On the Spread of Entanglement at Finite Cutoff

We study how entanglement spreads in the boundary duals of finite-cutoff three-dimensional theories with positive, negative and zero cosmological constant, the $T \bar{T} + \Lambda_{2}$ two-dimensional theories. We first study the Hawking-Page transition in all three cases, and find that there is a transition in all three scenarios at the temperature where the lengths of the two cycles of the torus are the same. We then study the entanglement entropy in the thermofield double states above the Hawking-Page transition, of regions symmetrically placed on the two boundaries. We consider the case where the region is one interval on each side, and the case where it is two intervals on each side. We give an entanglement tsunami interpretation of the time-evolution of the entanglement entropies.