{"id":"2e5710f7-a3e8-4fad-90cb-4e15f7bd2e6d","arxiv_id":"1204.4160","paper_version":3,"verdict":"ACCEPT","confidence":"HIGH","novelty_score":6.0,"correctness_risk":"low","formal_verification":"none","parameter_count":1,"one_line_summary":"For gapped (2+1)-dimensional theories the entanglement entropy's inverse-mass expansion contains only odd powers with coefficients built from even powers of extrinsic curvature, and for a holographic M-theory flow the renormalized entanglement entropy decreases monotonically.","lead":"Physicists computed how quantum entanglement entropy depends on the shape of the boundary region in two and three spatial dimensions, using both free-field and holographic methods. The results pin down which curvature terms can appear and confirm that a proposed measure of degrees of freedom decreases monotonically along a renormalization group flow.","discovery_kind":"unclear","skeptic_critique":{"model":"selfhosted/deepseek-v4.1-flash","headline":"The dimensional-reduction bridge (Eqs. 2.2-2.4) is the load-bearing step for the free-field coefficients, but the paper's own holographic route (Eqs. 3.21-3.22) independently reproduces the odd-power structure, so the concern does not overturn the central claim.","rationale":"The reader identified the right load-bearing assumption (Eqs. 2.2-2.4) and rated correctness risk low. I agree with the identification but would sharpen the weighting: the reduction bridge supports only the free-field coefficients (Eqs. 2.7, 2.8) and the conjectural Eq. 2.9-2.10, whereas the paper's headline structural claim in Eq. 1.6 is independently supported by the holographic large-Lambda expansion of Sec. 3.2, which yields Eqs. 3.21-3.22 with the even-curvature structure manifest and no torus involved. That redundancy is why the concern, even if it landed, would not move the verdict. The one place I would push harder than the reader is the unshown reduction of the two second-fundamental-form invariants to kappa^2(theta) between Eqs. 2.5 and 2.7, which fixes the relative factor of 3 between scalars and fermions in Eq. 2.7; that is a concrete arithmetic check rather than a structural worry. On the other results: the CGLP monotonicity check (Fig. 2) is correctly characterized as a confirmation of the general proof in [23], not a new proof, and the paper says so. The wedge result (Fig. 5) is a genuine check because the normalization a ~ 1.11 is fixed only at Omega -> 0 and the agreement at finite angle is then non-trivial; the cone factor-of-two discrepancy with the naive smooth-surface formula (Eqs. 5.21 vs 5.30) is flagged honestly by the authors, and the holographic derivation in Eqs. 5.26-5.30 is the more trustworthy of the two. The classifier reports no stated limitations, but the text does acknowledge the undetermined coefficients in Eq. 2.9 and the factor-of-two cone discrepancy, so the record understates the authors' candor. No concern here rises to the level of changing ACCEPT, and I would leave the verdict unchanged.","tokens_in":65936,"tokens_out":99514,"duration_ms":1819808,"concrete_test":"Re-derive Eq. 2.7 from scratch: take Sigma^2 = Sigma^1 x S^1 with the two normals given in the text (n1 timelike along t, n2 spacelike in the r-theta plane), compute k^a_{mu nu}k^{mu nu}_a and k^mu_{mu a}k^{nu nu}_a explicitly for the curve r = R(theta), and confirm both equal kappa^2(theta) with the sign conventions of Eq. 1.1. Then re-evaluate the c-anomaly integral to check the coefficient -1/480 (n_0 + 3 n_{1/2}), in particular the factor of 3 between scalar and Dirac contributions. If the coefficient shifts, Eq. 2.8's 1/m term changes and the free-field side of the structural claim weakens, though the holographic Eqs. 3.21-3.22 would be unaffected.