[Paper Review] Out-of-time-order correlator in weakly perturbed integrable systems

Tomer Goldfriend, Jorge Kurchan|arXiv (Cornell University)|Nov 10, 2019

Quantum chaos and dynamical systems4 citations

TL;DR

This paper investigates quantum Lyapunov exponents in weakly perturbed integrable systems via the out-of-time-order correlator (OTOC), deriving a linear superoperator equation analogous to classical tangent space dynamics. It finds that in the semi-classical limit, the quantum Lyapunov exponent scales as ε^{1/3}, matching the classical value, but quantum fluctuations suppress chaos in the highly quantal regime and further suppress the ε^{1/3} dependence for small perturbations.

ABSTRACT

Classical quasi-integrable systems are known to have Lyapunov times much shorter than their ergodicity time, but the situation for their quantum counterparts is less well understood. As a first example, we examine the quantum Lyapunov exponent -- defined by the evolution of the 4-point out-of-time-order correlator (OTOC) -- of integrable systems which are weakly perturbed by an external noise, a setting that has proven to be illuminating in the classical case. In analogy to the tangent space in classical systems, we derive a linear superoperator equation which dictates the OTOC dynamics. We find that {\em i)} in the semi-classical limit the quantum Lyapunov exponent is given by the classical one: it scales as $ε^{1/3}$, with $ε$ being the variance of the random drive, leading to short Lyapunov times compared to the diffusion time (which is $\sim ε^{-1}$). {\em ii)} in the highly quantal regime the Lyapunov instability is suppressed by quantum fluctuations, and {\em iii)} for sufficiently small perturbations the $ε^{1/3}$ dependence is also suppressed -- another purely quantum effect which we explain. Several numerical examples which demonstrate the theoretical predictions are given. The implication for the results to the behavior of real near-integrable systems, and for quantum limits on chaos are briefly discussed.

Motivation & Objective

• To understand the behavior of quantum Lyapunov exponents in weakly perturbed integrable systems, where classical counterparts exhibit short Lyapunov times.
• To bridge the gap between classical quasi-integrable dynamics and their quantum analogs, particularly regarding chaos and instability.
• To derive a quantum analog of the classical tangent space equation for OTOC dynamics in integrable systems under weak noise.
• To analyze the interplay between semi-classical behavior and purely quantum effects in the emergence of chaos.
• To provide numerical and analytical evidence for quantum suppression of Lyapunov instability in near-integrable quantum systems.

Proposed method

• Derive a linear superoperator equation governing the time evolution of the 4-point OTOC in weakly perturbed integrable quantum systems.
• Define the quantum Lyapunov exponent via the exponential growth rate of the OTOC, analogous to classical chaos indicators.
• Use the semi-classical limit to connect the quantum OTOC dynamics to classical Lyapunov behavior, particularly the ε^{1/3} scaling with noise variance ε.
• Analyze the system in the highly quantal regime to assess the role of quantum fluctuations in suppressing instability.
• Identify and explain the suppression of the ε^{1/3} dependence at small perturbations as a purely quantum effect.
• Perform numerical simulations to validate the theoretical predictions across different regimes of perturbation strength and quantal behavior.

Experimental results

Research questions

• RQ1 How does the quantum Lyapunov exponent behave in weakly perturbed integrable systems, and does it recover the classical ε^{1/3} scaling in the semi-classical limit?
• RQ2 To what extent are quantum fluctuations capable of suppressing Lyapunov instability in near-integrable quantum systems?
• RQ3 Why is the ε^{1/3} dependence on perturbation variance suppressed at small noise strengths, and what is the origin of this quantum effect?
• RQ4 How does the OTOC dynamics in quantum integrable systems compare to classical tangent space dynamics under weak perturbations?
• RQ5 What are the implications of these findings for the quantum limits on chaos and the behavior of real near-integrable systems?

Key findings

• In the semi-classical limit, the quantum Lyapunov exponent scales as ε^{1/3}, matching the classical value and leading to short Lyapunov times compared to the diffusion time ∼ε^{-1}.
• In the highly quantal regime, quantum fluctuations suppress the Lyapunov instability, indicating a fundamental difference from classical chaos.
• For sufficiently small perturbations, the ε^{1/3} dependence is suppressed, a purely quantum effect not present in classical dynamics.
• The linear superoperator equation derived for OTOC dynamics serves as the quantum analog of the classical tangent space equation.
• Numerical examples confirm the theoretical predictions across both semi-classical and highly quantal regimes.
• The results suggest that quantum effects impose fundamental limits on the emergence of chaos in near-integrable systems.