FLM Corrections and Holographic Use of Imported Generalized Entropy and QES

FLM and QES answer related but different questions. The FLM correction evaluates renormalized bulk entropy and local one-loop terms on the fixed classical RT/HRT surface at order $G_N^0$. A quantum extremal surface instead makes the full renormalized generalized entropy stationary and can move by $O(G_N)$; one must then compare all admissible QES saddles. Neither prescription permits adding a cutoff-dependent matter entropy to a renormalized area term in isolation. We import the semiclassical generalized-entropy and QES definitions from the curved-spacetime volume and apply them to holographic boundary entropy.

Required background. Generalized-entropy renormalization fixes the joint counterterms, and the semiclassical QES definition fixes the variational object and domain.

Helpful background. RT supplies the classical surface; higher-derivative entropy supplies local gravitational terms; and horizon entanglement explains the UV pairing across a surface.

Three orders, three statements

Let $X$ denote an anchored, homologous codimension-two surface and $\Sigma_X$ the bulk region between $A$ and $X$. The renormalized generalized entropy has the schematic expansion

$$S_{\rm gen}^{\rm ren}[X] =S_{\rm grav}^{\rm ren}[X] +S_{\rm bulk}^{\rm ren}(\Sigma_X) +S_{\rm edge}^{\rm ren}[X].$$

For Einstein gravity, $S_{\rm grav}$ begins with $\operatorname{Area}(X)/(4G_N^{\rm ren})$; higher-curvature and loop-induced local terms belong to the same renormalized functional.

The prescriptions separate as follows.

1. RT/HRT, order $G_N^{-1}$: extremize classical area and select the least admissible classical extremum $X_0$.
2. FLM, order $G_N^0$: evaluate the one-loop generalized-entropy correction on $X_0$, including the metric backreaction and local renormalization terms at the same order. The surface displacement does not contribute at this order because $X_0$ is classically extremal.
3. QES: solve $$\frac{\delta S_{\rm gen}^{\rm ren}}{\delta X^a(y)}=0$$ for every admissible branch, then select the branch with least generalized entropy according to the prescription and homology constraints.

Faulkner, Lewkowycz, and Maldacena derived the bulk-entanglement correction around the classical replica saddle (Faulkner, Lewkowycz, and Maldacena 2013, §§2–4). Engelhardt and Wall formulated the quantum-extremal-surface generalization (Engelhardt and Wall 2015, §§2–3). These statements share an expansion but must not be called the same prescription.

A regulated bulk-mode correction

As an explicit one-loop contribution, isolate one regulated pair of bulk wave-packet modes on opposite sides of the classical surface. Let their state be a two-mode squeezed vacuum

$$|\Psi_r\rangle =\frac{1}{\cosh r} \sum_{n=0}^{\infty}(\tanh r)^n |n\rangle_{\Sigma_{X_0}}|n\rangle_{\overline\Sigma_{X_0}}.$$

Tracing either mode gives mean occupation $\bar n=\sinh^2r$ and entropy

$$S_{\rm mode}(X_0) =(\bar n+1)\log(\bar n+1)-\bar n\log\bar n.$$

At fixed classical RT surface, this is an $O(G_N^0)$ contribution to the FLM term. In a full field theory one sums or integrates over regulated modes and adds the geometric counterterms; the single pair makes the state dependence and order transparent. If all other one-loop contributions are denoted $S_{\rm rest}^{(1)}$, then

$$S_A =\frac{\operatorname{Area}(X_0)}{4G_N^{\rm ren}} +S_{\rm mode}(X_0)+S_{\rm rest}^{(1)}(X_0) +O(G_N).$$

This is an FLM evaluation, not yet a QES calculation.

Deriving the quantum surface shift

Write the surface as $X=X_0+\delta X$ and let $H_{ab}(y,y')$ be the Hessian of the classical area at $X_0$. Since $\delta A/\delta X|_{X_0}=0$, quantum extremality gives at first nontrivial order

$$\frac{1}{4G_N} \int dy'\,H_{ab}(y,y')\delta X^b(y') +\frac{\delta S_{\rm bulk}^{\rm ren}} {\delta X^a(y)}\bigg|_{X_0} +\frac{\delta S_{\rm local}^{(1)}} {\delta X^a(y)}\bigg|_{X_0}=0.$$

On a nondegenerate branch,

$$\delta X^a =-4G_N(H^{-1})^{ab} \frac{\delta(S_{\rm bulk}^{\rm ren}+S_{\rm local}^{(1)})} {\delta X^b}\bigg|_{X_0} +O(G_N^2).$$

For the squeezed mode, if $r=r(X)$, its shape derivative is

$$\frac{dS_{\rm mode}}{dX} =\frac{d\bar n}{dX} \log\!\left(\frac{\bar n+1}{\bar n}\right),$$

which enters the source on the right-hand side. The shift is $O(G_N)$. Substituting it back into the area changes $S_A$ only at $O(G_N)$, because the linear area variation vanished. This explicitly explains why FLM at order $G_N^0$ uses the fixed surface while QES determines the next displacement.

If the Hessian has a zero mode or two branches nearly tie, the expansion is not uniform. One must solve and compare the full generalized entropy; a small matter term can then cause an $O(1)$ change of the dominant branch.

Renormalization adversarial test

Suppose a cutoff calculation gives

$$S_{\rm bulk}^{\rm bare}(X) =c_{\rm div}(\epsilon)\operatorname{Area}(X) +S_{\rm bulk}^{\rm finite}(X)+\cdots.$$

The same loops renormalize Newton’s coupling so that

$$\frac{1}{4G_N^{\rm ren}} =\frac{1}{4G_N^{\rm bare}}+c_{\rm div}(\epsilon),$$

in this sign convention, together with higher-curvature couplings for subleading geometric divergences. A scheme change $c_{\rm div}\to c_{\rm div}+\eta$ must be accompanied by the compensating coupling redefinition. If one changes only $S_{\rm bulk}$, the purported entropy and its shape derivative shift by

$$\Delta S_{\rm fake}=\eta\operatorname{Area}(X),$$

moving the alleged QES. This spurious scheme dependence directly falsifies the incomplete calculation. The renormalized sum, not either summand, is the observable semiclassical input.

Limits and saddle competition

FLM assumes a semiclassical state on a fixed classical entanglement wedge and computes through one loop. QES adds quantum extremization but still requires a code/algebra choice, a renormalized functional, homology, and comparison among all extrema. Replica-symmetry breaking, nonperturbative gravitational saddles, or order-one fluctuations can lie beyond the expansion.

Replica derivations explains when the area and bulk-entropy terms emerge from a gravitational path integral. Later wedge-reconstruction arguments may use the selected QES, but they do not replace its renormalization or saddle checks.

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

• Engelhardt, N., and Wall, A. C. (2015). “Quantum extremal surfaces: Holographic entanglement entropy beyond the classical regime.” Journal of High Energy Physics 2015(1), 073. DOI.
• Faulkner, T., Lewkowycz, A., and Maldacena, J. (2013). “Quantum corrections to holographic entanglement entropy.” Journal of High Energy Physics 2013(11), 074. DOI.