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The reader's weakest_assumption is well chosen: Eqs. 2.2-2.4 identify the 1/m^{2n+1} coefficient of the (2+1)-dimensional entropy with the log divergence of the (2n+4)-dimensional theory on Sigma x T^{2n+1}. Two things make this less fragile than it first appears. First, the bridge is not the only support for the structural claim. The holographic calculation of Sec. 3.2 derives the 1/m^3 coefficient (Eqs. 3.21-3.22) directly from the large-Lambda expansion of the minimal-surface functional, with no torus and no KK reduction; the result is manifestly built from kappa^4 and (d kappa/d s)^2, i.e. even powers of extrinsic curvature. So even if the reduction argument had a hidden assumption, the odd-power/even-curvature structure would survive on the holographic side. Second, the reduction step itself is standard and the paper is explicit about the one place it could bite: the fermion counting factor 2^{n+1} in the sentence after Eq. 2.4, and the fact that Eq. 2.9's coefficients (A_0, A_{1/2}, B_0, B_{1/2}) are left undetermined. That is an acknowledged gap, not a concealed one. The genuinely soft spot is narrower: Eq. 2.7's numerical coefficient -1/480 (n_0 + 3 n_{1/2}) rests on the 4d anomaly formula of [7] plus the claim that both k^a_{mu nu}k^{mu nu}_a and k^mu_{mu a}k^{nu nu}_a reduce to kappa^2(theta) for Sigma^2 = Sigma^1 x S^1. That reduction is asserted in the text between Eqs. 2.5 and 2.7 rather than shown term by term, and the relative factor of 3 between the scalar and fermion contributions is exactly what a sign or normalization slip there would corrupt. This is a checkable arithmetic step, not a conceptual hole.","agreement_with_reader":"agree"},"referee_report":{"model":"selfhosted/deepseek-v4.1-flash","summary":"The manuscript studies the shape dependence of entanglement entropy in (3+1)- and (2+1)-dimensional field theories. In (3+1)-dimensional CFT it analyzes the divergent terms for entangling surfaces with wedge and conical singularities, finding an L/ε term for the wedge and a log²(r_max/ε) term for the cone. In (2+1)-dimensional gapped theories it argues that the large-mass expansion contains only odd powers of 1/m, with coefficients built from even powers of the extrinsic curvature and its derivatives, and computes the first coefficients explicitly for free fields (Eq. 2.7) and holographically (Eqs. 3.14 and 3.21-3.22). It then applies the holographic formalism to the CGLP M-theory background, computes the renormalized entanglement entropy F(R), and confirms numerically that it is monotonic and approaches zero as ~1/R.","tokens_in":22315,"tokens_out":16021,"duration_ms":298498,"significance":"The paper is a careful and useful contribution. The explicit coefficients and the structural argument for odd powers are valuable, and the CGLP computation is a nontrivial check of the F-theorem in a strongly coupled, gapped theory with a known gravity dual. The observation that the normalized wedge and cusp functions agree is interesting and worth recording. The paper is clearly written and the derivations are transparent; the dimensional-reduction bridge in Eqs. 2.2-2.4 is standard and is supported by the cited literature, and the holographic route in Sec. 3.2 independently reproduces the odd-power/even-curvature structure without relying on it.","major_comments":[],"minor_comments":[{"comment":"The reduction of the two second-fundamental-form invariants k^a_{μν}k^{μν}_a and k^μ_{μa}k^{νν}_a to κ²(θ) for Σ² = Σ¹×S¹ is stated in a single sentence (``It follows that...'') rather than shown. Since the relative factor of 3 between n_0 and n_{1/2} in Eq. 2.7 depends on this reduction, a short explicit display of the two contractions would make the free-field coefficient self-contained and reproducible.","section":"§2, Eqs. 2.5-2.7","authors_acknowledge":null,"authors_acknowledge_p":null},{"comment":"The coefficients A_0, A_{1/2}, B_0, B_{1/2} in Eq. 2.9 are left undetermined, so the free-field c̃^Σ_{-3} in Eq. 2.10 is not actually computed. The text acknowledges this, but the abstract's ``we calculate... the expansion'' could be read as more than is delivered; consider stating in the abstract that only the first coefficient is computed explicitly for free fields, with c̃_{-3} obtained holographically.","section":"§2, Eqs. 