Island Selection and Quantum Extremal-Surface Competition

The island rule requires three separate operations: enumerate every admissible island satisfying boundary and homology conditions, extremize the renormalized generalized entropy for each topology, and only then choose the smallest valid saddle. A quantum extremal surface is stationary; it is not automatically dominant.

Required background. Evaporating Replicas, Radiation Entropy, and Entanglement-Wedge Transitions supplies the replica origin. Quantum Extremal Surfaces: Renormalized Semiclassical Definition supplies the imported definition.

Helpful background. FLM Corrections and Holographic Use of Imported Generalized Entropy and QES fixes perturbative order. Maximin Constructions and Extremal-Surface Existence supplies Lorentzian existence qualifications.

The island variational problem

For a nongravitating radiation region $R$ and candidate island $I$ in a semiclassical gravitating region,

$$S_{\mathrm{gen}}[I;R] =\frac{\operatorname{Area}(\partial I)}{4G_N^{\mathrm{ren}}} +S_{\mathrm{bulk}}^{\mathrm{ren}}(I\cup R) +S_{\mathrm{local}}^{\mathrm{ren}}(\partial I).$$

The local terms include higher-curvature and counterterm contributions at the same order. A QES satisfies

$$\frac{\delta S_{\mathrm{gen}}}{\delta X^a(y)}=0$$

for both independent normal deformations. The candidate must also be anchored and homologous as required by the replica boundary problem and lie in a regime where the semiclassical expansion is valid.

The prescription is

$$S(R)= \min_{I\in\mathfrak I_{\mathrm{admissible}}} \left\{ \operatorname*{ext}_{\partial I}S_{\mathrm{gen}}[I;R] \right\},$$

including the empty island. “Extremize, then minimize” is essential.

Application: an auditable endpoint model

To expose the calculation, consider a one-endpoint model with radiation endpoint $b>0$ and island coordinate $a>0$. Let the renormalized functional be

$$S_{\mathrm{gen}}(a) =S_0+\frac{\kappa}{a} +\frac{c}{6}\log\frac{(a+b)^2}{a\epsilon}, \qquad \kappa>0.$$

The $1/a$ term models a dilaton contribution and the logarithm a regulated CFT interval contribution. Stationarity gives

$$0=-\frac{\kappa}{a^2} +\frac{c}{3(a+b)}-\frac{c}{6a},$$

or

$$c\,a(a-b)=6\kappa(a+b).$$

The positive solution is

$$a_*= \frac{cb+6\kappa+ \sqrt{(cb+6\kappa)^2+24c\kappa b}}{2c}.$$

One must evaluate $S_{\mathrm{gen}}(a_*)$, test the second variation or the full saddle contour, and compare it with the empty-island value $S_{\mathrm{no}}(R)$. The existence of $a_*$ alone does not select it.

In full JT-plus-bath models the dilaton, conformal factors, state-dependent CFT correlator, and two null extremality equations replace this toy functional. The same workflow yields islands just inside or outside horizons depending on setup and time Almheiri et al. 2019.

Competing surfaces and renormalization check

Search for a second extremum, multi-interval islands, and replica-symmetry-breaking topologies. A local minimum can lose globally to another saddle or fail homology. In Lorentzian time dependence, maximin arguments require the appropriate global-hyperbolicity and energy assumptions; importing a static minimal surface is insufficient.

Change the cutoff $\epsilon\to e^\sigma\epsilon$. The bulk entropy shifts locally, and $G_N^{\mathrm{ren}}$ plus $S_{\mathrm{local}}^{\mathrm{ren}}$ must shift so that differences between candidate generalized entropies are invariant. If only the logarithm is transformed, the apparent Page time or selected surface becomes scheme-dependent.

Controlled scope

The island rule is reliable within a declared semiclassical code sector and perturbative order, with controlled curvature and matter entropy. It can fail near singularities, Planckian endpoints, or when omitted saddles compete. What this body of calculations licenses—and what it leaves open—is evaluated on What Island Calculations Establish—and What They Do Not.

The chapter overview contains the structure diagram and validity and failure diagram. They are embedded there once so that their shared chapter-level context is not repeated on every article.

For the chapter-wide comparison of assumptions, counterevidence, falsifiers, and claim ceilings, see the claim-domain table.

References

• Almheiri, A., N. Engelhardt, D. Marolf, and H. Maxfield. “The Entropy of Bulk Quantum Fields and the Entanglement Wedge of an Evaporating Black Hole.” Journal of High Energy Physics 2019, 12 (2019): 063. DOI.
• Engelhardt, N., and A. C. Wall. “Quantum Extremal Surfaces: Holographic Entanglement Entropy beyond the Classical Regime.” Journal of High Energy Physics 2015, 1 (2015): 073. DOI.