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On Shape Dependence and RG Flow of Entanglement Entropy

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Igor R. Klebanov, Tatsuma Nishioka, Silviu S. Pufu, Benjamin R. Safdi

High Energy Physics - Theory · Submitted 18 Apr 2012 · 47 references

Pith's verdict: A careful, honest paper: the general-contour 1/m expansion and the holographic c_{-3} are the real contributions, and the soft spots are narrow and mostly acknowledged.

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arXiv ID1204.4160 (version 3)
Pith IDCYBOXTRO, Pith's permanent record of this paper and its review
CitationBenjamin R. Safdi, Igor R. Klebanov, Silviu S. Pufu, Tatsuma Nishioka. "On Shape Dependence and RG Flow of Entanglement Entropy." arXiv:1204.4160 (2012). https://pith.science/paper/CYBOXTRO
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The paper

The paper claims

Entanglement entropy's shape-dependent terms are built from even powers of curvature

The reading

This paper asks what the entanglement entropy of a region actually depends on once the leading area law is stripped away. For a gapped $(2+1)$-dimensional theory, the authors argue that the entropy across an arbitrary smooth contour has the form $S_Σ = αℓ_Σ/ε + βmℓ_Σ - γ + Σ_n c_{-1-2n}/m^{2n+1}$, and that every shape-dependent coefficient $c_{-1-2n}$ is an integral along the contour of even powers of the extrinsic curvature $κ$ and its derivatives. The reason only odd powers of $1/m$ appear is that each such coefficient is tied, by dimensional reduction on a torus, to a logarithmic conformal anomaly in a higher even-dimensional theory, where odd powers of the curvature cannot appear. The authors check the structure two ways: with explicit free massive scalar and Dirac fields, and holographically in a general gapped supergravity background, where they compute $c_{-1}$ and $c_{-3}$ in closed form. They then run the whole machinery on a specific eleven-dimensional warped-throat background (CGLP), confirming that the renormalized entanglement entropy decreases monotonically along the flow and vanishes like $1/R$ in the infrared.

What carries the argument

The load-bearing device is the identification of the $1/m^{2n+1}$ term in the $(2+1)$-dimensional entropy with the logarithmic anomaly of the $(2n+4)$-dimensional theory on $Σ × T^{2n+1}$ (Eqs. 2.2–2.4). It converts a question about a gapped theory, where no conformal anomaly exists, into a question about an anomaly polynomial in a higher even dimension, where the answer is fixed by geometry: only even powers of the extrinsic curvature and its derivatives can appear, so only odd powers of $1/m$ can appear in three dimensions. On the holographic side the corresponding machinery is the minimal-surface area functional in a background of the form $ds^2 = α(u)[du^2 + β(u)dx_μdx^μ] + g_{ij}dy^idy^j$ with a collapsing internal volume $V(u_0)=0$; expanding the embedding $r(u,θ) = ΛR(θ) + δ(u,θ)/Λ$ in inverse powers of the contour scale produces the coefficients $c_{-1}$ and $c_{-3}$ as integrals of $g(u)$ and $β(u)$ over the radial direction. The renormalized entanglement entropy $F(R) = -S(R) + R S'(R)$, whose derivative is $R S''(R)$, is the quantity used to test monotonicity along the CGLP flow.

Extended reading notes

Core claim

The central claim is a structural statement about entanglement entropy in gapped $(2+1)$-dimensional theories: for any smooth entangling contour $Σ$, the expansion in inverse powers of the mass gap contains only odd powers, and the coefficient of each $1/m^{2n+1}$ is an integral over $Σ$ of a local density built from even powers of the extrinsic curvature $κ$ and its derivatives. The authors derive this by a dimensional-reduction argument: place the theory on $Σ × T^{2n+1}$ with a small mass $M$, so that the Kaluza-Klein tower reproduces the $(2+1)$-dimensional entropy integrated over masses; the term scaling as $1/m^{2n+1}$ then becomes precisely the logarithmic conformal anomaly of the $(2n+4)$-dimensional theory, whose anomaly is a polynomial in even powers of the curvature. For $n=0$ this yields the explicit free-field coefficient $c_{-1} = -[(n_0 + 3n_{1/2})/480]∮ ds κ^2$ and the entropy $S = αℓ/ε - m(n_0+n_{1/2})ℓ/12 - (n_0+3n_{1/2})/(480m)∮ds κ^2 + O(1/m^3)$. Holographically, the authors solve the minimal-surface problem in a general gapped background with a shrinking internal space and read off $c_{-1}$ and $c_{-3}$ as integrals over the background warp factors, finding $c_{-3}$ to be a combination of $∮ds κ^4$ and $∮ds (dκ/ds)^2$. The paper also treats singular surfaces in $(3+1)$-dimensional CFT: a wedge of opening angle $Ω$ produces an $L/ε$ divergence whose coefficient, after a single normalization constant, coincides numerically with the known cusp function, and a cone produces a $∼ -[π cos^2Ω/(8 sinΩ)] log^2(r_max/ε)$ divergence, a factor of two smaller than the naive extrapolation of the smooth-surface anomaly formula.

