Replica wormholes and capacity of entanglement

TL;DR

The paper defines and computes the capacity of entanglement within a two-dimensional dilaton gravity framework with a large central charge matter sector, deriving a gravity formula $C=-\sum_i \partial_n\Phi^{(n)}(w)|_{n=1}+C_{\rm mat}$ that supplements the island (quantum extremal surface) prescription for entanglement entropy. It extends the replica-trick formalism to refine Rényi entropies and analyzes how topology changes between black hole and replica-wormhole saddles can induce a discontinuity in the capacity at the Page time, in contrast to the continuous generalized entropy. The authors apply the formalism to an analytically tractable AdS$_2$ eternal black hole with a flat bath in the high-temperature limit, where conformal welding is simplified and the island saddle always dominates. They find that both entanglement entropy and capacity of entanglement saturate the thermal counterparts of the black hole, supporting a thermodynamic interpretation and highlighting capacity as a probe of replica-wormhole formation and island dynamics.

Abstract

We consider the capacity of entanglement as a probe of the Hawking radiation in a two-dimensional dilaton gravity coupled with conformal matter of large degrees of freedom. A formula calculating the capacity is derived using the gravitational path integral, from which we speculate that the capacity has a discontinuity at the Page time in contrast to the continuous behavior of the generalized entropy. We apply the formula to a replica wormhole solution in an eternal AdS black hole coupled to a flat non-gravitating bath and show that the capacity of entanglement is saturated by the thermal capacity of the black hole in the high temperature limit.

Figures (5)

• Figure 1: In the conformal welding problem two regions parametrized by $|w|\leq1$ and $|v|\geq1$ are glued together along their boundaries, by finding holomorphic functions: $G(w),\, F(v)$ subject to the boundary condition \ref{['eq:welding maps']}. In the JT gravity on an AdS${}_2$ black hole coupled to a flat bath region, we parametrize the former by $w$ and the latter by $v$.
• Figure 2: This figure shows the single interval $[P,T]$ in the JT gravity on AdS${}_2$ plus a non-gravitating bath. As we will see in section \ref{['ss:entropy']}, we can interpret this as the Euclidean island/replica wormhole configuration of our main interest throughout this paper.
• Figure 3: The AdS$_2$ eternal black hole in Lorentzian signature (colored in blue). We introduce three coordinates $y_R^{\pm}$, $y_L^{\pm}$ and $W^{\pm}$. We show the patch described by $y_R^{\pm}$, $y_L^{\pm}$ as the shaded regions. $W^{\pm}$ covers inside the dashed square.
• Figure 10: Two ways of describing a conical singularity. In the $w=e^{\frac{2\pi}{\beta}(\sigma+{\rm i}\theta)}$ coordinates, the orbifold ${\cal M}_n$ has an $n$ dependent metric with the inverse temperature $\beta$. On the other coordinates $\tilde{w} = e^{\frac{2\pi}{\beta}(\tilde{\sigma}+{\rm i}\tilde{\theta})}$, the geometry is uniformized and its metric realizes an AdS$_2$ disk with the inverse temperature $\beta/n$.
• Figure 11: The Euclidean geometry of the model is the semi-infinite cylinder anchored on the boundary of the gravitating disk region [Left]. It can be conformally mapped to the finite cylinder geometry [Right], which is used to calculate the thermal free energy.