2.9-2.10","authors_acknowledge":null,"authors_acknowledge_p":null},{"comment":"The claim that the normalized wedge function f̃^{(hol)}_wedge agrees with f^{(hol)}_cusp rests on a numerical comparison with a one-parameter normalization tuned at Ω→0. The text should quantify the agreement (for example, the maximum fractional deviation over the plotted range) rather than stating ``within the numerical accuracy,'' so the reader can judge how nontrivial the match is.","section":"§5.1.2, Fig. 5","authors_acknowledge":null,"authors_acknowledge_p":null},{"comment":"The factor-of-2 discrepancy between the naive application of (1.1) and the precise holographic result is explained, but the text should state explicitly that the log² coefficient is not claimed to be universal and that (1.1) is being used only as a heuristic for singular surfaces. This would prevent the naive result in Eq. 5.21 from being read as a competing prediction.","section":"§5.2, Eqs. 5.21 and 5.30","authors_acknowledge":null,"authors_acknowledge_p":null},{"comment":"The numerical values 0.1959 and 1.845×10^{-2} in Eq. 4.8 are quoted without an estimate of numerical error, and the behavior of F(R) across R_crit (the topology-changing transition) is not quantified. A brief statement of the numerical precision and of the continuity of F at R_crit would strengthen the monotonicity check.","section":"§4, Fig. 2 and Eq. 4.8","authors_acknowledge":null,"authors_acknowledge_p":null},{"comment":"The symbol ε is used both for the UV cutoff (elsewhere ϵ) and for the Stenzel deformation parameter in Eq. 4.1; consider distinguishing them to avoid confusion. There are also a few typographical slips, including ``entangelement'' in the discussion around Eq. 1.5, ``Preforming'' in §5.1.1, and ``Cvetiˇ c'' for Cvetič in §1.","section":"§1 and §4","authors_acknowledge":null,"authors_acknowledge_p":null},{"comment":"Reference [6] carries an unusual bibliographic entry (``Latex, 30 pages, invited review for Classical and Quantum Gravity''); please clean this up to a standard journal citation.","section":"References","authors_acknowledge":null,"authors_acknowledge_p":null},{"comment":"The introduction states ``we find f_wedge(Ω) = f_cusp(Ω) both in the free scalar field theory and in the holographic calculations.'' In the holographic case the equality holds only after the one-parameter normalization described in §5.1.2; the introduction should say ``after normalization'' to match the body of the paper.","section":"§1, Introduction","authors_acknowledge":null,"authors_acknowledge_p":null}],"recommendation":"accept","confidential_remarks":null},"author_rebuttal":null,"desk_editor":{"model":"selfhosted/deepseek-v4.1-flash","letter":"This one is worth your time if you work on entanglement entropy in gapped or holographic theories. The core result is the general-contour expansion in Eq. 1.6 — for a gapped (2+1)-dimensional theory, S = alpha*l/epsilon + beta*m*l - gamma + sum c_n/m^{2n+1}, with each coefficient an integral of even powers of the extrinsic curvature and its derivatives. The odd-power structure was argued before (their ref. [37]), but the explicit coefficients are new: c_{-1} for free scalars and fermions (Eq. 2.7), and both c_{-1} and c_{-3} holographically (Eqs. 3.14, 3.21, 3.22). The c_{-3} derivation is the nicest piece of technical work in the paper — a controlled large-Lambda expansion of the minimal-surface functional that lands on kappa^4 and (d kappa/ds)^2 with no torus and no Kaluza-Klein machinery. Section 5 is also solid: the wedge picks up an L/epsilon divergence whose coefficient matches the (2+1)-dimensional cusp anomaly after normalization, and the cone's log^2 divergence comes out a factor of two below what the naive smooth-surface formula gives. They flag that factor of two themselves rather than burying it, which I appreciate. The CGLP renormalized entropy curve (Fig. 2) is a check of monotonicity already proved generally in [23], not a new proof, and they present it as such. On the soft spots: the dimensional-reduction bridge in Eqs. 