Real-time dynamics of matrix quantum mechanics beyond the classical approximation

TL;DR

This paper introduces a Gaussian state approximation method for simulating real-time dynamics of many-body quantum systems, surpassing classical limits and revealing temperature-dependent quantum chaos behavior.

Contribution

The authors develop a Gaussian state approximation method that extends classical dynamics to include quantum effects in many-body systems, applicable to matrix quantum mechanics.

Findings

Gaussian approximation is accurate at lower field strengths and longer times.

Quantum Lyapunov exponents are generally smaller than classical ones.

System behavior aligns with finite-temperature phase transition and respects chaos bounds.

Abstract

We describe a numerical method which allows us to go beyond the classical approximation for the real-time dynamics of many-body systems by approximating the many-body Wigner function by the most general Gaussian function with time-dependent mean and dispersion. On a simple example of a classically chaotic system with two degrees of freedom we demonstrate that this Gaussian state approximation is accurate for significantly smaller field strengths and longer times than the classical one. Applying this approximation to matrix quantum mechanics, we demonstrate that the quantum Lyapunov exponents are in general smaller than their classical counterparts, and even seem to vanish below some temperature. This behavior resembles the finite-temperature phase transition which was found for this system in Monte-Carlo simulations, and ensures that the system does not violate the Maldacena-Shenker-Stanford bound $\lambda_L < 2 \pi T$, while the classical dynamics inevitably breaks the bound.