Scrambling in charged hairy black holes and the Kasner interior

Abstract: We analyze how the axion parameter, the Einstein-Maxwell-Scalar (EMS) coupling constant, and the charge density affect the chaotic properties of a charged hairy black hole, as characterized by the quantum Lyapunov exponent. We inject charged shock waves from the asymptotic boundary and compute the out-of-time-ordered correlators (OTOCs). Due to the relevant deformation in the boundary theory induced by a bulk scalar field, the bulk solution flows to a more general Kasner spacetime near the black hole singularity. We examine the behavior of chaotic parameters, including the Lyapunov exponent, butterfly velocity, and scrambling time delay, under this deformation. We find that as the deformation parameter increases, the ratio of the quantum Lyapunov exponent to the surface gravity decreases. For sufficiently large deformation, the Lyapunov exponent in the deformed geometry can exceed that of the axion Reissner-Nordstrom case. We observe that boundary deformation generally reduces the scrambling time delay, with the EMS coupling having a significant effect on the delay. These results provide further insight into the role of boundary deformations in modifying chaotic properties in charged hairy black holes.