BBulk entanglement and its shape dependence
Zhong-Ying Fan Department of Astrophysics, School of Physics and Materials Science,Guangzhou University, Guangzhou 510006, ChinaABSTRACTWe study one-loop bulk entanglement entropy in even spacetime dimensions using the heatkernel method, which captures the universal piece of entanglement entropy, a logarithmicallydivergent term in even dimensions. In four dimensions, we perform explicit calculations forvarious shapes of boundary subregions. In particular, for a cusp subregion with an arbi-trary opening angle, we find that the bulk entanglement entropy always encodes the sameuniversal information about the boundary theories as the leading entanglement entropy inthe large N limit, up to a fixed proportional constant. By smoothly deforming a circle inthe boundary, we find that to leading order of the deformations, the bulk entanglemententropy shares the same shape dependence as the leading entanglement entropy and hencethe same physical information can be extracted from both cases. This establishes an inter-esting local/nonlocal duality for holographic CFT . However, the result does not hold forhigher dimensional holographic theories. Email: [email protected] . a r X i v : . [ h e p - t h ] J a n ontents D = 4 dimension 11 D = 4 dimension . . . . . . . . . . . . . . . . . . . . 194.2 Deformed spheres in the D = 6 dimension . . . . . . . . . . . . . . . . . . . 20 (106) Entanglement entropy measures how closely entangled a given wave function is in a quantummechanical system. It plays an important role in our exploration for a better understandingof quantum systems. In the past two decades, holographic description of entanglemententropy in gauge/gravity duality attracted a lot of attentions since the pioneer work [1,2]. The exciting development in this area gives people confidence that this quantity mayprovide a bridge to connect several different research areas: quantum gravities, quantumfield theories and quantum information theories. In particular, recent progress in blackhole evaporation [3, 4, 5] shows that entanglement entropy might be the correct quantityto characterize evaporation of black holes since it follows the Page curve and hence resolvesthe information loss paradox argued by Hawking [6, 7].2t was first proposed in [1, 2] that for a boundary subregion A, holographic entanglemententropy is given by one quarter of the area of the minimal area surface Σ, which is anchoredon the boundary and is homologous to A ( ∂ Σ = ∂A ) S = Area(Σ)4 G N . (1)Without confusion, we always omit the subscript A for Σ and S since we will only studythe entanglement entropy for a single subregion in the boundary. This is referred to as RTformula in the literature. It looks very similar to that of Bekenstein-Hawking entropy ofblack holes. For the latter case, the entropy formula can be discovered by introducing athermodynamic interpretation for Euclidean gravity solutions, which have a U (1) isometry[8]. The partition function for the gravity systems can be evaluated as a Euclidean functional Z ( β ) = (cid:90) [ D ϕ ][ D g µν ] exp (cid:104) − I grav − I matt (cid:105) , (2)where I grav and I matt denotes the Euclidean action of gravity sector and matter sector,respectively. In the saddle point approximation, the Euclidean gravitational action can beconsidered as I = I grav + I matt = − log Z ( β ) , (3)and the entropy is derived as S = − (cid:0) β∂ β − (cid:1) log Z ( β ) (cid:12)(cid:12)(cid:12) β = β H , (4)where β H = 1 / πT , T is the temperature of black holes. Application of the above relationsto stationary solutions with a U (1) isometry indeed leads to the entropy formula.In [9], the approach was successfully generalised to gravity solutions, which in generaldo not have a U (1) isometry. The basic idea is to perform replica trick in the boundaryand extend it to the bulk: the original gravity solution M with the boundary subregion A is replicated to M n , which is defined by taking n copies of the original manifold, cuttingthem apart at A and gluing them together in a cyclic order. The entanglement entropy isevaluated as S = (cid:0) n∂ n − (cid:1) I [ M n ] (cid:12)(cid:12)(cid:12) n → = ∂ n I [ ˆ M n ] (cid:12)(cid:12)(cid:12) n → , (5)where ˆ M n stands for the orbifold geometry M n /Z n and the second equality follows fromthe relation I [ M n ] = n I [ ˆ M n ] because of Z n symmetry of the replica geometry M n . Thena careful examination of the Euclidean action in the orbifold geometry leads to the RTformula [9]. Derivation of holographic entanglement entropy for higher derivative gravitieswas studied by several different authors using the same approach [10, 11, 12, 13, 14, 15].3roof of the formula for some special cases can be found in [16, 17, 18]. The covariantversion of the formula [2] was proved in [19].In this paper, we are interested in studying bulk one-loop corrections to holographicentanglement entropy (we do not consider backreaction effects from bulk quantum fluctua-tions, which changes holographic entanglement entropy at order unity as well). This topicwas early discussed in [20, 21] and was analyzed extensively in AdS / CFT correspondence[22, 23, 24, 25]. Quantum corrections to holographic mutual information was studied in[26, 27].In this paper, we will study bulk entanglement entropy in higher D ≥ D = 4 dimension for several different shapes ofboundary subregions. In particular, for a cusp subregion with an arbitrary opening angle,we find that the bulk entanglement