Screened volume law of Holographic Entanglement Entropy in Holographic Spontaneous Vectorization model

We present a holographic study of spontaneous vectorization in the background of an isotropic asymptotically AdS black brane. By extending spontaneous scalarization to vector fields, we demonstrate how the effective mass of the vector field drives tachyonic instability, leading to a transition from the AdS-RN phase to a vectorized phase. Thermodynamic analysis reveals the critical temperature $ T_c $ and coupling $ \alpha_c $ for this transition, with the vectorized phase exhibiting lower free energy. A central discovery is the emergence of a ``screened volume law'' in the holographic entanglement entropy (HEE), a novel phenomenon where the entanglement entropy scales with the subregion size via a screened entropy density distinct from thermal entropy. This arises from a geometric constraint tied to the vanishing of the Christoffel symbol $ \Gamma^z{}_{xx} $, which defines an effective boundary outside the horizons. Unlike conventional ``entanglement shadows'' in black hole systems, this surface acts as a boundary for minimal surfaces in a translationally invariant geometry. This screening effect suggests the inability of entanglement measure to fully probe the Hilbert space of this thermal system. Additionally, the HEE in the vectorized phase displays non-monotonic temperature dependence. These results establish spontaneous vectorization as a mechanism for generating novel entanglement structures in holographic systems, with implications for quantum information and critical phenomena in strongly coupled systems.