Entanglement entropy in higher derivative holography

Aninda Sinha; Apratim Kaviraj; Arpan Bhattacharyya

REVIEW

Entanglement entropy in higher derivative holography

Arpan Bhattacharyya, Apratim Kaviraj, Aninda Sinha

High Energy Physics - Theory · Submitted 29 May 2013 · 35 references

Pith's verdict: A useful, honest check that Lewkowycz-Maldacena reproduces the Gauss-Bonnet area functional — but the 'regime' where it works is asserted rather than derived, and the general four-derivative claim is conditional on assumptions the authors themselves call problematic.

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CitationAninda Sinha, Apratim Kaviraj, Arpan Bhattacharyya. "Entanglement entropy in higher derivative holography." arXiv:1305.6694 (2013). https://pith.science/paper/XHJS7NIW

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Editor's note

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The thing to know: this paper runs Lewkowycz-Maldacena on Gauss-Bonnet and gets back the Hung-Myers-Smolkin area functional, plus the T_BY_tt = 0 condition, in the small-extrinsic-curvature corner. That is a real check and worth having in the literature. The second result — that for general four-derivative gravity the ww divergence carries a ∇²K piece that no f(R,K) area functional can produce — is a clean structural argument and probably the most durable thing here. The counterterm observation in §4 is interesting but is a matching, not a derivation of necessity: they add a two-parameter family and find α = κ = 1 cancels the power laws. The self-citation to [16] is fine; it is a derivation being compared against, not a fitted input, so the check is not circular. The sphere exactness in §2.2 is a genuine result, not a restatement of [14,15]. Where it is soft: the 'regime where this method is applicable' is never quantified, and the justification in §2.2 — K ∝ 1/z0, so K³ is subleading at large z0 — compares K³ to K, not to the retained λL²(RK − 2R_ij K^ij). Both scale as λL²/z0³. The actual separation condition is R ≫ K², which fails for the sphere (R ~ K²) and is marginal for the cylinder. The sphere is saved because the difference eq (24) vanishes identically on f = sqrt(z0² − z²); that is a real cancellation, not a controlled approximation. For the cylinder the difference eq (27) is nonzero and the paper says so — solutions agree to z⁴ ln z but not near z = zh. So the agreement is exact where the discriminating terms cancel and leading-order elsewhere. The paper should say this plainly instead of invoking large z0. The general four-derivative conclusion is honestly hedged but rests on stacked assumptions (small K, small Q_ij, α²/ε ≪ 1) that the authors themselves call problematic; fine as a conditional statement, though the abstract's 'we compute the entangling surface in four derivative gravity' oversells it slightly. Appendix B's cylinder counterterm needs a fitted ζ = 3/(2π) with no independent derivation, which the authors flag. Who it is for: people working on holographic entanglement entropy in higher-derivative gravity, and anyone who wants a reference for LM-in-GB. Worth a serious referee — the GB check is solid, the negative result is clean, and the fixes needed are presentational plus one regime clarification.

Referee report

Major objections (3)

