Phase transition in a doubly holographic model of closed $\mathbf{dS_{2} }$ spacetime

Double holography has been proved to be a powerful method in comprehending the spacetime entanglement. In this paper we investigate the doubly holographic construction in ${\mathrm{dS_{2}} }$ spacetime. We find that in this model there exists a new extremal surface besides the Hartman-Maldacena surface and the island surface, which could lead to a more complex phase structure. We then propose a generalized mutual entropy to interpret the phase transition. However, this extremal surface has a subtle property that the length of a part of the geodesic is negative when this saddle is dominant. This is because the negative part of the geodesic is within the horizon of the bulk geometry. We move to the ${\mathrm{AdS_{2}} }$ spacetime and find this subtlety still exists. We purpose a simple solution to this issue.