Lyapunov Exponents, Phase Transitions, and Chaos Bound of ModMax AdS Black Holes

TL;DR

This work links the thermodynamic phase structure of ModMax AdS black holes to dynamical chaos by analyzing Lyapunov exponents of massless and massive test particles in unstable circular orbits. The authors derive the Lyapunov exponent expressions, show that the exponent's thermal profile exhibits multivalued behavior below the critical charge and serves as an order parameter with a mean-field critical exponent $\delta=1/2$, supported by numerical verification. They also explore the chaos bound violation, finding it occurs for horizon radii below a threshold $r_0$ within the thermodynamically stable small-black-hole phase, with $r_0$ decreasing as the ModMax parameter $\eta$ increases and increasing with angular momentum $L$; such violations can persist even without a phase transition. Overall, the study demonstrates a deep connection between black-hole thermodynamics, nonlinear electrodynamics, and chaotic dynamics, offering a universal critical behavior and highlighting the conditions under which the chaos bound can be violated in ModMax AdS backgrounds.

Abstract

We study the thermodynamic phase transition of ModMax anti-de Sitter (AdS) black holes using Lyapunov exponents of massless and massive particles in unstable circular orbits. Our results demonstrate that the thermal profile of the Lyapunov exponent serves as an efficient probe of the black hole's phase structure. We calculate the discontinuity in the Lyapunov exponent across the transition and show that it acts as an order parameter, exhibiting a critical exponent $δ=1/2$ in the vicinity of the critical point. Furthermore, we explore the violation of the chaos bound, finding that the bound is violated when the horizon radius falls below a threshold value. We also examine how the ModMax parameter and the particle's angular momentum modify this threshold, revealing their role in controlling the onset of chaos bound violation.

Figures (9)

• Figure 1: Hawking temperature as a function of horizon radius
• Figure 2: Gibbs free energy as a function of temperature
• Figure 3: Phase structure in the parameter space
• Figure 4: Lyapunov exponent $\lambda$ of massless particles as a function of temperature and horizon radius
• Figure 5: Lyapunov exponent $\lambda$ of massive particles as a function of temperature and horizon radius