Abstract
We consider entanglement entropies of finite spatial intervals in Minkowski radiation baths coupled to the eternal black hole in JT gravity, and the related problem involving free fermion BCFT in the thermofield double state. We show that the non-monotonic entropy evolution in the black hole problem precisely matches that of the free fermion theory in a high temperature limit, and the results have the form expected for CFTs with quasiparticle description. Both exhibit rich behaviour that involves at intermediate times, an entropy saddle with an island in the former case, and in the latter a special class of disconnected OPE channels. The quantum extremal surfaces start inside the horizon, but can emerge from and plunge back inside as time evolves, accompanied by a characteristic dip in the entropy also seen in the free fermion BCFT. Finally an entropy equilibrium is reached with a no-island saddle.
Highlights
We show that the nonmonotonic entropy evolution in the black hole problem precisely matches that of the free fermion theory in a high temperature limit, and the results have the form expected for conformal field theory (CFT) with quasiparticle description
The quantum extremal surfaces start inside the horizon, but can emerge from and plunge back inside as time evolves, accompanied by a characteristic dip in the entropy seen in the free fermion boundary conformal field theory (BCFT)
The actual magnitudes of the coordinates of the Quantum Extremal Surfaces (QES) do not contribute at leading order to the final result for the high temperature entropy, but how big they are relative to coordinates of other points determines whether we are in regime (III) or (IV)
Summary
We consider BCFT on the half-line and take two copies CFTL and CFTR prepared in the thermofield double state. We define Kruskal-Szekeres (KS) coordinates w± that cover the whole of the spacetime, as well as null coordinates x± which are Minkowski coordinates in the baths and Schwarzschild coordinates in the AdS2 region, related by w± = ±e±2πx±/β. The Schwarzschild time in the left region includes a sign change and an imaginary shift relative to the coordinate on the right:. The imaginary shift is an important feature of the BCFT set up where we identify the two copies of the CFT’s with the bath regions of the spacetime. In the BCFT picture, each copy of the CFT is defined on a half-Minkowski space that is isomorphic to the left or right bath region in the black hole geometry with Lorentzian null coordinates x± as in (2.2): see figure 3. The imaginary shift on the left implements the necessary analytic continuation to obtain correlators in the TFD state from Euclidean thermal correlators (3.1)
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