[Section 2.2 and Section 3] The justification for dropping the O(K^3) terms is not correct as stated. The paper argues (Section 2.2, after eq. 24; Section 3, after eq. 33) that since K ~ 1/z0, the O(K^3) terms are subleading to the O(K) terms in the large-z0 limit. But the retained correction is lambda L^2 (R K - 2 R_ij K^ij), which for a surface of size z0 scales as lambda L^2 / z0^3, and the dropped term lambda L^2 (-K^3 + 3 K K2 - 2 K3) also scales as lambda L^2 / z0^3. Large z0 therefore does not separate them; both are suppressed relative to the leading K by the same factor lambda L^2 / z0^2. The correct separation condition is R >> K^2 (intrinsic curvature large compared to extrinsic curvature squared), which fails for the sphere where R ~ K^2. The sphere is nevertheless an exact solution of both eq. (15) and eq. (23) because the difference eq. (24) vanishes identically on f = sqrt(z0^2 - z^2); this is a special cancellation, not a controlled approximation. For the cylinder the difference eq. (27) is nonzero and the paper itself states the solutions agree only up to z^4 ln z and differ near z = zh. Since the claim that the LM equation coincides with S_JM and T_BY_tt = 0 is load-bearing, the regime of validity should be stated correctly and quantitatively, and the special status of the sphere should be made explicit.
[Section 3 (general four-derivative case)] The conclusion that no area functional exists for the Weyl-squared combination rests on assumptions the authors themselves describe as problematic. Deriving eq. (39) requires dropping the O(epsilon/w^2) terms, which the text says needs alpha^2/epsilon << 1 and notes 'Making this assumption already appears problematic', and also requires Q_ij << w^2/epsilon. The abstract's statement that 'We use this method to compute the entangling surface in four derivative gravity' overstates what is actually established for the general case. The body is appropriately hedged ('it is unlikely that one exists'), but the abstract and the framing of question 3 in the Discussion should be softened to match the conditional nature of the derivation.
[Section 4 and Appendix B] The claim that S_JM 'arises as a counterterm' is stronger than what is demonstrated. In Section 4 the coefficients alpha and kappa are fixed by demanding cancellation of the power-law divergences, and they come out to alpha = kappa = 1; the nontrivial content is that the required counterterm has the S_JM structure with the specific R coefficient, which is worth stating clearly. However, the identification of the finite piece with the entanglement entropy uses the thermodynamic analogy T I_tot = -T S_EE with T = 1/(2 pi z0), which is an analogy rather than a derivation. In Appendix B the cylinder case is left incomplete: the temperature scale zeta is undetermined and the text calls this 'the key missing link'. The abstract's 'we demonstrate' should be qualified accordingly.

Minor objections (4)

[Appendix A, eq. (52)] The claimed vanishing of the combination K2 K_tt - 2 K K_ct K^tc - K_tt K^2 + 2 K_ta K^ab K_bt = 0 should be checked and the index conventions stated explicitly. For a diagonal induced metric with K_ti = 0, the first and third terms cancel identically and the remaining terms reduce to 2 K_tt K^tt (K_tt - K) = -2 K_tt K^tt K_ii, which does not vanish unless K_tt = 0 or K_ii = 0. If a different convention or a non-diagonal metric is intended, this should be spelled out, since eq. (54) and hence eq. (22) depend on it.
[Section 3, eqs. (31) and (35)] Footnote 7 notes that the coefficients of the RK and R_ab K^ab terms in eq. (35) had typos in previous versions. Given that history, the coefficients in eq. (31) and eq. (35) should be re-checked and, if possible, cross-checked against an independent computation, since these are the equations from which the central GB result eq. (34) follows.
[Section 2.2] The statement that 'for a general surface, we have K proportional to 1/z0' should be supplemented with the scaling of R, so that the reader can see which combinations are being compared. As written, the comparison of K^3 to K rather than to R K is the source of the ambiguity noted in the major comments.
[Section 2.2 and Discussion] The T_BY_tt = 0 check relies on ref. [16], which is a self-citation. The paper should state explicitly that this is an independent construction (a stress-tensor condition rather than a replica-based derivation) so that the reader can judge the independence of the check.

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The paper

The paper claims

Higher-derivative gravity gets one consistent entropy rule, but only for Gauss-Bonnet

The reading

Holographic entanglement entropy identifies the entropy of a region in a quantum field theory with the area of a minimal surface dipping into a higher-dimensional bulk. That prescription is well tested for ordinary Einstein gravity but not for the fourth-order curvature corrections that string theory predicts. This paper asks whether a recently proposed first-principles derivation of the surface equation, the Lewkowycz–Maldacena replica argument, survives when those corrections are added. For Gauss-Bonnet gravity the answer is yes: the derived equation coincides with the one obtained by extremizing the modified area functional $S_{JM} = (2/4G_N)$-style proposal $S_{JM} = 2π/ℓ_P^3 ∫ d^3x √Σ (1 + λL^2 R)$ and with the condition that the time-time component of the Brown-York stress tensor vanish on the entangling surface. For a general four-derivative theory the authors find the derived equation contains a $∇^2 K$ term that no purely geometric area functional appears able to reproduce. A final section shows the Gauss-Bonnet area functional also acts as the counterterm that cancels power-law divergences in the Euclidean action, leaving only the universal logarithmic piece.

