Chaos in Matrix Gauge Theories with Massive Deformations

TL;DR

This paper investigates chaotic dynamics in a deformed $SU(N)$ matrix quantum mechanics model, analyzing Lyapunov exponents and their relation to energy, fixed points, and the MSS bound, revealing conditions for chaos and bounds on temperature.

Contribution

It introduces a family of effective Hamiltonians with fuzzy sphere configurations and numerically studies their chaotic behavior, providing new insights into Lyapunov exponents in matrix gauge theories.

Findings

Lyapunov exponents scale as E^{1/4} or (E - E_n)_F^{1/4}

Bounds on temperature for MSS bound compliance and violation

Chaotic dynamics linked to unstable fixed points

Abstract

Starting from an $SU(N)$ matrix quantum mechanics model with massive deformation terms and by introducing an ansatz configuration involving fuzzy four- and two-spheres with collective time dependence, we obtain a family of effective Hamiltonians, $H_n \,, (N = \frac{1}{6}(n+1)(n+2)(n+3))$ and examine their emerging chaotic dynamics. Through numerical work, we model the variation of the largest Lyapunov exponents as a function of the energy and find that they vary either as $\propto (E-(E_n)_F)^{1/4} $ or $\propto E^{1/4}$, where $(E_n)_F$ stand for the energies of the unstable fixed points of the phase space. We use our results to put upper bounds on the temperature above which the Lyapunov exponents comply with the Maldacena-Shenker-Stanford (MSS) bound, $2 \pi T $, and below which it will eventually be violated.