entropy always encodes the same universal informa-tion about the boundary theories as the leading entanglement entropy in the large N limit,up to a fixed constant proportional to the central charges of the boundary. Furthermore,by studying a smoothly deformed circle in the boundary, we find that the bulk entangle-ment entropy, which is non-geometric, shares the same shape dependence with the leading,geometric RT formula. The former captures local information about the O (1) degrees offreedoms (d.o.fs) whilst the latter encodes nonlocal information about the O ( N ) d.o.fs inthe boundary. Hence, our result establishes a local/nonlocal duality for the shape depen-dence of entanglement entropy for holographic CFT . We extend our discussions to the D = 6 dimension. However, we find that the result in general does not hold any longer forhigher dimensions.The remaining of this paper is organized as follows. In section 2, we apply the heatkernel method to massless scalar fields and discuss the one-loop bulk entanglement entropyin diverse dimensions. We also briefly review the derivations for integrals of curvatureinvariants in the orbifold geometry. In section 3, we perform the one-loop calculationsexplicitly in the D = 4 dimension. In section 4, we extend the calculations to the D = 64imension. We conclude in section 5. Extending (5) to include the contribution of quantum fluctuations in a fixed background,the one-loop bulk entanglement entropy is given by S q = (cid:0) n∂ n − (cid:1) I eff [ M n ] (cid:12)(cid:12)(cid:12) n → = ∂ n I eff [ ˆ M n ] (cid:12)(cid:12)(cid:12) n → , (6)where I eff [ M n ] stands for the one-loop effective action in the replica geometry for variousquantum fluctuations. In this section, we will adopt heat kernel method to evaluate I eff aswell as the bulk entanglement entropy. For convenience, we will work in Euclidean signaturethroughout this paper. Without loss of generality, we consider a bulk massless scalar field in general dimensions I matt = − (cid:90) d D x √ G (cid:0) ∂ Φ (cid:1) . (7)The one-loop effective action is formally given by I eff [ M ] = 12 log det (cid:3) , (cid:3) = ∇ µ ∇ µ . (8)In DeWitt-Schwinger proper time representationlog det (cid:3) = − (cid:90) ∞ (cid:15) dss Tr (cid:0) e − s (cid:3) (cid:1) , (9)where the heat kernel Tr (cid:0) e − s (cid:3) (cid:1) can be expanded asTr (cid:0) e − s (cid:3) (cid:1) = 1 (cid:0) πs (cid:1) D/ ∞ (cid:88) (cid:96) =0 a (cid:96) s (cid:96) . (10)The expansion coefficients a (cid:96) ’s can be expressed as geometric invariants in the backgroundmanifold. We present some lower lying examples a = (cid:90) d D x √ G ,a = 16 (cid:90) d D x √ G R ,a = (cid:90) d D x √ G (cid:16) R − R µν R µν + 1180 R µναβ R µναβ − (cid:3) R (cid:17) . (11)5he higher order coefficients a i ( i ≥
3) have much lengthy expressions, see for example [31]and the references therein. Notice that the i -th order coefficient a i involves integrals of i -th order curvature polynomials as well as derivatives of curvatures with the same lengthdimensions. For other types of massless fields, such as photons and gravitons, the coefficients a i ’s are simply changed by constant factors associated to each curvature invariants in theintegrals. As a consequence, our discussions are easily generalised to all massless fields. Formassive fields, more terms should be included associated to the mass. Yet, generalisationto this case is straightforward as well and will not change our main results in this paper.According to the heat kernel expansion (10), one has to relevant ordersTr (cid:0) e − s (cid:3) (cid:1) = 1 (cid:0) πs (cid:1) D/ (cid:0) a + a s + a s + · · · + a [ D/ s [ D/ + · · · (cid:1) , (12)where at the [ D ]-th order, there will be a logarithmically divergent term to the effectiveaction in even spacetime dimensions. In this case, the coefficient a [ D/ is independent ofthe cut-off and hence contains universal information about the underlying theories. Onefinds I eff [ M ] = − (cid:0) π (cid:1) D/ (cid:104) a D(cid:15) D + a ( D − (cid:15) D − + · · · + a [ D/ log (cid:0) Λ (cid:15) (cid:1)(cid:105) + · · · , (13)where Λ is an infrad cut-off and the dots outside the square bracket stands for regular terms.However, in odd dimensions, the a [ D/ term just gives a least divergent term ∼ a [ D/ /(cid:15) tothe effective action. In this case, the physical information is encoded in the constant term,which however cannot be extracted using the above method ( the constant term of bulkentanglement entropy across hemispheres was derived in [30] by using full heat kernel inAdS space ).To proceed, we need evaluate the heat kernel coefficients in the replica geometry M n (or its orbifold ˆ M n = M n /Z n ) and derive their derivative with respect to the replicaparameter ( ∂ n a (cid:96) ) n =1 . This is similar to the derivations of holographic entanglement entropy[10, 11, 12, 13, 14, 15], see also [32, 33, 34, 35]. The major result is each of the coefficients a (cid:96) can be expressed as a regular part a reg(cid:96) and a singular part a (cid:96) ,n a (cid:96) = n (cid:0) a reg(cid:96) + a (cid:96) ,n (cid:1) . (14)The regular part is expressed as integrals in the smooth region of ˆ M n and hence a reg(cid:96) isindependent of n . These terms will not contribute to the bulk entanglement entropy S q .On the other hand, the singular part a (cid:96) ,n is evaluated in the cone region of the orbifoldgeometry and depends on n nontrivially. Its derivative ∂ n a (cid:96) ,n in the n → a (cid:96) = (cid:82) d D x √ G L ( G , R , ∇ R , · · · ), one has a reg(cid:96) = (cid:90) M d D x √ G L ( G , R , ∇ R , · · · ) ,a (cid:96) ,n = (cid:90) ˆ M n d D x √ G L ( G , R , ∇ R , · · · ) . (15)Compared to (6), one finds (we set D = d + 2)( n∂ n − a (cid:96) (cid:12)(cid:12)(cid:12) n =1 = ∂ n a (cid:96) ,n (cid:12)(cid:12)(cid:12) n =1 = 2 π (cid:90) Σ d d y √ γ a , (16)where explicit results about the surface density a will be presented in the next subsection.Here we would like to point out that the first coefficient a does not have a singular part. As aconsequence, the leading order contribution to the bulk entanglement entropy is determinedby a S q = λ A Σ (cid:15) D − + · · · . (17)where λ is a constant depending on the cut-off. This is the well-known are law for entan-glement entropy. Combined with the RT formula, the above divergence can be absorbed byrenormalzing the Newton constant as1 G ren = 1 G + λ(cid:15) D − . (18)Likewise, subleading order divergences associated to a k with k < [ D/
2] can be absorbedby higher order coupling constants in the gravity (counter term) action. For the sake ofconvenience, we will not repeat this step in the remaining of this paper any longer. Instead,we focus on computing the coefficient a [ D/ , which contains universal information aboutthe boundary theories. We introduce S q = · · · + s q log (cid:0) Λ (cid:15) (cid:1) + · · · ,s q = − π ) D/ (cid:0) ∂ n a [ D/ ,n (cid:1) n =1 . (19)Without confusion, s q will be briefly referred to as bulk entanglement through the remainingof this paper. However, it should be emphasized that s q diverges in the asymptotic AdSboundary since it will be determined as surface integrals of certain geometric quantitiesevaluated on the minimal area surface. The divergence structure is similar to that of theleading entanglement entropy in the boundary. Since it is interesting to compare the physicalinformation extracted from both cases, we may set S = S/c eff , c eff = (cid:96) dAdS G , (20)7here c eff is referred to as the effective central charge for the boundary theories. One hasfor smooth entangling surfaces S = R d − δ d − + R d − δ d − + · · · + Rδ + ( − d s univ ,s q = R d − δ d − + R d − δ d − + · · · + Rδ + s univ q , (21)where R denotes a characteristic length scale of the boundary subregion (it should notbe confused with the Ricci scalar) and δ is the UV cutoff at asymptotic AdS boundary.Universal information about the boundary theories is essentially contained in the aboveconstant pieces. However, for singular subregions, as will be shown in sec.3.4, one hasinstead S = R d − δ d − + R d − δ d − + · · · + Rδ − a (Ω) log (cid:16) Rδ (cid:17) + cons ,s q = R d − δ d − + R d − δ d − + · · · + Rδ − b (Ω) log (cid:16) Rδ (cid:17) + cons , (22)where Ω stands for the opening angle of the cone in the boundary subregion. In this case,the constant terms are regulator dependent. Instead, the physical information is encodedin the functions a (Ω) , b (Ω). To calculate ( ∂ n a (cid:96) ) n =1 , let us briefly review the derivation of integrals of curvature invariantsin the orbifold geometry ˆ M n . More details can be found in the literature [10, 11, 12, 13,14, 15].Close to a codimension − M n looks like a product form C n × Σ. In the adapted coordinates,one has ds = e φ ˆ g ab dx a dx b + (cid:0) γ ij ( y ) + 2 K aij x a + Q abij x a x b + · · · (cid:1) dy i dy j + · · · , (23)where K aij are the extrinsic curvatures of Σ and Q abij ≡ ∂ a K bij . We will express the twodimensional cone C n in three types of coordinates: the Cartesian coordinates ( x , x ), thecylindrical coordinates ( ρ , τ ) and the complex coordinates ( z , ¯ z ). One hasˆ g ab dx a dx b = ( dx ) + ( dx ) = dρ + ρ dτ = dzd ¯ z , (24)where x = ρ cos τ , x = ρ sin τ, z = ρe iτ , ¯ z = ρe − iτ . The function φ is given by φ = − (cid:15) log ρ = − (cid:15) log ( z ¯ z ), where (cid:15) = 1 − /n . For the above metric, the Riemann tensor to8eading order can be computed as R abcd = e φ ˆ r abcd , R iabc = 0 , R aijk = D k K aij − D j K aik ,R ijab = γ kl ( K bik K aj(cid:96) − K aik K bj(cid:96) ) ,R iajb = Γ cab K cij + γ k(cid:96) K ajk K bi(cid:96) − Q abij ,R ijk(cid:96) = R ijk(cid:96) + e − φ ( K ai(cid:96) K ajk − K aik K aj(cid:96) ) , (25)where ˆ r abcd , R ijkl stand for the Riemann tensor associated to ˆ g ab and γ ij respectively. Moreresults about curvatures and their covariant derivatives in the orbifold geometry can befound in the literature [10, 11, 12, 13, 14, 15]. For self-consistency, we collect some relevantresults in our Appendix A.For simplicity, let us first consider the general higher order Riemannian gravities I = (cid:90) d D x √ G L ( G µν ; R µνρσ ) . (26)This case was studied very carefully in [12]. One has ∂ n I [ ˆ M n ] (cid:12)(cid:12)(cid:12) n =1 = 2 π (cid:90) Σ (cid:104) ∂L∂R z ¯ zz ¯ z + (cid:88) α (cid:16) ∂ L∂R izjz ∂R k ¯ zl ¯ z (cid:17) α K zij K ¯ zkl q α + 1 (cid:105) , (27)where (cid:82) Σ = (cid:82) d d y √ γ and q α is a constant, counting the total number of Q zzij , Q ¯ z ¯ zij andpairs of K in each term of the derivative ∂ L∂R izjz ∂R k ¯ zl ¯ z , which is expanded according to (25).Notice that the second term in the square bracket, should be evaluated in the originalgeometry, namely taking the limit n → − π (cid:90) Σ ∂L∂R µνρσ ε µν ε ρσ . (28)The second term, referred to as anomaly in [12], can be written asAnomaly = 2 π (cid:90) Σ (cid:16) ∂ L∂R µ ρ ν σ ∂R µ ρ ν σ (cid:17) α K λ ρ σ K λ ρ σ q α + 1 × (cid:104) ( n µ µ n ν ν − ε µ µ ε ν ν ) n λ λ + ( n µ µ ε ν ν + ε µ µ n ν ν ) ε λ λ (cid:105) , (29)where K βρσ = n ( a ) β K aρσ ,ε µν = n ( a ) µ n ( b ) ν (cid:15) ab ,n µν = n ( a ) µ n ( b ) ν g ab , (30)9here (cid:15) ab is the usual Levi-Civita tensor.Application of above results to some simple cases is give by [12]1) : L = L ( R ) = ⇒ ∂ n I [ ˆ M n ] (cid:12)(cid:12)(cid:12) n =1 = − π (cid:90) Σ ∂L∂R ,