What carries the argument

The load-bearing object is the modified area functional $S_{JM}$, which replaces the Ryu–Takayanagi area by $∫d^3x√Σ(1 + λL^2 R)$ — the area of the entangling surface plus a Gauss-Bonnet coupling times the Ricci scalar of the induced three-dimensional metric. Varying it with respect to the embedding coordinates, with the variation of the induced metric and the identity $∇_i(δL/δΣ_{ij}) = 0$, produces the surface equation $K + λL^2(RK - 2R^{ij}K_{ij}) = 0$. The second piece of machinery is the Lewkowycz–Maldacena expansion of the five-dimensional metric in Gaussian normal coordinates around the surface, $g_{ij} = h_{ij} + w{}^{(w)}K_{ij} + O(w^2)$, with $ρ = -(ε/2)log(x_1^2+x_2^2)$; the coefficients of $ε/w$ in the $ww$ component of the equations of motion then have to vanish, and for Gauss-Bonnet they vanish precisely into the same combination that $S_{JM}$ extremization produces. The third piece is the Brown-York stress tensor of Gauss-Bonnet gravity with its known Gibbons-Hawking surface term, whose $tt$ component reduces, after a 3+1 split, to $K + λL^2[(RK - 2R^{ij}K_{ij}) + (1/3)(-K^3 + 3KK_2 - 2K_3)]$.

Extended reading notes

Core claim

The central claim is that three independent constructions of the entangling surface in Gauss-Bonnet gravity agree. Extremizing the Jacobson–Myers/Hung–Myers–Smolkin area functional $S_{JM} = 2π/ℓ_P^3 ∫ d^3x √Σ (1 + λL^2 R)$ gives $K + λL^2(RK - 2R^{ij}K_{ij}) = 0$; setting the $tt$ component of the Brown-York tensor to zero on the co-dimension one surface gives the same equation up to $O(K^3)$ terms; and demanding that the $n = 1+ε$ replica solution satisfy the linearized Gauss-Bonnet equations of motion near the conical singularity gives exactly the same equation, $^{(w)}K + λL^2[^{(w)}K R - 2R_{ij}{}^{(w)}K^{ij}] = 0$. The agreement holds in the regime where the extrinsic curvature is small, so that $O(K^3)$ terms and the divergence structure in the mixed and transverse components can be dropped. Extending the same analysis to a general four-derivative Lagrangian, the authors find divergences of order $ε/w^2$ in the $ww$ component, driven by $∇∇R$ terms, that survive the approximation and produce an equation containing $∇^2{}^{(w)}K$; they state they have not been able to find an area functional yielding it and consider it unlikely that one exists. Finally, evaluating the Euclidean action outside a spherical entangling surface, the $O(1/ε^4)$ and $O(1/ε^2)$ divergences cancel only when a term proportional to $S_{JM}$ is added, with coefficients $α = κ = 1$; what remains is $4π^2L^3(1-6f_∞λ)/(f_∞^{3/2}ℓ_P^3) ln(z_0/ε)$, the universal piece of the entanglement entropy.

Load-bearing premise

The derivation assumes the extrinsic curvature of the entangling surface is small enough that $O(K^3)$ terms and the divergences in the mixed and transverse components of the equations of motion can be dropped, an approximation the paper invokes but does not quantify.

