Chaotic Dynamics of the Mass Deformed ABJM Model

Abstract

We explore the chaotic dynamics of the mass-deformed Aharony-Bergman-Jafferis-Maldacena model. To do so, we first perform a dimensional reduction of this model from $2+1$ to $0+1$ dimensions, considering that the fields are spatially uniform. Working in the 't Hooft limit and tracing over ansatz configurations involving fuzzy 2-spheres, which are described in terms of the Gomis-Rodriguez-Gomez-Van Raamsdonk-Verlinde matrices with collective time dependence, we obtain a family of reduced effective Lagrangians and demonstrate that they have chaotic dynamics by computing the associated Lyapunov exponents. In particular, we focus on how the largest Lyapunov exponent, $\lambda_L$, changes as a function of $E/N^2$. Depending on the structure of the effective potentials, we find either $\lambda_L \propto (E/N^2)^{1/3}$ or $\lambda_L \propto (E/N^2 - \gamma_N)^{1/3}$, where $\gamma_N(k, \mu)$ are constants determined in terms of the Chern-Simons coupling $k$, the mass $\mu$, and the matrix level $N$. Noting that the classical dynamics approximates the quantum theory only in the high-temperature regime, we investigate the temperature dependence of the largest Lyapunov exponents and give upper bounds on the temperature above which $\lambda_L$ values comply with the Maldacena-Shenker-Stanford bound, $ \lambda_L \leq 2 \pi T $, and below which it will eventually be not obeyed.