Load-bearing premise

The argument rests on the dimensional-reduction step that equates the $1/m^{2n+1}$ coefficient of the three-dimensional entropy with the logarithmic anomaly of the $(2n+4)$-dimensional theory on $Σ × T^{2n+1}$, which assumes the entropy factorizes cleanly over the compact torus directions and that the replica trick commutes with Kaluza-Klein reduction.

Editorial extensions

If this is right

  • In any gapped $(2+1)$-dimensional theory with a mass gap, the shape dependence of the entanglement entropy at order $1/m$ is fixed up to a single number by $∮ds κ^2$, so a measurement or lattice computation of that coefficient tests the whole framework rather than an arbitrary function of shape.
  • The vanishing of even powers of $1/m$ in the expansion gives a sharp, checkable prediction: any smooth contour's entropy, plotted against $1/m$, should fall on an odd-power curve with no constant or $1/m^2$ contamination beyond the topological term.
  • For the CGLP flow, the renormalized entanglement entropy is monotonic and falls off like $1/R$ in the infrared, with a second-order transition at $R_{crit} ≈ 0.73$ where the bulk minimal surface changes topology from disk-type to cylinder-type.
  • In $(3+1)$-dimensional CFT, a conical entangling surface produces a $\log^2(r_{max}/ε)$ divergence with coefficient $-\pi\cos^2Ω/(8\sinΩ)$ in the holographic case, half the value obtained by naively applying the smooth-surface anomaly formula.
  • The wedge $L/ε$ coefficient and the $(2+1)$-dimensional cusp coefficient appear to be the same function of opening angle once normalized, in both free scalar theory and holography.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If the dimensional-reduction bridge holds, the shape dependence of gapped entanglement entropy is not an independent dynamical question at all but a bookkeeping exercise in higher-dimensional anomaly polynomials; that would explain why free fields and strongly coupled holographic theories show the same functional form of shape dependence even when their coefficients differ.
  • The factor-of-two mismatch between the naive smooth-surface formula and the exact conical computation suggests that anomaly-based formulas for singular surfaces should be trusted only up to order-one coefficients, so any proposed extraction of central charges from singular entangling surfaces needs an independent normalization.
  • The topology-changing transition at $R_{crit}$ in the CGLP background is a candidate signature for gapped holographic theories more broadly: a discontinuity or kink in $F'(R)$ at the radius where the minimal surface first reaches the bottom of the throat would be a distinctive, searchable feature.
  • Because the coefficients are integrals of local curvature invariants, one could in principle invert the logic: measure the entropy of several differently shaped contours at fixed large mass and fit for the anomaly coefficients of the parent higher-dimensional theory.
Original abstract

We use a mix of field theoretic and holographic techniques to elucidate various properties of quantum entanglement entropy. In (3+1)-dimensional conformal field theory we study the divergent terms in the entropy when the entangling surface has a conical or a wedge singularity. In (2+1)-dimensional field theory with a mass gap we calculate, for an arbitrary smooth entanglement contour, the expansion of the entropy in inverse odd powers of the mass. We show that the shape-dependent coefficients that arise are even powers of the extrinsic curvature and its derivatives. A useful dual construction of a (2+1)-dimensional theory, which allows us to exhibit these properties, is provided by the CGLP background. This smooth warped throat solution of 11-dimensional supergravity describes renormalization group flow from a conformal field theory in the UV to a gapped one in the IR. For this flow we calculate the recently introduced renormalized entanglement entropy and confirm that it is a monotonic function.

The compiled record

The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Compiled by classifier-1.13-20260917 via OpenRouter, question bank abstract-p1-20260923, 2026-09-23: 1 request, 5,642 input tokens, $0.0002 reported.

What the abstract itself states, as probabilities: claims first 0.03 · states limitation 0.06 · human participants 0.01 · controlled comparison 0.02 · randomized assignment 0.01 · states quantitative result 0.14 · compares with prior results 0.14 · mentions code or data availability 0.01.

The abstract, sentence by sentence

arXiv listing abstract v3. Results and the main claim are underlined; .