Editorial extensions

If this is right

For Gauss-Bonnet holography, the modified area functional is the correct entropy functional in the small-extrinsic-curvature regime, since a first-principles replica derivation lands on the same surface equation.
The universal coefficients of four-dimensional entanglement entropy — proportional to $a$ for a sphere and to $c$ for a cylinder — are reproduced by $S_{JM}$ through the logarithmic terms $S_{EE} = -4a\ln(z_0/ε)$ and $S_{EE} = -(c/2)(H/z_0)\ln(z_0/ε)$.
The area functional can be reinterpreted as a counterterm: adding it to the Euclidean action of Gauss-Bonnet gravity removes all power-law divergences, so the finite part of the action is the universal entanglement entropy, in analogy with $TI_{tot} = E - TS_{EE}$.
At four-derivative order, only Gauss-Bonnet appears to admit an area functional; a general combination of $R^2$, $R_{μν}R^{μν}$ and $R_{μνρσ}R^{μνρσ}$ gives a surface equation with a $\nabla^2 K$ term that no local geometric functional of the induced metric reproduces.
The equivalence between the minimal-surface condition and $T^{BY}_{tt} = 0$ holds in Gauss-Bonnet only up to $O(K^3)$ terms; the residual difference is $2f_∞λ(z + f(z)f'(z))(f'(z) + f'(z)^3 - zf''(z))$ for the sphere.

Reading between the lines

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

If the small-extrinsic-curvature restriction is the real boundary of the method rather than a technical convenience, then holographic entanglement entropy for small regions (large extrinsic curvature at the turning point) in Gauss-Bonnet gravity is not yet derived from first principles, and the agreement found here is a large-surface statement only.
The counterterm reading of $S_{JM}$ suggests a diagnostic for other higher-derivative theories: any candidate entropy functional should be the object whose addition cancels power-law divergences in the on-shell Euclidean action, which is a check independent of the replica construction.
The authors' remark that a field redefinition can recast four-derivative terms into Gauss-Bonnet order-by-order in the coupling hints that the obstruction found for Weyl-squared may be an artifact of treating the coupling non-perturbatively; a perturbative expansion might restore an area functional at each order.
The failure to find an area functional for general four-derivative gravity, if confirmed, would mean that entanglement entropy in such theories is not determined by the induced geometry alone, paralleling how Wald entropy for black holes depends on the full curvature tensor and not just on area.

Original abstract

We consider holographic entanglement entropy in higher derivative gravity theories. Recently Lewkowycz and Maldacena arXiv:1304.4926 have provided a method to derive the equations for the entangling surface from first principles. We use this method to compute the entangling surface in four derivative gravity. Certain interesting differences compared to the two derivative case are pointed out. For Gauss-Bonnet gravity, we show that in the regime where this method is applicable, the resulting equations coincide with proposals in the literature as well as with what follows from considerations of the stress tensor on the entangling surface. Finally we demonstrate that the area functional in Gauss-Bonnet holography arises as a counterterm needed to make the Euclidean action free of power law divergences.

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

We consider holographic entanglement entropy in higher derivative gravity theories. Recently Lewkowycz and Maldacena arXiv:1304.4926 have provided a method to derive the equations for the entangling surface from first principles. We use this method to compute the entangling surface in four derivative gravity. Certain interesting differences compared to the two derivative case are pointed out. For Gauss-Bonnet gravity, we show that in the regime where this method is applicable, the resulting equations coincide with proposals in the literature as well as with what follows from considerations of the stress tensor on the entangling surface. Finally we demonstrate that the area functional in Gauss-Bonnet holography arises as a counterterm needed to make the Euclidean action free of power law divergences.