2) : L = R µν = ⇒ ∂ n I [ ˆ M n ] (cid:12)(cid:12)(cid:12) n =1 = − π (cid:90) Σ (cid:0) R a a − K a K a (cid:1) , (31)3) : L = R µνρσ = ⇒ ∂ n I [ ˆ M n ] (cid:12)(cid:12)(cid:12) n =1 = − π (cid:90) Σ (cid:0) R ab ab − K aij K aij (cid:1) , where R a a = R µν n µν , R ab ab = 12 R µλνρ ε µλ ε νρ = R µλνρ n µν n λρ . (32)Moreover, for the most general gravitational action I = (cid:82) M d D x √ g L ( g µν , R , ∇ R , · · · ),there will be more singular terms emerging in the covariant derivatives of Riemann curva-tures. In this case, one must consider all the terms involving ∂ z ∂ ¯ z φ as well as ∂ z φ∂ ¯ z φ . As amatter of fact, the metric expansion around the bulk surface Σ should be considered morecarefully, including all the relevant subleading order terms ds = e φ (cid:104) dzd ¯ z + e φ T (¯ zdz − zd ¯ z ) (cid:105) + 2 ie φ V i (¯ zdz − zd ¯ z ) dy i + ( γ ij + Q ij ) dy i dy j , (33)where T = T + T a x a + O ( x ) ,V i = U i + V ai x a + O ( x ) ,Q ij = 2 K aij x a + Q abij x a x b + P abcij x a x b x c + O ( x ) . (34)It was established in [15] that holographic entanglement entropy can be formally evaluatedas ∂ n I [ ˆ M n ] (cid:12)(cid:12)(cid:12) n =1 = 2 π (cid:90) Σ (cid:104) δLδ∂ z ∂ ¯ z φ + (cid:88) α β α δδ∂ ¯ z φ (cid:16) δLδ∂ z φ (cid:17) ∂ z ∂ ¯ z φ =0 (cid:105) (cid:15) =0 , (35)where β α is a constant. In the above result, the first term is referred to as the generalisedWald entropy whilst the second term gives the general anomaly terms [15].As an example, for six derivative gravities I = (cid:82) M d D x √ g L ( g µν , R , ∇ R ), the gener-alised Wald entropy was derived explicitly as [15]Wald = − π (cid:90) Σ (cid:104) δLδR µνρσ ε µν ε ρσ − ∂L∂ ∇ α R µρνσ K βρσ (cid:0) n βµ n αν − ε βµ ε αν (cid:1)(cid:105) , (36) Here we have ignored the so-called splitting problems discussed in [15] , since it does not effect our mainresults in this paper. We refer the interested readers to that paper for details. ∂ z φ∂ ¯ z φI = (cid:88) α (cid:90) dzd ¯ z C α e − β α φ ∂ z φ∂ ¯ z φ + · · · , (37)then according to (35), the anomaly term is given byAnomaly = 2 π (cid:90) Σ C α β α . (38)Evaluation of the anomaly terms for several special cases can be found in [15]. We willadopt the results therein for six derivative gravities to our D = 6 dimensional calculations. D = 4 dimension Now let us calculate the one-loop bulk entanglement entropy in the D = 4 dimension, wherethe relevant heat kernel coefficient is a . Using (31), we deduce (cid:0) ∂ n a ,n (cid:1) n =1 = − π (cid:90) Σ d y √ γ (cid:104) R − R µν n µν + 190 R µλνρ n µν n λρ − K aij K aij + 1360 K a K a (cid:105) . (39)Here it is worth emphasizing that the total derivative term (cid:3) R in a does not contributeto (cid:0) ∂ n a ,n (cid:1) n =1 , as shown in [38]. The result can be even more simplified by using Gauss-Codazzi identity R = R Σ + 2 R µν n µν − R µλνρ n µν n λρ − K a K a + K aij K aij , (40)where R Σ is the scalar curvature of the bulk surface Σ. One has (cid:0) ∂ n a ,n (cid:1) n =1 = − π (cid:90) Σ d y √ γ (cid:104) R Σ + 160 R + 160 R µν n µν − K a K a (cid:105) , (41)where the last term in the square bracket vanishes for minimal area surfaces. Moreover,for vacuum solutions to Einstein’s gravity (including Schwarzschild black holes), curvaturestake particularly simple forms R µν = − ( D − (cid:96) − AdS g µν , (42)where (cid:96) AdS is AdS radius. This greatly simplifies our calculations in the D = 4 dimension.One finds (cid:0) ∂ n a ,n (cid:1) n =1 = − π (cid:90) Σ d y √ γ (cid:104) R Σ − (cid:96) − AdS (cid:105) . (43)11t implies that to compute the one-loop bulk entanglement entropy, we just need evaluatethe scalar curvature of the RT surface in four dimension. Of course, the same result canbe obtained by constructing the adapted coordinates for the RT surface and extracting theextrinsic curvatures explicitly, see section 4.2 for more details. The simplest way to testthis is checking the Gauss-Codazzi identity, which implies for RT surfaces in AdS vacuum R Σ + K aij K aij = − d ( d − (cid:96) − AdS . (44) Consider a hemisphere in
AdS D vacuum. For later purpose, we keep our discussions asgeneral as possible and will return to the D = 4 dimension when necessary. The readersshould not be confused. In the boundary, the entangling surface is a ( d − A = { t E = 0 , ≤ r ≤ R } , (45)where R denotes the radius of the sphere. Under the boundary spherical coordinates( r , Ω d − ), the bulk metric reads ds = (cid:96) AdS dξ + dt E + dr + r d Ω d − ξ , (46)where d Ω d − is the metric of a unit sphere S d − . According to the RT formula, the bulkminimal surface is derived as [1] Σ : ξ + r = R , (47)which describes a hemisphere. The induced metric on Σ reads ds = (cid:96) AdS ξ − (cid:0) R dr + ξ r d Ω d − (cid:1) . (48)In fact, this describes a uniform d -dimensional hyperbolic space with curvature radius (cid:96) AdS .To see this, we introduce a new coordinate θ : r = R sin θ , ξ = R cos θ and the hyperboliccoordinate u as: sinh u ≡ tan θ . The induced metric becomes ds = (cid:96) AdS cos θ (cid:0) dθ + sin θd Ω d − (cid:1) = (cid:96) AdS (cid:0) du + sinh ud Ω d − (cid:1) . (49)It might