2.2-2.4 is standard and, more to the point, is not load-bearing for the structural claim, since the holographic route gets the same structure independently. The genuinely soft spot is narrower — the step between Eqs. 2.5 and 2.7, where both k^a k_a and (k^mu_mu)^2 are asserted to collapse to kappa^2(theta) for Sigma^1 x S^1. That is checkable arithmetic, and the relative factor of 3 between scalar and fermion contributions is exactly what a slip there would corrupt. Worth ten minutes with a pencil. The free-field c_{-3} is left as a structural conjecture with undetermined coefficients (Eqs. 2.9-2.10); they say so. The wedge/cusp match uses one tuned normalization a ~ 1.11 fixed at small angle, so the agreement at other angles is a real check but a one-parameter one. None of this threatens the paper. Send it out.","headline":"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.","tokens_in":23155,"tokens_out":1199,"would_cite":true,"duration_ms":23391,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"pith_extraction":{"msc":[],"pacs":["11.25.Tq","03.67.Mn"],"model":"selfhosted/deepseek-v4.1-flash","headline":"Entanglement entropy's shape-dependent terms are built from even powers of curvature","keywords":["entanglement entropy","holographic entanglement entropy","renormalization group flow","F-theorem","extrinsic curvature","conformal anomaly","gapped field theory","CGLP background"],"falsifier":"A direct numerical or lattice computation of the $1/m^3$ coefficient for a free massive field on a contour with non-vanishing $dκ/ds$ would settle it: the paper's Eq. (2.10) predicts this coefficient is a specific combination of $∮ds κ^4$ and $∮ds (dκ/ds)^2$, so a computed value that requires an odd power of $κ$, or that fails to fit that two-term form, would break the structural claim.","tokens_in":23211,"feed_emoji":"🌀","tokens_out":3424,"duration_ms":67134,"temperature":0.7,"pith_summary":"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.","feed_headline":"Entanglement entropy's shape terms are even in curvature","feed_subtitle":"For gapped 2+1-dimensional theories, every inverse-mass coefficient is an integral of even powers of extrinsic curvature.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Supplies the geometric form of the smooth-surface entanglement entropy in $(3+1)$-dimensional CFT, Eq. (1.1), which the paper extends to singular surfaces.","marker":"[7]"},{"why":"Provides the cusp entanglement entropy and the function $f_{cusp}(Ω)$ in $(2+1)$ dimensions, the benchmark the wedge result is compared against.","marker":"[8]"},{"why":"Gives the holographic cusp calculation whose normalized coefficient the wedge computation is matched to.","marker":"[10]"},{"why":"Original Ryu-Takayanagi proposal, the holographic minimal-surface prescription used for all gravity computations here.","marker":"[11]"},{"why":"Establishes $S''(R) ≤ 0$ and hence monotonicity of the renormalized entanglement entropy, the property the CGLP computation confirms numerically.","marker":"[23]"},{"why":"Introduces the renormalized entanglement entropy $F(R) = -S(R) + RS'(R)$ and the large-$R$ limit to $F_{IR}$ that this paper computes for CGLP.","marker":"[24]"},{"why":"Defines the CGLP eleven-dimensional warped-throat background that serves as the concrete dual of the RG flow studied.","marker":"[26]"},{"why":"Casini and Huerta's method for extracting the $1/m$ expansion of entanglement entropy for massive free fields, which the paper generalizes from circles to arbitrary contours.","marker":"[4]"},{"why":"Argues that shape-dependent coefficients in gapped phases are even powers of extrinsic curvature, the structural claim this paper derives and computes explicitly.","marker":"[37]"}],"fun_headline_variants":["Only even powers of extrinsic curvature enter gapped entropy's shape terms","Gapped 2+1D entropy expands in odd inverse-mass powers with even curvature coefficients","Renormalized entanglement entropy is monotonic along the CGLP flow","Wedge and cone surfaces sharpen entanglement entropy divergences