Probing the Factorized Island Branch with the Capacity of Entanglement in JT Gravity

Abstract: Black hole islands are usually diagnosed through the von Neumann entropy, but the full replica saddle contains more information than survives in the limit $n \to 1$. In this paper we show that the capacity of entanglement can detect that extra structure already within the controlled factorized island branch of JT gravity coupled to a large-$c$ bath. In the late-time high-temperature regime, the entropy plateau remains unchanged at the first nontrivial order, while the capacity acquires a definite correction. This provides a sharp semiclassical example in which nearby replica data are physically meaningful even when the entropy itself appears rigid. Our result shows that the factorized island saddle already carries finite-$n$ information beyond the entropy, and that the capacity is a natural observable for exposing it. More broadly, it highlights that the physics of island saddles is not exhausted by the $n=1$ limit: the surrounding replica geometry can contain additional, and observable, information about how the semiclassical saddle is assembled.

• The paper demonstrates that the capacity of entanglement captures finite-n information in JT gravity, exposing corrections invisible to the von Neumann entropy.
• It employs a perturbative high-temperature approach with a small κ, leveraging the factorized island branch to simplify replica geometry analysis.
• The analytic results, validated by numerical checks, reveal an unambiguous O(κ²) negative shift in the entanglement capacity, providing deeper insights into modular structure.

Probing Finite-$n$ Replica Structure in JT Islands via the Capacity of Entanglement

Introduction

This work investigates the structure of replica saddles in Jackiw-Teitelboim (JT) gravity coupled to a large-$c$ non-gravitating bath by examining the capacity of entanglement as a probe for finite-$n$ information that is not accessible to the von Neumann entropy. The study is motivated by the observation that the von Neumann entropy ($n \to 1$ limit of the R\'enyi entropies) samples only a restricted sector of the full replica geometry, potentially missing physically significant finite replica-index ($n$) features. The capacity of entanglement, corresponding to the second derivative of the replica partition function, provides a sharper probe of these features. The analysis is realized within a perturbative high-temperature regime where the island branch of the saddle factorizes, enabling controlled, analytic results.

Factorized Island Branch and the Limits of Entropy as a Probe

The standard semiclassical understanding of black hole evaporation and the Page curve derives from the competition between Hawking and island replica saddles, resolved via the von Neumann entropy. However, away from $n=1$, the full replica partition function retains more intricate, $n$-dependent structure. The primary technical regime is characterized by small $\kappa$, with $$\kappa \equiv \frac{c\,\beta\,G_N}{6\pi \phi_r} \ll 1$$, and a late-time separation $e^{-t_0}\ll \kappa \ll 1$, where $c$0. In this regime, the two-sided geometry factorizes into building blocks equivalent to one-sided QES problems, simplifying the analysis.

Physically, the entropy probes only the $c$1 point of the replica partition function. The central question posed is whether the factorized island saddle already contains nontrivial, physically meaningful finite-$c$2 information, and if so, whether such information is accessible via the capacity of entanglement even when the entropy appears rigid.

JT Setup and the Structure of Refined Observables

The analysis proceeds by considering the universal two-dimensional JT gravity action, coupled to a large-$c$3 conformal matter sector with transparent coupling to non-gravitating flat baths. The key observables in this context are the refined (modular) entropy $c$4 (the $c$5-derivative of a $c$6-normalized free energy functional) and the capacity of entanglement $c$7, with $c$8. The capacity is operationally the variance of the modular Hamiltonian and, in the replica "thermodynamics" language, serves as a heat capacity.

In the factorized late-time regime, the entropic and modular observables can be computed via a controlled expansion, leveraging the analytical tractability of the conformal welding problem and the dominant boundary reparametrization modes.

Analytic Computation in the Factorized Regime

The computation leverages the fact that, in the high-temperature factorized regime, only the dominant $c$9 boundary mode (non-gauge) significantly perturbs the saddle. The welding maps $n$0 (exterior) and $n$1 (interior) are thus constructed perturbatively, and the matter entropy for a one-QES configuration is given by

$n$2

where $n$3 is the QES position and $n$4 is the bath endpoint. These maps and positions are systematically expanded in $n$5, retaining terms up to order $n$6.

The key technical result is that the first nontrivial $n$7 correction to the generalized modular entropy (at fixed $n$8) is encoded in a coefficient $n$9 (with $n \to 1$0), which vanishes at $n \to 1$1, but whose $n \to 1$2-derivative, $n \to 1$3, is nonzero. Thus, while the island branch entropy remains uncorrected at this order, the capacity of entanglement receives a definite negative shift.

(Figure 1)

Figure 1: Numerical fit check of $n \to 1$4, showing excellent agreement between the fitted points and the analytic expression across the displayed range.

For the full two-sided geometry, the analytic result for the capacity $n \to 1$5 on the island branch in the factorized regime is:

$n \to 1$6

There is no $n \to 1$7 correction to the entropy plateau, but the capacity has an unambiguous $n \to 1$8 shift stemming from finite-$n \to 1$9 replica structure.

Interpretation and Theoretical Implications

This analysis demonstrates that the factorized island saddle, even before including genuinely non-factorizing (two-QES) corrections, retains observable finite-$n$0 information. The capacity of entanglement functions as a sensitive probe for this data, which is invisible at the entropy level. Thus, the capacity is a powerful diagnostic of modular and replica structure in semiclassical gravity and black hole evaporation.

This result confirms that finite-$n$1 observables capture physics that is not fully constrained by the Page curve or the $n$2 limit. The theoretical implications are significant for gravitational replica methods: the vicinity of the entropy plateau harbors modular information about the assembly of replica saddles. Practically, this extends the set of tools available for diagnosing semiclassical gravity and quantum extremal surfaces beyond entropy.

This finding also demarcates local/factorized finite-$n$3 effects from genuinely global, two-sided corrections, which are suppressed by $n$4 in the late-time hierarchy. The explicit $n$5 capacity shift calculated here is insensitive to these corrections, providing a robust semiclassical handle on finite-$n$6 geometry.

Broader and Future Directions

The demonstration here, in the analytically controlled JT gravity setup, motivates investigating finite-$n$7 diagnostics in other two-dimensional dilaton gravity models (cf. RST gravity (Arias et al., 10 Mar 2026)) and higher-dimensional analogues. The universality of the capacity as a probe of non-entropic replica structure is a pertinent subject for future work, with potential connections to modular Hamiltonian polynomial invariants and information-theoretic monotones.

Further progress would involve computing the first non-factorizing corrections beyond the factorized branch, as well as examining the behavior of other finite-$n$8 observables (e.g., higher modular moments) in both equilibrium and evaporating setups.

Conclusion

This work presents a controlled, semiclassical analysis of the factorized island branch in JT gravity, uncovering clear evidence that the capacity of entanglement accesses finite-$n$9 information that the von Neumann entropy does not. The analytic computation, validated numerically, reveals a capacity plateau shift of $n=1$0, with no corresponding entropy correction at the same order. This challenges the notion that the $n=1$1 limit exhausts the physical content of replica methods and establishes the capacity as a sharp probe of local replica geometry in semiclassical gravitational systems.