We use a mix of field theoretic and holographic techniques to elucidate various properties of quantum entanglement entropy. In (3+1)-dimensional conformal field theory we study the divergent terms in the entropy when the entangling surface has a conical or a wedge singularity. In (2+1)-dimensional field theory with a mass gap we calculate, for an arbitrary smooth entanglement contour, the expansion of the entropy in inverse odd powers of the mass. We show that the shape-dependent coefficients that arise are even powers of the extrinsic curvature and its derivatives. A useful dual construction of a (2+1)-dimensional theory, which allows us to exhibit these properties, is provided by the CGLP background. This smooth warped throat solution of 11-dimensional supergravity describes renormalization group flow from a conformal field theory in the UV to a gapped one in the IR. For this flow we calculate the recently introduced renormalized entanglement entropy and confirm that it is a monotonic function.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#sentencerolethe authors' own workstates the main conclusionstrength
s0 We use a mix of field theoretic and holographic techniques to elucidate various properties of quantum entanglement entropy. method 0.64 0.90 0.05 asserted 0.95
s1 In (3+1)-dimensional conformal field theory we study the divergent terms in the entropy when the entangling surface has a conical or a wedge singularity. contribution 0.99 0.91 0.05 asserted 0.91
s2 In (2+1)-dimensional field theory with a mass gap we calculate, for an arbitrary smooth entanglement contour, the expansion of the entropy in inverse odd powers of the mass. contribution 0.94 0.94 0.08 asserted 1.00
s3 We show that the shape-dependent coefficients that arise are even powers of the extrinsic curvature and its derivatives. result 1.00 0.92 0.22 asserted 1.00
s4 A useful dual construction of a (2+1)-dimensional theory, which allows us to exhibit these properties, is provided by the CGLP background. method 0.91 0.47 0.05 asserted 0.99
s5 This smooth warped throat solution of 11-dimensional supergravity describes renormalization group flow from a conformal field theory in the UV to a gapped one in the IR. method 0.68 0.24 0.07 asserted 1.00
s6 For this flow we calculate the recently introduced renormalized entanglement entropy and confirm that it is a monotonic function. result 0.98 0.92 0.27 asserted 1.00

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 0.99.

What was examined, and what was not

sources read
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not examined
  • full text
  • figures and tables
  • references and citing sentences
  • relations to other papers
levels run
  • P abstract
not run
  • S sections
  • T body sentences
  • C citations
  • R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

Figures
Figure 1. (a) Numerical solutions to the equation of motion for the holographic entangling surface, given by r(τ ), in the CGLP theory. The dotted red line indicates the critical value Rcrit, where the solutions change from disk-type to cylinder-type. (b) A zoomed-in plot of the UV region, with disk-type solutions, where we plot the AdS approximation in (4.4) in dotted black. (c) A zoomed-in plot of the IR region, with cylinder-type solutions, with the analytic approximation given by δ(τ ) in (3.11) plotted in dotted black. of R < Rcrit and R > Rcrit. In the far UV the solution for r(τ )…
Figure 2. The renormalized entanglement entropy F(R) along the RG flow in the CGLP theory plotted in orange. The left dotted black curve is the asymptotic UV approximation to F(R) given in (4.7). The right dotted black curve is the IR approximation to F(R) given in (4.8). The dotted red line marks the value R = Rcrit. 4.1 The renormalized entanglement entropy in the UV and the IR In the far UV we can treat the CGLP M-theory background as a perturbation of the AdS4 × V5,2 background. From (A.7), we know that the UV fixed point has a renormal￾14
Figure 3. (a) A plot of an example entangling surface Σ1, described by the function RUV(θ) in polar coordinates, in the CGLP theory. (b) The extrinsic curvature κUV(θ) for the entangling surface RUV(θ). The extrinsic curvature is small over the whole curve. 1 2 3 4 5 6 Θ -0.006 -0.004 -0.002 0.002 0.004 RUVHΘL-RIRHΘL
Figure 4. The function RUV(θ) − RIR(θ) in the CGLP theory, with RUV(θ) plotted in fig. 3. The solid orange curve is computed by numerically solving the equation of motion for the holographic entangling surface Σ2. The dotted black curve is an analytic approximation, which is equal to −δ(τ = 0, θ), with δ(0, θ) given in (3.20). both numerically and using (3.20), which is done in
Figure 5. The entanglement entropy for the wedge has a 1

Figures reproduced from arXiv:1204.4160 by the authors.

References
How later papers used it

57 later papers cite it

Evidence: substantial. 53 of the citing papers share no author with it, and 5 of those carry a confident label from Pith's citation classifier.

