Probing thermodynamic phase transitions via Lyapunov exponent in AdS black hole with perfect fluid dark matter

We revisit the large-small black hole phase transitions in Reissner–Nordström (RN) AdS black holes coupled with perfect fluid dark matter. Besides the free energy, we examine the features of Lyapunov exponents for both massive and massless particles, which can characterize the chaos and instability of a particle’s motion orbiting the black hole. The presence of dark matter increases the critical temperature, making it more challenging for the phase transition to occur in comparison to an RN AdS black hole. Our analysis reveals that during the transition in phase structure, the Lyapunov exponents manifest multivalued behavior, with exclusive branches associated with distinct phases of black holes. Additionally, we observe that the discontinuous changes in the Lyapunov exponent act as an order parameter, characterized by a critical exponent of 1/2 near the critical point. These phenomena are similar to those found in RN AdS black holes and further suggest that the Lyapunov exponent can provide a valuable framework for probing the phase structure of black holes.