state, the irreversible energy absorption scales as $\langle \Delta E \rangle = \frac{k_B T}{2} I_{\bar{v}}(\mu) \bar{v}^2$. This provides a direct link between chaos, entropy production, and measurable energy dissipation.
4. Numerical Validation:
   • Quantum Example: The 1D mixed-field Ising model. The integrable case ($h_z=0$) shows vanishing drag as $\mu \to 0$, while the chaotic case ($h_z \neq 0$) exhibits the predicted $1/\mu$ divergence, consistent with the low-frequency plateau in its spectral function.
   • Classical Example: The $\beta$-Fermi-Pasta-Ulam-Tsingou ($\beta$-FPUT) model. The integrable harmonic chain ($\beta=0$) is adiabatically stable, whereas the strongly nonlinear chaotic case ($\beta=10$) shows divergent drag and a finite low-frequency spectral weight.
   • Non-Hamiltonian Extension: The framework is extended to the Lorenz-96 system, where the stability of stationary states under slow driving is similarly linked to the low-frequency spectral behavior of the divergence of the perturbation.

Significance and Claims
The authors claim that this approach offers a unified, operational notion of chaos that treats classical and quantum systems on equal footing. Unlike traditional metrics that rely on specific system properties (trajectories or level spacing), this framework posits that regular or chaotic behavior is a property of the system, its state, and the relevant observable (reflected in its low-frequency spectral behavior).

The paper emphasizes that this method complements traditional diagnostics by providing a criterion for the presence of chaos that is directly accessible in numerical and experimental setups via the measurement of thermodynamic drag or energy fluctuations. It further suggests that the framework can distinguish between different types of chaotic behavior (thermalizing vs. non-thermalizing) and describes how chaos manifests across different physical observables and timescales, particularly noting that certain observables (like currents) may exhibit greater stability than others.