background contribution method result interpretation main claim

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#	sentence	role	the authors' own work	states the main conclusion	strength
s0	We consider holographic entanglement entropy in higher derivative gravity theories.	background 0.72	0.58	0.04	asserted 0.64
s1	Recently Lewkowycz and Maldacena arXiv:1304.4926 have provided a method to derive the equations for the entangling surface from first principles.	background 0.97	0.03	0.03	asserted 1.00
s2	We use this method to compute the entangling surface in four derivative gravity.	method 0.53	0.95	0.11	asserted 0.99
s3	Certain interesting differences compared to the two derivative case are pointed out.	result 0.89	0.77	0.08	asserted 0.87
s4	For Gauss-Bonnet gravity, we show that in the regime where this method is applicable, the resulting equations coincide with proposals in the literature as well as with what follows from considerations of the stress tensor on the entangling surface.	result 1.00	0.89	0.22	asserted 1.00
s5	Finally we demonstrate that the area functional in Gauss-Bonnet holography arises as a counterterm needed to make the Euclidean action free of power law divergences.	result 0.95	0.93	0.28	asserted 1.00

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

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arXiv listing abstract v3

not examined

full text
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references and citing sentences
relations to other papers

levels run

P abstract

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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.** Ttt = 0 determines the shape which also implies the ground state of the configuration. This has a similarity with the MERA procedure in which all the states in a particular level are in the ground state. We wish to ask if setting Ttt = 0 in the Gauss-Bonnet case gives rise to the same equations or not. 5This condition is reminiscent of the way that Einstein equations emerge from thermodynamical considerations [25]. 7

Figures reproduced from arXiv:1305.6694 by the authors.

References

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Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

A Quantum Focussing Conjecture
hep-th 2015-06 conditional novelty 8.0 of 10

A single conjectured inequality on generalized entropy is shown to imply the Bousso bound, the generalized second law, and a new Quantum Null Energy Condition on stress-energy.
Holographic Entanglement Entropy for General Higher Derivative Gravity
hep-th 2013-10 accept novelty 8.0 of 10

A general formula for holographic entanglement entropy in higher-derivative gravity: Wald's entropy plus extrinsic-curvature corrections whose coefficients come from would-be logarithmic divergences in the replica con...
The Ryu-Takayanagi Formula from Quantum Error Correction
hep-th 2016-07 accept novelty 7.0 of 10

In any quantum error-correcting code with complementary recovery, the Ryu-Takayanagi formula, subregion duality, and relative-entropy equivalence are proven to be equivalent statements.
Reconstruction of Bulk Operators within the Entanglement Wedge in Gauge-Gravity Duality
hep-th 2016-01 accept novelty 7.0 of 10

A short quantum-information theorem shows that bulk operators inside the entanglement wedge of a boundary region can always be reconstructed as operators supported on that region.
Universality of corner entanglement in conformal field theories
hep-th 2015-05 accept novelty 7.0 of 10

The ratio of the corner entanglement coefficient a(theta) to the stress-tensor central charge C_T is nearly theory-independent across many 3d CFTs and is conjectured to equal pi^2/24 exactly as theta approaches pi.

Show 11 more citing papers

A Second Law for Higher Curvature Gravity
gr-qc 2015-04 conditional novelty 7.0 of 10

Black hole entropy increases monotonically under linearized perturbations of stationary horizons in arbitrary higher-curvature gravity, and the increasing quantity matches Dong's holographic entanglement entropy at li...
Generalized entropy and higher derivative Gravity
hep-th 2013-10 conditional novelty 7.0 of 10

For curvature-squared gravity, holographic entanglement entropy is Wald's entropy plus a term quadratic in the extrinsic curvature of the entangling surface, with coefficient the second derivative of the Lagrangian wi...
Quantum corrections to holographic entanglement entropy
hep-th 2013-07 accept novelty 7.0 of 10

The one-loop correction to holographic entanglement entropy is the bulk entanglement entropy between the region behind the Ryu-Takayanagi surface and its complement, plus local Wald-like and counterterm contributions.
Renormalized entanglement entropy
hep-th 2016-04 conditional novelty 6.0 of 10

A covariant counterterm procedure renormalizes holographic entanglement entropy, and for deformations by operators of dimension between 3/2 and 5/2 it implies the F quantity grows along the renormalization group flow.
Universal Terms of Entanglement Entropy for 6d CFTs
hep-th 2015-03 conditional novelty 6.0 of 10