in 3+1D CFT","Shape dependence of gapped entropy reduces to even extrinsic curvature integrals"],"cache_read_input_tokens":0,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Only even powers of extrinsic curvature enter gapped entropy's shape terms","Gapped 2+1D entropy expands in odd inverse-mass powers with even curvature coefficients","Renormalized entanglement entropy is monotonic along the CGLP flow","Wedge and cone surfaces sharpen entanglement entropy divergences in 3+1D CFT","Shape dependence of gapped entropy reduces to even extrinsic curvature integrals"]},"model":"selfhosted/deepseek-v4.1-flash","effort":"low","attempts":1,"cost_usd":0.0,"raw_usage":{"total_tokens":2337,"prompt_tokens":1159,"completion_tokens":1178,"prompt_tokens_details":null,"completion_tokens_details":{"reasoning_tokens":1090}},"tokens_in":1159,"tokens_out":1178,"duration_ms":21419,"image_pages":[],"temperature":1.0,"reasoning_budget":50,"reasoning_tokens":1090,"transport_provider":"selfhosted","cache_read_input_tokens":0,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-11-10T14:52:27.671696+00:00","model_set":{"reader":"selfhosted/deepseek-v4.1-flash"},"falsifier":"A direct numerical or lattice computation of the $1/m^3$ coefficient for a free massive field on a contour with non-vanishing $dκ/ds$ would settle it: the paper's Eq. (2.10) predicts this coefficient is a specific combination of $∮ds κ^4$ and $∮ds (dκ/ds)^2$, so a computed value that requires an odd power of $κ$, or that fails to fit that two-term form, would break the structural claim.","supporting_citations":[{"cited_title":"Entanglement entropy, conformal invariance and extrinsic geometry","cited_arxiv_id":"0802.3117","evidence_quote":"Supplies the geometric form of the smooth-surface entanglement entropy in $(3+1)$-dimensional CFT, Eq. (1.1), which the paper extends to singular surfaces."},{"cited_title":"Universal terms for the entanglement entropy in 2+1 dimensions","cited_arxiv_id":"hep-th/0606256","evidence_quote":"Provides the cusp entanglement entropy and the function $f_{cusp}(Ω)$ in $(2+1)$ dimensions, the benchmark the wedge result is compared against."},{"cited_title":"AdS/CFT and Strong Subadditivity of Entanglement Entropy","cited_arxiv_id":"hep-th/0608213","evidence_quote":"Gives the holographic cusp calculation whose normalized coefficient the wedge computation is matched to."},{"cited_title":"Holographic Derivation of Entanglement Entropy from AdS/CFT","cited_arxiv_id":"hep-th/0603001","evidence_quote":"Original Ryu-Takayanagi proposal, the holographic minimal-surface prescription used for all gravity computations here."},{"cited_title":"On the RG running of the entanglement entropy of a circle","cited_arxiv_id":"1202.5650","evidence_quote":"Establishes $S''(R) ≤ 0$ and hence monotonicity of the renormalized entanglement entropy, the property the CGLP computation confirms numerically."},{"cited_title":"A refinement of entanglement entropy and the number of degrees of freedom","cited_arxiv_id":"1202.2070","evidence_quote":"Introduces the renormalized entanglement entropy $F(R) = -S(R) + RS'(R)$ and the large-$R$ limit to $F_{IR}$ that this paper computes for CGLP."},{"cited_title":"Ricci-flat Metrics, Harmonic Forms and Brane Resolutions","cited_arxiv_id":"hep-th/0012011","evidence_quote":"Defines the CGLP eleven-dimensional warped-throat background that serves as the concrete dual of the RG flow studied."},{"cited_title":"Entanglement entropy in free quantum field theory","cited_arxiv_id":"0905.2562","evidence_quote":"Casini and Huerta's method for extracting the $1/m$ expansion of entanglement entropy for massive free fields, which the paper generalizes from circles to arbitrary contours."},{"cited_title":"Entanglement Entropy of Gapped Phases and Topological Order in Three dimensions","cited_arxiv_id":"1108.4038","evidence_quote":"Argues that shape-dependent coefficients in gapped phases are even powers of extrinsic curvature, the structural claim this paper derives and computes explicitly."}],"review_version":1}