Entanglement of defect subregions in double holography

In the framework of double holography, we investigate the entanglement behavior of a brane subregion in AdS spacetime coupled to a bath on its boundary and also extract the contribution from the quantum matter within this subregion. From the boundary perspective, the brane subregion serves as the entanglement wedge of a subsystem on the conformal defect. In the ground state, we find the subsystem undergoes an entanglement phase transition induced by the degrees of freedom on the brane. With subcritical parameters, the wedge and entanglement entropy sharply decrease to zero. In contrast, in the supercritical regime, both the wedge and entropy stabilize, enabling analysis of both entanglement and reflected entropy. In this phase, we derive formulas for entanglement measures based on defect and bath central charges in the semi-classical limit. For entanglement entropy, the classical geometry only contributes a subleading term with logarithmic divergence, but the brane-bath CFT entanglement exhibits a dominant linear divergence, even in the semi-classical limit. Regarding reflected entropy within the defect subsystem, classical geometry contributes a leading term with logarithmic divergence, while the quantum matter within the entanglement wedge only contributes a finite term.