Entanglement entropy of an annulus in holographic thermalization

The thermalization process of the holographic entanglement entropy (HEE) of an annular domain is investigated in the Vaidya-AdS geometry. We determine numerically the Hubeny-Rangamani-Takayanagi (HRT) surface, which may be a hemi-torus or two disks, depending on the ratio of the inner radius to the outer radius of the annulus. More importantly, for some fixed ratio of the two radii, the annulus undergoes a phase transition, or a double phase transition, during thermalization from a hemi-torus to a two-disk configuration, or vice versa. The occurrence of various phase transitions is determined by the ratio of the two radii of the annulus. The rate of entanglement growth is also investigated during the thermal quench. The local maximal rate of entanglement growth occurs in the region with a double phase transition. Finally, if the quench process is sufficiently slow, which may be controlled by the thickness of the null shell, the region with a double phase transition vanishes.