The universal logarithmic entanglement entropy of 6d CFTs is derived holographically and field-theoretically, with exact agreement on the C^2 and C k^2 terms and agreement on k^4 terms only after assuming a conical-me...
Holographic Entanglement Entropy for the Most General Higher Derivative Gravity
hep-th 2014-11 conditional novelty 6.0 of 10

A generalized Wald entropy plus an anomaly-of-entropy term gives holographic entanglement entropy for gravity actions containing derivatives of the curvature, and the extra term resolves the Hung-Myers-Smolkin mismatch.
On entanglement entropy functionals in higher derivative gravity theories
hep-th 2014-05 conditional novelty 6.0 of 10

For general R² and quasi-topological gravity, the surface equation from the proposed entropy functional does not match the one from the generalized gravitational entropy method.
On short interval expansion of R\'enyi entropy
hep-th 2013-09 conditional novelty 6.0 of 10

Extending the twist-operator OPE to level six yields a new order-1/c contribution to the Rényi entropy of two short disjoint intervals, proposed as the two-loop bulk correction.
Distributional Geometry of Squashed Cones
hep-th 2013-06 conditional novelty 6.0 of 10

A geometric regularization procedure shows that integrals of quadratic curvature invariants over squashed conical singularities acquire surface terms depending on the extrinsic curvatures of the singular surface.
Corner contributions to holographic entanglement entropy
hep-th 2015-05 accept novelty 5.0 of 10

The ratio of the corner coefficient sigma in entanglement entropy to the stress-tensor central charge C_T equals pi squared over 24 for all holographic models studied and for free scalars and fermions, suggesting a un...
On generalized gravitational entropy, squashed cones and holography
hep-th 2013-08 conditional novelty 5.0 of 10

The Fursaev-Patrushev-Solodukhin squashed cone regularization, combined with the Lewkowycz-Maldacena generalized gravitational entropy procedure, reproduces the known universal entanglement entropy terms for spherical...

How the literature uses it

52 arXiv papers cite this in 83 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 · 51 sentences without a confident label

Contests it 2 papers

This argument would lead to the proposal ([ref]) beyond the spherical shape [ref] [The subsequent papers [ref] [ref] point out there are some mismatches for higher derivative gravity theory, but the mismatch is removed by the paper [ref] ].

Holographic R\'enyi Entropy and Generalized Entropy method · Introduction · contest 0.71
We will explain that a modification to the order of limits needs to be done to the use of the LM method in [ref] for deriving the surface equation.

On generalized gravitational entropy, squashed cones and holography · Introduction · contest 0.80

Uses its method or tool 2 papers

In order to proceed, we could assume the following as in [ref] that $O() ∼αr/ϵ$ where $α≪1$, then the $ij$, $ir$ components go to zero.

On generalized gravitational entropy, squashed cones and holography · Equation for the entangling surface · 0.91
Here we will use the gravitational entropy proposed by Dong [ref] and Camps [ref] , which motivated to describe the gravitational dual of the entanglement entropy of CFT in the AdS/CFT setup (see also [ref] ).

Cutoff Scale of Quadratic Gravity from Quantum Focusing Conjecture · Gravitational Entropy · 0.80

Compares with it 3 papers

Then the difference ([ref]) becomes -2ł[ K_(2) K_(2) K_(2) -3K_(2) K_(2)ij K_(2)^ij +2K_(2)ij K_(2)^jk K_(2)k^(2)i ], which is in accord with the result in [ref] .

Note on generalized gravitational entropy in Lovelock gravity · Further investigation · 0.81
* We also showed that the entangling surface equations are the same as what comes from the JM entropy functional without the small extrinsic curvature condition needed in [ref] .

