Chaos bound for spinning particles in a hairy black hole with two photon spheres

Abstract

In this work, we investigate the unstable circular orbits of spinning test particles and the associated classical orbital chaos bound in a hairy black hole spacetime admitting a double photon sphere structure. This background supports two distinct branches of unstable circular orbits, referred to as the inner and outer branches. We find that, within the parameter ranges considered, the inner branch always satisfies the chaos bound, whereas violations can occur on the outer branch. This behavior contrasts with many previous studies focusing primarily on near-horizon violations of the chaos bound and reveals a clear dependence of the bound on the orbital branch. Our analysis further shows that the total angular momentum of the particle has a pronounced effect on the local orbital instability and can enlarge the bound-violating region on the outer branch. Meanwhile, the deformation parameter of the background geometry regulates the validity of the bound by modifying both the orbital Lyapunov exponent and the black hole surface gravity. By contrast, the effect of the particle spin is comparatively weak and mainly shifts the existing unstable circular orbits and the corresponding critical boundaries through the spin-curvature coupling.