Probing Hawking radiation through capacity of entanglement

TL;DR

This work shows that the capacity of entanglement, defined via the replica parameter as $C_A = obreak ext{lim}_{n o1} n^2 \partial_n^2 \log \text{Tr}\,\rho_A^{n}$, provides a sensitive diagnostic of the black hole evaporation process and replica wormhole topology changes. Through two tractable models—the end-of-the-world brane setup in JT gravity and moving mirrors in 2D CFTs—the authors demonstrate that $C_A$ exhibits a pronounced peak near the Page time and, in holographic contexts, can undergo discontinuities when the dominant replica saddle switches topology, signaling island formation. In microcanonical and canonical ensembles of the EOW brane model, the capacity shows a Page-time peak and, depending on the ensemble, approaches a nonzero late-time value $C_{\rm BH}$, reflecting the underlying thermodynamics and island contributions. The moving mirror analyses reinforce that $C_A$ can reveal phase transitions between disconnected and connected replica wormhole configurations, with universal patterns such as jumps of $\pm 2S_{\rm bdy}$ at Page-time transitions, highlighting the capacity as a practical probe of Hawking radiation and quantum gravitational phase structure. Overall, the paper argues that capacity of entanglement is a valuable, topology-sensitive diagnostic for black hole evaporation and replica wormhole dynamics with potential broader applicability to non-perturbative gravitational phenomena.

Abstract

We consider the capacity of entanglement in models related with the gravitational phase transitions. The capacity is labeled by the replica parameter which plays a similar role to the inverse temperature in thermodynamics. In the end of the world brane model of a radiating black hole the capacity has a peak around the Page time indicating the phase transition between replica wormhole geometries of different types of topology. Similarly, in a moving mirror model describing Hawking radiation the capacity typically shows a discontinuity when the dominant saddle switches between two phases, which can be seen as a formation of island regions. In either case we find the capacity can be an invaluable diagnostic for a black hole evaporation process.

Figures (12)

• Figure 1: A typical shape of the entanglement entropy (orange) and an asymptotic form of the capacity of entanglement (blue) at early and late times for black holes with the Hawking radiation. The entropy grows linearly and saturates after the Page time while the capacity is vanishing at early time and approaches some value at late time.
• Figure 2: [Left] The geometry of $\langle \psi_i | \psi_j\rangle_B$ in the EOW brane model. The hyperbolic disk (blue) has the asymptotic boundary and terminates on the EOW brane (orange). [Right] The replica geometries for the moment \ref{['eq:nth moment']} of $\rho_R$ with $n=3$. There are three ways to connect the EOW branes in the bulk region. The planar replica wormholes with $l$-dotted loops and $m_b$-disconnected disk regions with $b$-asymptotic boundaries provide the factors $k^l \prod_{m_b, b} (Z_{b})^{m_b}$ respectively.
• Figure 3: The entanglement entropy $S_R$\ref{['eq:SRmc2']} [Left] and capacity of entanglement $C_R$\ref{['eq:CRmc2']} [Right] for the EOW brane model with $\mathbf{S} = 2$ in the microcanonical ensemble. The capacity has a peak at the Page time $\log k = \mathbf{S}$, which clearly shows the crossover from the fully disconnected to fully connected replica wormhole solutions describing the Hawking radiation in an evaporating black hole.
• Figure 4: The entropy (blue) and capacity (orange) of the EOW brane model in the canonical ensembles. Both approach the asymptotic values $S_\text{BH}$ and $C_\text{BH}$ shown in the dashed lines in the large $k$ limit.
• Figure 5: A moving mirror model in the $(t,x)$-coordinate [Left] and in the tilde coordinates [Right]. A CFT lives outside the mirror $v\geq p(u)$ ($x\geq z(t)$), which can be mapped to the RHP by a conformal transformation. The red region $(\tilde{x}\geq 0,\,\tilde{t}\geq \tilde{x})$ in the right panel has no counterpart in the original coordinate system and can be seen as a black hole interior.