Chaos in $SU(2)$ Yang-Mills Chern-Simons Matrix Model

TL;DR

This paper investigates how adding a Chern-Simons term affects chaos in an $SU(2)$ Yang-Mills matrix model, revealing parameter-dependent chaotic behavior and phase transitions between chaotic and non-chaotic regimes.

Contribution

It introduces the analysis of chaos in a minimal $SU(2)$ Yang-Mills matrix model with a Chern-Simons term, highlighting the impact of coupling and symmetry parameters on system dynamics.

Findings

Lyapunov exponents increase with Chern-Simons coupling for certain parameter ranges.

Chaotic behavior depends sensitively on the product of coupling and conserved momentum.

Critical exponents characterize the transition from chaos to order as parameters vary.

Abstract

We study the effects of addition of Chern-Simons (CS) term in the minimal Yang Mills (YM) matrix model composed of two $2 \times 2$ matrices with $SU(2)$ gauge and $SO(2)$ global symmetry. We obtain the Hamiltonian of this system in appropriate coordinates and demonstrate that its dynamics is sensitive to the values of both the CS coupling, $\kappa$, and the conserved conjugate momentum, $p_\phi$, associated to the $SO(2)$ symmetry. We examine the behavior of the emerging chaotic dynamics by computing the Lyapunov exponents and plotting the Poincar\'{e} sections as these two parameters are varied and, in particular, find that the largest Lyapunov exponents evaluated within a range of values of $\kappa$ are above that is computed at $\kappa=0$, for $\kappa p_\phi < 0$. We also give estimates of the critical exponents for the Lyapunov exponent as the system transits from the chatoic to non-chaotic phase with $p_\phi$ approaching to a critical value.