Pith read 25 of the 119 citing sentences, choosing only those whose citation markers point at this paper alone, and wrote the paraphrases from them. The summary, the counts, the evidence level, the links and the quotes come from code; the summary counts the papers listed below. A later paper is listed under a label when Pith's citation classifier gives one of its sentences that label at probability 0.8 or more (0.95 for support), Pith's reading finds that the citing paper itself did it, and a second check confirms the paraphrase against the sentence. This section lists uses only. It does not list criticism or disagreement, so their absence here says nothing about how the paper has been received. Papers that share an author with this one are listed apart, under "By the same authors". A quote is the citing authors' own sentence, shown when their paper's licence is CC BY, CC BY-SA or CC0. JSON

Builds on it 1 of 2 papers with this label · 1 by the same authors

By the same authors

  • Extended this paper's work for a disk to study the holographic entanglement entropy of the annulus in a confining gauge theory with a gap described by the CGLP background.

    Entanglement Entropy of Annulus in Three Dimensions (2015) (same authors) · 0.99 · paraphrased: the licence does not allow quoting

Uses its method or tool 1 paper · 1 by the same authors

By the same authors

  • Carried out the large mass expansion following this paper to fix the order 1/m term of the entropy for a general entangling surface.

    Renormalized Entanglement Entropy on Cylinder (2015) (same authors) · 0.98 · paraphrased: the licence does not allow quoting

Uses its data or values 1 of 2 papers with this label · 1 by the same authors

By the same authors

  • Used the known coefficients beta and gamma in this paper's entropy, beta=-1/12 and gamma=0, for a free massive scalar field on the annulus.

    Entanglement Entropy of Annulus in Three Dimensions (2015) (same authors) · 0.87 · paraphrased: the licence does not allow quoting

Compares with it 1 of 5 papers with this label · 1 by the same authors

By the same authors

  • Found that its own result for the coefficient c_1^Sigma reproduces this paper's in the R to infinity and Theta to 0 limit with R sin Theta kept fixed.

    Renormalized Entanglement Entropy on Cylinder (2015) (same authors) · 0.93 · paraphrased: the licence does not allow quoting

How the literature uses it

54 arXiv papers cite this in 116 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers citing it · 89 sentences without a confident label

Contests it 1 paper
Builds on it 2 papers
Uses its method or tool 1 paper
  • On the other hand, the large mass expansion will be carried out following [ref] and the order $1/m$ term of the entropy will be fixed for a general entangling surface.

    Renormalized Entanglement Entropy on Cylinder · Free massive scalar field · 0.98

Uses its data or values 2 papers
  • The calculation of the regularised volume of $γ_A$ with UV cutoff $r>ϵ$ and IR cutoff $ρ<l$ for the case, [$r$ is the Poincaré coordinate pointing into the interior of $AdS_5$ and $ρ$ is the Euclidean distance from the tip of the cone.] where $A$ is an infinite cone with opening angle $2Ω$, yields [ref] S_ϵ,l = =1/4G_N^(5)(π sinΩ/2l^2/ϵ^2 - π cosΩ Ω/8 log^2ϵ/l + O(logϵ)) .

    On the logarithmic divergent part of entanglement entropy, smooth versus singular regions · Introduction · 0.85

  • Let us apply the general discussion on the annulus entropy in Section 2 to a free massive scalar field whose action is defined by [equation] In this case, the coefficients $β$ and $γ$ in the entropy [ref] are known to be $β=-1/12$ and $γ=0$.[The topological entanglement entropy vanishes because there is an empty theory in the IR of the massive scalar theory.] The coefficients $c^Σ_2n+1$ are calculated [ref] up to $n=1$, being local integrals of functions of the extrinsic curvature $κ$ and $κ$'s derivatives on the $Σ$.

    Entanglement Entropy of Annulus in Three Dimensions · Free massive scalar field · 0.87

Compares with it 4 papers
  • Based on a naive identification of the two regularisation scales ($ϵ=ϵ'$), the comparison of the factor for the logarithmic divergence, given by Solodukhin's formula, in the limit where $∂A$ develops a conical singularity with the direct holographic calculation for the singular $∂A$ yields agreement up to a numerical mismatch factor of 2 [ref] .

    On the logarithmic divergent part of entanglement entropy, smooth versus singular regions · Conclusions · 0.95

  • A short calculation shows that $=2/R^2$, $_aa=1/R^2$, $_abab=0$, and [equation] Combining with [ref] and [ref], we find [equation] A few comments are in order: * Our result [ref] for the coefficient $c_1^Σ$ reproduces that of [ref] in the $R→∞$ and $Θ→0$ limit with $RsinΘ$ kept fixed, under which the entangling surface becomes a curve on $^2$.

    Renormalized Entanglement Entropy on Cylinder · Large mass expansion · 0.93

  • This result agrees with those in [ref] .

    Probing renormalization group flows using entanglement entropy · Asymptotic expansion of the REE · 0.99

Cites it as background 15 papers

Discussion

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