Fuzzy Spheres in Stringy Matrix Models: Quantifying Chaos in a Mixed Phase Space

TL;DR

This paper investigates the classical and quantum chaos in a truncated BMN matrix model with two fuzzy spheres, revealing a mixed phase space with integrable and chaotic regions, and analyzes quantum chaos indicators like eigenvalue spacing and out-of-time-ordered correlators.

Contribution

It provides a detailed analysis of chaos in a string-inspired matrix model, connecting classical phase space structures with quantum chaos indicators, and explores the transition between integrable and chaotic regimes as the mass parameter varies.

Findings

Classical phase space contains both integrable islands and chaotic regions.

Eigenvalue spacing follows a Brody distribution, indicating a mixed quantum system.

Quantum chaos indicators show weak correlation with classical phase space features.

Abstract

We consider a truncation of the BMN matrix model to a configuration of two fuzzy spheres, described by two coupled non-linear oscillators dependent on the mass parameter $\mu$. The classical phase diagram of the system generically ($\mu \neq 0$) contains three equilibrium points: two centers and a center-saddle; as $\mu \to 0$ the system exhibits a pitchfork bifurcation. We demonstrate that the system is exactly integrable in quadratures for $\mu=0$, while for very large values of $\mu$, it approaches another integrable point characterized by two harmonic oscillators. The classical phase space is mixed, containing both integrable islands and chaotic regions, as evidenced by the classical Lyapunov spectrum. At the quantum level, we explore indicators of early and late time chaos. The eigenvalue spacing is best described by a Brody distribution, which interpolates between Poisson and Wigner distributions; it dovetails, at the quantum level, the classical results and reemphasizes the notion that the quantum system is mixed. We also study the spectral form factor and the quantum Lyapunov exponent, as defined by out-of-time-ordered correlators. These two indicators of quantum chaos exhibit weak correlations with the Brody distribution. We speculate that the behavior of the system as $\mu \to 0$ dominates the spectral form factor and the quantum Lyapunov exponent, making these indicators of quantum chaos less effective in the context of a mixed phase space.