be a surprise that the hemisphere coincides with the event horizon of a hyperbolicblack hole with radius (cid:96) AdS . In fact, it has vanishing extrinsic curvatures K aij = 0 aswell. The physical meaning of this was clarified in [16]: for conformal field theories (not12ecessarily holographic ones) defined in a spherical ball-shaped region with radius R , thevacuum state can be unitarily transformed into a thermal bath in a hyperbolic space withcurvature radius R . Hence, the entanglement entropy across a spherical entangling surface S d − is equal to the thermal entropy on the hyperbolic space H d − . For holographic CFTs,the latter is given by the entropy of a ( d + 2)-dimensional AdS-hyperbolic black holes withtemperature T = 1 / πRds = (cid:16) ρ R − (cid:17) dt E + dρ ρ R − ρ (cid:0) du + sinh ud Ω d − (cid:1) , (50)where R = (cid:96) AdS .The area of the hemisphere is given by A Σ = Ω H d (cid:96) dAdS , (51)where Ω H d is the area of a unit hyperbolic plane H d . The induced scalar curvature turnsout to be a constant R Σ = − d ( d − (cid:96) − AdS . (52)Return to the D = 4 dimension, the leading entanglement reads S = A Σ /(cid:96) AdS = 2 πRδ − π . (53)Evaluating (43) yields (cid:0) ∂ n a ,n (cid:1) n =1 = 58 π S , (54)which leads to s q = − S π = − R δ + 29180 . (55)This simple example clearly shows that s q shares the same divergence structure as theleading entanglement entropy in the boundary. Furthermore, the relation (55) inspires usthat the universal information encoded in the bulk entanglement may have a simple relationto that in the leading entanglement entropy. This will be said more precisely when we studythe entanglement entropy for deformed spheres in sec.4. Finite temperature states of the boundary is dual to AdS black holes, described by ds = (cid:96) AdS ξ (cid:16) f ( ξ ) dt E + dξ f ( ξ ) + dr + r dφ (cid:17) , f ( ξ ) = 1 − (cid:0) ξξ h (cid:1) , (56)13here ξ h denotes the location of event horizon and ( r , φ ) are the boundary polar coordinates.The temperature of the black hole is given by T = 3 / πξ h .In this case, the bulk minimal surface will be deformed away from the hemispherebecause of temperature corrections. Nevertheless, it still respects the boundary symmetryand is characterized by a function ξ = ξ ( r ). The induced metric of Σ becomes ds = (cid:96) AdS ξ (cid:16)(cid:0) ξ (cid:48) ( r ) f ( ξ ) (cid:1) dr + r dφ (cid:17) , (57)so that the area functional is given by A Σ = 2 π(cid:96) AdS (cid:90) R dr rξ (cid:114) ξ (cid:48) ( r ) f ( ξ ) . (58)Variation of the functional leads to ξξ (cid:48)(cid:48) + (cid:16) − ξf ξ f (cid:17) ξ (cid:48) + (cid:16) ξ (cid:48) f (cid:17) ξξ (cid:48) r + 2 f = 0 , (59)where f ξ ≡ df /dξ . We would like to analytically solve the equation for a large thermalscale β = 1 /T with β/R >> ξ ( r ) = (cid:112) R − r − π R β ( R − r )(2 R − r ) R + · · · . (60)The hemisphere in AdS vacuum is deformed by a leading order correction ∼ R /β . Thearea of the minimal surface becomes A Σ = Ω H (cid:96) AdS + 32 π R β (cid:96) AdS + · · · , (61)which indeed receives an extra contribution proportional to R /β . Likewise, the scalarcurvature of Σ is corrected at the same order R Σ = − (cid:96) AdS (cid:16) − π R β (cid:0) R − r (cid:1) / (2 R − r ) R + · · · (cid:17) . (62)Substituting these results into (43) and (19), we arrive at s q = − R δ + (cid:16) − π R β (cid:17) + · · · . (63)It is also interesting to compare it to the leading entanglement S = A Σ /(cid:96) AdS = 2 πRδ − (cid:16) π − π R β (cid:17) + · · · . (64)It is intriguing to notice that in both cases, the universal term decreases as the temperatureincreases. We may expect that the bulk entanglement entropy signals quantum/thermalphase transitions as the leading entanglement entropy [39, 40, 41, 42, 43]. This may deservefurther investigations. 14 .3 Strips We move to consider a striped subregion, preserving ( d − A = { y ∈ R d | y ∈ ( − a , a ) , y i ∈ (0 , L ) for i = 2 , , · · · , d } . (65)Holographic entanglement entropy for this case has been widely studied in the literature,see for example [1]. In AdS vaccum, the bulk minimal surface is given by ξ = ξ ( y ) ξ (cid:48) ( y ) = (cid:112) ξ d ∗ − ξ d ξ d , ξ ∗ = a Γ (cid:16) d (cid:17) √ π Γ (cid:16) d +12 d (cid:17) , (66)where ξ ∗ is the turning point of the minimal surface. Solving the above equation gives y ( ξ ) = ξ d +1 ( d + 1) ξ d ∗ F (cid:16) , d + 12 d , d + 12 d , ξ d ξ d ∗ (cid:17) − a . (67)The induced metric on Σ is given by ds = (cid:96) AdS ξ (cid:16) ξ d ∗ ξ d ( dy ) + ( dy ) + · · · + ( dy d ) (cid:17) . (68)Evaluating the area of Σ yields A Σ = 2 (cid:96) dAdS L d − (cid:90) ξ ∗ δ dξ ξ d ∗ ξ d (cid:112) ξ d ∗ − ξ d = 2 (cid:96) dAdS L d − d − (cid:16) δ d − − (cid:0) aξ ∗ (cid:1) d a d − (cid:17) . (69)In the D = 4 dimension, it gives S = 2 Lδ − La (cid:16) aξ ∗ (cid:17) . (70)On the other hand, the scalar curvature of the induced metric (68) is given by R Σ = ξ (cid:96) AdS ξ ∗ (cid:0) ξ (cid:48) + ξξ (cid:48)(cid:48) (cid:1) = − (cid:96) AdS (cid:16) (cid:0) ξξ ∗ (cid:1) (cid:17) . (71)Substituting the result into (43) and (19), we deduce s q = − π S − L πa (cid:0) aξ ∗ (cid:1) = − L πδ + L πa (cid:0) aξ ∗ (cid:1) . (72)Again s q has a linear relation to S and hence the two share the same divergence structurein this case. 15 .4 Singular shapes Next, we consider singular shapes of entangling surfaces. We choose a cusp subregion inthe boundary with opening angle Ω, which is specified as A = { t E = 0 , r > , | θ | ≤ Ω / } . (73)We shall introduce a large distance cutoff for the subregion r max = R . The leading entan-glement entropy for this case was extensively studied in [44, 45, 46]. Following these papers,we parameterized the bulk surface as ξ = ξ ( r , θ ) = rϕ ( θ ). The separation of variables isdue to scaling symmetries of AdS vacuum (and there is no other scale in the problem). Thefunction ϕ ( θ ) satisfies ˙ ϕ (0) = 0 , ϕ ( ± Ω /
