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We propose a doubly holographic version of the semi-classical island formulafor the entanglement negativity in the framework of the defect AdS/BCFTcorrespondence where the AdS bulk contains a defect conformal matter theory. Inthis context, we propose a defect extremal surface (DES) formula for computingthe entanglement negativity modified by the contribution from the defect mattertheory on the end-of-the-world brane. The equivalence of the DES proposal andthe semi-classical island formula for the entanglement negativity isdemonstrated in AdS$_3$/BCFT$_2$ framework. Furthermore, in the time-dependentAdS$_3$/BCFT$_2$ scenarios involving eternal black holes in the lowerdimensional effective description, we investigate the time evolution of theentanglement negativity through the DES and the island formulae and obtain theanalogues of the Page curves.

Defect extremal surfaces for entanglement negativity