Replica Wormholes, Modular Entropy, and Capacity of Entanglement in JT Gravity

TL;DR

This paper investigates how the replica parameter influences modular entropy and entanglement capacity in JT gravity models, revealing how multiple copies lead to more purified Hawking radiation and extending the island formula for finite replicas.

Contribution

It develops the replica geometry for finite $n$ in JT gravity models and generalizes the island formula to finite $n$, providing new insights into entanglement in gravitational systems.

Findings

Modular entropy and entanglement capacity decrease with increasing $n$ at late times.

Single island configurations mimic thermal entropy and heat capacity behaviors.

More replica copies lead to more connected geometries that purify Hawking radiation.

Abstract

By employing the replica trick we study the impact of the replica parameter $n$ on the modular entropy and the capacity of entanglement in the End of the World (EoW) model and the island model, respectively. For the EoW model, we present $n$-dependent evolution curves of the modular entropy and the capacity of entanglement under both microcanonical and canonical ensembles. In particular, in the canonical ensemble, all quantities decrease as $n$ increases at late times. For the island model, we develop the replica geometry for finite $n$ and re-evaluate the modular entropy and the capacity of entanglement in a two-sided eternal Jackiw-Teitelboim black hole coupled with a thermal bath. In the case of a single island configuration, the modular entropy and capacity of entanglement closely resemble the thermal entropy and the heat capacity, respectively, yielding results analogous to those obtained in the canonical ensemble for the EoW model. The analysis of the results from these two models strongly indicates that in geometries with a greater number of $n$ copies, more connected geometries effectively purify thermal Hawking radiation. In addition, we compare these findings with statistical mechanics and provide an interpretation for the replica parameter $n$. Finally, we generalize the island formula to accommodate the finite $n$ case under this interpretation.