2) = 0, where ˙ ϕ ≡ dϕ/dθ .The induced metric on the surface is given by ds = (cid:96) AdS r (cid:16) ϕ (cid:17) dr + (cid:96) AdS ϕ (cid:0) ϕ (cid:1) dθ + 2 (cid:96) AdS ˙ ϕrϕ drdθ . (74)Evaluation of the area functional yields A Σ = 2 (cid:96) AdS (cid:90)
Rδ/ϕ drr (cid:90) Ω / − ε dθ (cid:112) ϕ + ˙ ϕ ϕ , (75)where ϕ ≡ ϕ (0) stands for the maximum value of the function. Here the angular cutoff ε is defined such that ξ = δ , ϕ (Ω / − ε ) = δ/r . Variation of the area functional determinesthe minimal surface as [44] 1 + ϕ ϕ (cid:112) ϕ + ˙ ϕ = (cid:112) ϕ ϕ . (76)To compute the area of the minimal surface, it is more convenient to introduce a variable µ = (cid:112) /ϕ − /ϕ . One has [45, 46] S = 2 (cid:90) Rδ/ϕ drr (cid:90) √ r /δ − /ϕ dµ (cid:115) ϕ (1 + µ )2 + ϕ (1 + µ )= 2 Rδ − a (Ω) log (cid:16) Rδ (cid:17) + · · · , (77)where dots stands for constant terms, which are now regulator dependent. Instead, physicalinformation about the boundary theories is encoded in the function a (Ω), given by [45, 46] a (Ω) = 2 (cid:90) ∞ dµ (cid:104) − (cid:114) ϕ (1+ µ )2+ ϕ (1+ µ ) (cid:105) . (78)This is an implicit function of the opening angle Ω through the dependence of ϕ on Ω.One has Ω = (cid:90) +Ω / − Ω / dθ = (cid:90) ϕ dϕ ϕ (cid:112) ϕ (cid:112) ϕ (cid:113) ( ϕ − ϕ ) (cid:0) ϕ + (1 + ϕ ) ϕ (cid:1) . (79)16ull numerical solution of the function a (Ω) was presented in [45, 46] and we reproduce itin the left panel of Fig.1. Moreover, careful examination of the function for both the smallopening angle Ω → → π , one finds to leading order [45, 46]Ω → , a (Ω) = 2Γ (cid:0) (cid:1) / ( π Ω) + · · · , Ω → π , a (Ω) = ( π − Ω) / π + · · · . (80)It was established in [45, 46] that some universal information about the boundary theoriescan be read off from the above asymptotic expansions.Return to the bulk entanglement entropy. We deduce the scalar curvature of the RTsurface Σ as R Σ = − (cid:96) AdS ˙ ϕ + (1 + 2 ϕ ) ˙ ϕ − ϕ (1 + ϕ ) ¨ ϕ (1 + ϕ + ˙ ϕ ) = − (cid:96) AdS ϕ (1 + 2 ϕ ) + (1 + ϕ + ϕ ) ϕ ϕ (1 + ϕ ) , (81)where in the second line we have adopted the relation (76). Using these results and evalu-ation of the one-loop bulk entanglement entropy yields s q = 1180 π (cid:90) Rδ/ϕ drr (cid:90) √ r /δ − /ϕ dµ Y Ω ( µ ) , (82)where Y Ω ( µ ) = 2(1 + ϕ ) + 29 (cid:0) ϕ (1 + µ ) (cid:1) (cid:112) ϕ (1 + µ ) (cid:0) ϕ (1 + µ ) (cid:1) / . (83)The subscript of the function Y Ω ( µ ) reminds us that it depends on the opening angle Ωimplicitly. Note that the right hand side of the equation (82) diverges in the short distancelimit δ → Y Ω ( µ ) →
29 as µ → ∞ . To isolate this divergence, we treat (82) as s q = 1180 π (cid:90) Rδ/ϕ drr (cid:90) ∞ dµ (cid:104) Y Ω ( µ ) − (cid:105) + 29180 π (cid:90) Rδ/ϕ drr (cid:115) r δ − ϕ = 29 R πδ − b (Ω) log (cid:16) Rδ (cid:17) + · · · , (84)where again the dots stands for constant terms which are regulator dependent. The function b (Ω) is defined as b (Ω) = 1180 π (cid:90) ∞ dµ (cid:104) − Y Ω ( µ ) (cid:105) . (85)It is interesting to notice that the result (84) takes a similar form to (77) for the leadingorder entanglement entropy. Full numerical solution of the function b (Ω) is shown in the17 .0 0.2 0.4 0.6 0.802468 W (cid:144) p a H W L W (cid:144) p b H W L Figure 1: The plots for the function a (Ω) (left panel) and b (Ω) (right panel). For small open-ing angle Ω →
0, one has to leading order a (Ω) = 2Γ( ) / ( π Ω) and b (Ω) = 3Γ( ) / (cid:0) π Ω (cid:1) (shown in dashed red) whilst in the smooth limit Ω → π , a (Ω) = ( π − Ω) / π and b (Ω) = 3( π − Ω) / π (show in dashed green). In both limits, b (Ω) /a (Ω) = 3 / π .right panel of Fig.1. At first sight, we find that it behaves very similar to the function a (Ω).In addition, examination of the function for a small opening angle as well as the smoothlimit tells us that in both cases, their ratio is a constant b (Ω) a (Ω) = 340 π . (86)This motives us to study the function b (Ω) more carefully. As a matter of fact, we cananalytically show that the ratio is valid for an arbitrary opening angle Ω, despite that theintegrated function in (85) looks quite different from that in (78). The proof is straightfor-ward by making use of a mathematical identity (cid:90) ∞ dµ f ( µ ) = µ (cid:16) − (cid:114) ϕ (1+ µ )1+ ϕ (1+ µ ) (cid:17)(cid:12)(cid:12)(cid:12) ∞ = 0 , (87)where f ( µ ) = 1 − (cid:114) ϕ (1+ µ )2+ ϕ (1+ µ ) − ϕ √ ϕ (1+ µ ) (cid:0) ϕ (1+ µ ) (cid:1) / . (88)This strongly implies that the same information about holographic CFT can be read offfrom both the bulk entanglement entropy and the leading entanglement entropy of singularshapes, except for a constant ratio. It is remarkable that the result does not depend on theopening angle of the cone in the boundary. Having studied the bulk entanglement entropy for various shapes of boundary subregions,we would like to further investigate its shape dependence to see whether the same physical18nformation can be extracted in this case as that in the leading entanglement entropy. Wefirst consider the D = 4 dimension and then extend the discussions to the D = 6 dimension. D = 4 dimension Consider a