Extremal curves in single-trace $T\bar{T}$-holography

In this paper, we continue the study of single-trace $T\bar{T}$-holography where the boundary field theory can be realized as a CFT$_2$ deformed by a single-trace irrelevant operator of dimension $(2,2)$ and dual spacetime geometry is $AdS_3$ smoothly glued to flat spacetime with a linear dilaton near the boundary. In this non-AdS holographic framework, we propose that the length of real extremal curves connecting the two boundaries of an eternal black hole at fixed boundary time captures the time-evolved entanglement entropy of an entangled, quenched boundary system. At late times, we find two analytic extremal solutions in the complexified geometry, which become real in complementary temperature regimes. Focusing only on the real solutions leads to a non-analyticity at a critical temperature $T_c$, which we interpret as a second-order phase transition separating a local (CFT$_2$) phase from a non-local (Little String Theory) phase.