Lyapunov Exponents and Phase Transitions in Four-Dimensional AdS Black Holes with a Nonlinear Electrodynamics Source

We investigate the relationship between dynamical instability and thermodynamic phase transitions in four-dimensional Anti–de Sitter black holes in Einstein gravity coupled to a nonlinear power-law electromagnetic field with exponent $p = 3/4$. In the canonical ensemble, we identify a critical electric charge $Q_c$ separating a regime exhibiting a first-order small/large black-hole (SBH/LBH) phase transition from a regime with a single thermodynamically stable phase. For both massless and massive probes, the thermal profile of the Lyapunov exponent $λ(T)$ becomes multivalued in the SBH/LBH coexistence region and exhibits a finite discontinuity at the transition temperature. This jump vanishes continuously as $Q \to Q_c$, signaling the termination of the first-order transition at a second-order critical point. Near criticality, the Lyapunov discontinuity obeys a universal mean-field scaling law with critical exponent $1/2$. For massless probes, we further analyze the critical impact parameter $b_c$, which displays the same multivalued structure and critical behavior as the Lyapunov exponent. We also demonstrate that the spinodal temperatures, defined by the extrema of the $T(r_h)$ curve where the heat capacity at fixed charge diverges, coincide with singular features in the Lyapunov exponent. Our results identify the Lyapunov exponent as a unified dynamical probe capable of capturing both first-order phase coexistence and second-order critical behavior in black-hole thermodynamics.

💡 Research Summary

In this work the authors explore the interplay between dynamical instability, quantified by the Lyapunov exponent λ, and thermodynamic phase transitions of four‑dimensional anti‑de Sitter (AdS) black holes sourced by a nonlinear power‑law electromagnetic field with exponent p = 3/4. The study is carried out in the canonical ensemble, i.e., at fixed electric charge Q.

First, the Einstein‑Power‑Maxwell (PMI) action is introduced, and for p = 3/4 the static, spherically symmetric black‑hole solution is written explicitly. The metric function f(r) contains contributions from the mass M, the AdS curvature radius ℓ, and a term proportional to Q^{3/2} that reflects the sub‑linear nature of the electromagnetic field. By solving f(r_h)=0 the horizon radius r_h is obtained, and the Hawking temperature T(r_h,Q) as well as the Helmholtz free energy F(T,Q) are derived. After rescaling all quantities to dimensionless variables (denoted with a tilde), the critical charge \tilde Q_c and critical temperature \tilde T_c are found from the simultaneous conditions ∂\tilde T/∂\tilde r_h = ∂^2\tilde T/∂\tilde r_h^2 = 0. The resulting critical values separate two distinct regimes:

• For \tilde Q > \tilde Q_c the T–r_h curve is monotonic, implying a single thermodynamically stable black‑hole branch for each temperature.
• For \tilde Q < \tilde Q_c the curve develops a swallow‑tail, i.e., two local extrema, which signals the coexistence of three branches (small, intermediate, large black holes). The free‑energy plot exhibits the characteristic “double‑well” shape, and the intersection of the two minima defines the first‑order small/large black‑hole transition temperature T_p.

Next, the authors turn to the dynamical side. They consider both massless (null) and massive (timelike) test particles moving in the equatorial plane of the black‑hole spacetime. The Lagrangian L = ½