generally smooth entangling surface in the boundary, obtained by slightly de-forming a circle r ( φ ) /R = 1 + ˆ (cid:15) ∞ (cid:88) (cid:96) =1 (cid:16) a (cid:96) cos ( (cid:96)φ ) √ π + b (cid:96) sin ( (cid:96)φ ) √ π (cid:17) , (89)where ˆ (cid:15) is a small parameter. We have properly chosen the expansion coefficients a (cid:96) , b (cid:96) sothat the Fourier modes are normalized to unity.As shown in [47], the bulk minimal surface in this case becomes a deformed hemisphereand the lowest order correction to the leading entanglement entropy appears at ˆ (cid:15) . Weshall briefly review these results in the following and then calculate the one-loop bulk en-tanglement entropy. We will show that the lowest order correction to the bulk entanglementappears at ˆ (cid:15) as well.For later convenience, we introduce new coordinates ( ρ , θ ), under which the bulk metricreads ds = (cid:96) AdS ρ cos θ (cid:16) dt E + dρ + ρ (cid:0) dθ + sin θ dφ (cid:1)(cid:17) , (90)where ρ = (cid:112) ξ + r , θ = arctan (cid:0) rξ (cid:1) , (91)where θ ∈ [0 , π/
2] and θ = π/ ρ slices describe bulk hemispheres. Hence, a deformed hemisphere can beparameterized as ρ/R = 1 + ˆ (cid:15) ρ ( θ , φ ) + O (ˆ (cid:15) ) , (92)where the higher order terms O (ˆ (cid:15) ) do not contribute to the leading entanglement entropy[47] as well as the bulk entanglement at ˆ (cid:15) order, as will be shown later.The induced metric of the bulk surface is given by ds = (cid:96) AdS ρ cos θ (cid:104)(cid:0) ρ θ + ρ (cid:1) dθ + 2 ρ θ ρ φ dθdφ + (cid:0) ρ φ + ρ sin θ (cid:1) dφ (cid:105) , (93)where ρ θ ≡ ∂ θ ρ , ρ φ = ∂ φ ρ . The deformations ρ ( θ , φ ) can be solved analytically by mini-mizing the area functional A Σ = (cid:96) AdS (cid:90) π/ − ε dθ (cid:90) π dφ ρ cos θ (cid:113) ρ φ + (cid:0) ρ θ + ρ (cid:1) sin θ , (94)19here the angular cutoff ε is related to the short distance cutoff δ at the asymptotic bound-ary as δ = R cos( π/ − ε ). Then straightforward calculations lead to [47, 48] ρ ( θ , φ ) = ∞ (cid:88) (cid:96) =1 (cid:16) a (cid:96) cos ( (cid:96)φ ) √ π + b (cid:96) sin ( (cid:96)φ ) √ π (cid:17) tan (cid:96) (cid:0) θ (cid:1) (1 + (cid:96) cos θ ) . (95)Evaluation of the area functional gives S = 2 πRδ (cid:16) (cid:15) (cid:88) (cid:96) (cid:96) ( a (cid:96) + b (cid:96) )4 (cid:17) − (cid:16) π + ˆ (cid:15) (cid:88) (cid:96) (cid:96) ( (cid:96) − a (cid:96) + b (cid:96) ) (cid:17) . (96)To compute the one-loop bulk entanglement, we deduce the scalar curvature for thedeformed hemisphere R Σ = − (cid:96) − AdS (cid:16) (cid:15) (cid:96) ( (cid:96) − ( a (cid:96) + b (cid:96) ) cot θ tan (cid:96) (cid:0) θ (cid:1)(cid:17) . (97)It is clear that the lowest order corrections appears at (cid:15) . We obtain s q = − R δ (cid:16) (cid:15) (cid:88) (cid:96) (cid:96) ( a (cid:96) + b (cid:96) )4 (cid:17) + (cid:16) (cid:15) (cid:88) (cid:96) π (cid:96) ( (cid:96) − a (cid:96) + b (cid:96) ) (cid:17) . (98)It is interesting to observe that the bulk entanglement shares the same shape dependencewith the leading entanglement entropy in the boundary, except for a constant factor π .This is highly non-trivial since in the boundary, the universal piece of the leading entangle-ment entropy is a constant, encoding nonlocal information about the underlying theorieswhile the bulk entanglement just contains local information about O (1) degrees of freedomsin the boundary. This local/nonlocal duality implies that for holographic CFT , the samephysical information can be extracted from the entanglement entropy at either the leadingorder O ( N ) or the subleading order O (1) (this is said in the sense that interactions betweenthe O (1) and O ( N ) degrees of freedoms is ignored since we have not included backreactioneffects of bulk fluctuations ).Last but not least, we realize that the particular value of the ratio π is the same as (86)for the entanglement entropy of singular shapes. This is easily explained for the smooth limitΩ → π . It was shown in [49] that the universal term of the leading entanglement entropy(77) for singular shapes can be derived from that for the smoothly deformed spheres (96).It is clear that this will also be the case for the bulk entanglement due to (98) and hencethe coincidence of the ratio π is not a surprise. D = 6 dimension Inspired by the interesting results in four dimension, we would like to investigate the shapedependence of bulk entanglement in higher dimensions to see whether the local/nonlocal20uality is still valid. We focus on the D = 6 dimension in this subsection. The relevantheat kernel coefficient is a , given by [31] a = 17! (cid:90) d D x √ g (cid:104) R − RR µν R µν + 143 RR µνρσ R µνρσ − R µλ R µρ R λρ − R µν R λρ R µλνρ − R µλ R µαβρ R λαβρ − R λρµν R αβλρ R µναβ − R λ ρµ ν R α βλ ρ R µ να β + der (cid:105) , (99)where der stands for derivative terms of curvatures, given by der = 18 (cid:3)(cid:3) R + 17 (cid:0) ∇ R (cid:1) − ∇ σ R µν ∇ σ R µν − ∇ σ R µν ∇ µ R νσ +9 ∇ σ R µνλρ ∇ σ R µνλρ + 28 R (cid:3) R − R µν (cid:3) R µν (100)+24 R µλ ∇ µ ∇ ρ R λρ + 12 R µνρσ (cid:3) R µνρσ . However, since total derivative terms do not contribute to ( ∂ n a ,n ) n =1 [38], we can do anintegration by parts and drop all these terms. This simplifies der as der = − (cid:0) ∇ R (cid:1) + 6 ∇ σ R µν ∇ σ R µν − ∇ σ R µνλρ ∇ σ R µνλρ +4 (cid:0) R µλ R µρ R λρ − R µν R λρ R µλνρ (cid:1) . (101)The coefficient a can be reorganized as a = a (1)3 + a (2)3 , (102)where a (1)3 = 17! (cid:90) d D x √ g (cid:104) R − RR µν R µν + 143 RR µνρσ R