On generalized gravitational entropy, squashed cones and holography · Discussion · 0.88
In [ref] the same approach was investigated for Gauss-Bonnet gravity, and it was shown that demanding $T_tt=0$ imposes the following equation on the profile of the co-dimension one hypersurface: [equation] Although not equal, this equation bears a remarkable similarity with the divergent part of the $qq$-component [ref] derived above from the conical boundary condition approach.

Conditions on holographic entangling surfaces in higher curvature gravity · Testing the conditions · 0.96

Cites it as background 21 papers

Remarkably enough, a general formula for any theory involving arbitrary contractions of the Riemann tensor $ℒ(R_μνρσ)$, which seems to satisfy several consistency checks, has been recently proposed by Dong [ref] (see also, e.g. [ref] )).

Higher-curvature corrections to holographic entanglement entropy in geometries with hyperscaling violation · Introduction · 0.84
Interest in CFTs with bulk Lovelock duals extends to holographic computations of entanglement entropy [ref] .

Extended First Law for Entanglement Entropy in Lovelock Gravity · Introduction · 0.89
In adapted coordinates —see subsection [ref] below for our conventions— these two terms read[Interesting additional developments and explorations include [ref] .][Generalizations of dongf to the case in which covariant derivatives of the Riemann appear in the action have also been presented [ref] .] [equation] In principle, the generalized holographic surface $Γ_A$ should be obtained by extremizing the new functional [ref] .

Holographic entanglement entropy for perturbative higher-curvature gravities · Introduction · 0.89

What it cites · 29 sentences without a confident label

Contests 2 papers

Fursaev suggested a derivation [ref] of the Ryu-Takayanagi prescription which appeared to have some problems since it led to the wrong Renyi entropies [ref] (there is a proposed resolution to this in [ref] ).

Entanglement Renyi entropies in holographic theories · Introduction · contest 0.87
Fursaev suggested a derivation [ref] of the Ryu-Takayanagi prescription which appeared to have some problems since it led to the wrong Renyi entropies [ref] (there is a proposed resolution to this in [ref] ).

Entanglement Renyi Entropies in Conformal Field Theories and Holography · Introduction · contest 0.87

Uses the method or tool of 2 papers

Since the metric perturbation is periodic, we will need to argue that the contribution from the perturbations are zero [ref] .

Generalized gravitational entropy · Generalized gravitational entropy method · 0.83
At this stage, following [ref] , we choose the following coordinate system [equation] where $y^l$ denotes the two transverse directions to the surface.

A Covariant Holographic Entanglement Entropy Proposal · General minimal surface condition · 0.95

Compares with 3 papers

In particular they were in agreement with [ref] .

Holographic Entanglement Entropy in Lovelock Gravities · Discussion · 0.99
In particular they were in agreement with [ref] .

On Holographic Entanglement Entropy and Higher Curvature Gravity · Discussion · 0.99
Further in this regime they also agreed with the $T_tt^BY=0$ condition [ref] .

Entanglement entropy from the holographic stress tensor · Discussion · 0.98

Cites as background 7 papers

In the context of holography, Ryu and Takayanagi [ref] proposed a remarkably simple prescription to compute entanglement entropy (EE) for the field theory dual to Einstein gravity.

Holographic Entanglement Entropy: An Overview · Introduction · 0.98
Recently, Lewkowycz and Maldacena [ref] have proposed a way to set up the calculation of entanglement entropy in what appears to be a self-consistent manner.

Generalized gravitational entropy · Introduction · 0.84
For Gauss-Bonnet gravity, $λ_1=λ ,λ_2=-4λ , λ_3=λ .$ The equations of motion (these can be found in many places, eg [ref] ) are, G_αβ-6/L^2g_αβ- L^2/2H_αβ=T_αβ where, [equation] and [equation] [equation] [equation] Here $∇̂$ is the usual covariant derivative made of the full five dimensional metric.

Effect of curvature squared corrections in AdS on the viscosity of the dual gauge theory · Generalized gravitational entropy method · 0.87

Discussion

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