µνρσ − R µλ R µρ R λρ − R µν R λρ R µλνρ − R µλ R µαβρ R λαβρ − R λρµν R αβλρ R µναβ − R λ ρµ ν R α βλ ρ R µ να β (cid:105) , (103) a (2)3 = (cid:90) d D x √ g (cid:16) − (cid:0) ∇ R (cid:1) + 1840 ∇ σ R µν ∇ σ R µν − ∇ σ R µνλρ ∇ σ R µνλρ (cid:17) , where the first term involves third order curvature polynomials whilst the second termcontains first order derivatives of curvatures. Since the contributions of the above twoterms are highly different, we shall deal with them separately.According to (27), the cubic Riemannian term a (1)3 can be evaluated as ∂ n a (1)3 ,n (cid:12)(cid:12)(cid:12) n =1 = Wald + Anomaly , (104)21here the shorthand notations on the r.h.s are specified in (28) and (29). We quote themas follows Wald = − π (cid:90) Σ E µνρσ ε µν ε ρσ , Anomaly = 2 π (cid:90) Σ (cid:88) α (cid:16) ∂ L∂R izjz ∂R k ¯ zl ¯ z (cid:17) α K zij K ¯ zkl q α + 1 , (105)where E µνρσ = ∂L/∂R µνρσ . To proceed, we need derive the tensor E µνρσ = ∂L/∂R µνρσ for each of the cubic curvature polynomials as well as the second order derives ∂ L∂R izjz ∂R k ¯ z(cid:96) ¯ z .The calculations are straightforward but a bit lengthy. We refer the readers to AppendixB for details. The results for general cases are quite involved but will be greatly simplifiedfor minimal area surfaces in AdS vacuum. Here we just present the final result ∂ n a (1)3 ,n (cid:12)(cid:12)(cid:12) n =1 = − π (cid:96) AdS (cid:90) Σ (cid:0) (cid:96) − AdS + K aij K aij (cid:1) . (106)The second term a (2)3 in (103), as a gravitational action, its contribution to holographicentanglement entropy was studied carefully in [15] for general coupling constants I = (cid:90) d D x √ g (cid:16) λ ∇ µ R ∇ µ R + λ ∇ σ R µν ∇ σ R µν + λ ∇ σ R µνλρ ∇ σ R µνλρ (cid:17) . (107)In our case (103), λ = − / , λ = 1 / , λ = − / ∂ n I [ ˆ M n ] (cid:12)(cid:12)(cid:12) n =1 = Wald + Anomaly , (108)where the generalised Wald entropy term is given by (36). For this particular action (107),one has [15]Wald = 4 π (cid:90) Σ λ (cid:3) R + λ n µν (cid:3) R µν + λ ε µν ε ρσ (cid:3) R µνρσ + (cid:0) λ ∇ α R µν K β + 2 λ ∇ α R µρνσ K βρσ (cid:1)(cid:0) n βµ n αν − ε βµ ε αν (cid:1) . (109)However, this term will vanish for RT surfaces in AdS vacuum, where the Riemann curvaturetakes a particularly simple form R µλνρ = − (cid:96) − AdS (cid:0) g µν g λρ − g µρ g νλ (cid:1) . (110)Derivation of the anomaly terms is much more involved and the results have lengthy ex-pressions as well, see the Appendix A of [15]. Fortunately, in this subsection, we focus ondeformed hemispheres, which just has small extrinsic curvatures K ∼ ˆ (cid:15) . We have knownthat in this case, the lowest order correction of the smooth deformations to the universal22erms of entanglement entropy appears at the quadratic order O ( K ). Thus, we can dropall the higher order terms presented in [15]. We find ∇ µ R ∇ µ R : A nomaly = O ( K ) . (111)For ∇ σ R µν ∇ σ R µν ,A nomaly = − π (cid:90) Σ (cid:16) K aij K ijb Q cd − Q ab Q cd − (cid:0) R ai R ib + R ci R id (cid:1)(cid:17)(cid:0) n ac n bd − ε ac ε bd (cid:1) + O ( K ) , (112)and for ∇ σ R µναβ ∇ σ R µναβ ,A nomaly = 4 π (cid:90) Σ (cid:104) (cid:0) Q abij Q ijcd − P abcij K ijd (cid:1)(cid:0) n ac n bd − ε ac ε bd (cid:1) +4 K ia(cid:96) K a(cid:96)j Q b bij + (cid:0) Q b b − T (cid:1) K aij K aij +2 D (cid:96) K aij D (cid:96) K aij + 2 R aijk R aijk (cid:105) + O ( K ) . (113)However, here we have not considered the relation between Q abij , P abcij and the extrinsiccurvatures. By comparing the Riemann tensor in the metric expansion (33) with that forpure AdS, one finds [15] T = − (cid:96) AdS , V i = 0 , Q zzij = K zi(cid:96) K (cid:96)zj ,Q ¯ z ¯ zij = K ¯ zi(cid:96) K (cid:96) ¯ zj , Q z ¯ zij = (cid:96) AdS g ij + K zi(cid:96) K (cid:96) ¯ zj ,P zz ¯ zij = (cid:96) AdS K zij + O ( K ) , P ¯ z ¯ zzij = (cid:96) AdS K ¯ zij + O ( K ) . (114)Using these relations, we find that remarkably all the terms in (112) are of quartic order O ( K ) and hence are irrelevant for our discussions. Finally, the anomaly terms for thederivative action (107) greatly simplify toA nomaly = 8 πλ (cid:90) Σ (cid:0) D(cid:96) − AdS K aij K aij + D (cid:96) K aij D (cid:96) K aij (cid:1) . (115)Combing all the results together, we deduce ∂ n a ,n (cid:12)(cid:12)(cid:12) n =1 = − π S − (cid:90) Σ (cid:16) π (cid:96) AdS K aij K aij + π D (cid:96) K aij D (cid:96) K aij (cid:17) . (116)To proceed, we need construct the adapted coordinates around the deformed hemi-spheres ρ Σ = 1 + (cid:15)ρ (Ω d ) and read off the extrinsic curvatures. This is achieved by con-structing geodesics emanating from the surface [48]. We denote the affine parameter of thegeodesics by s and s = 0 at the surface. We set ∂ s · ∂ t = sin τ and the starting point onthe surface is ω d = ( θ , ω d − ). The geodesics can be constructed by solving the geodesic23quation in a power series of s and hence covering a small neighborhood of the surface usingthe new coordinates ( τ , s , ω d ). We present the linear in s peace as t = s sin τ cos θ ρ Σ ( ω d ) + · · · ,ρ = ρ Σ ( ω d ) + s cos τ cos θ ρ Σ ( ω d ) + · · · , Ω d = ω d − s cos τ cos θ ∂ ω d ρ Σ ( ω d ) + · · · , (117)where ∂ ω d = ˜ g ij ∂ j , where ˜ g ij is the metric of unit S d . Finally, we change the variables to z = se iτ , ¯ z = se − iτ . The extrinsic curvatures can be read off straightforwardly from theconstruction for each dimension. We find that the results, valid to general dimensions, canbe expressed compactly as K zij = K ¯ zij = 12 (cid:16) D i m j + γ (0) ij (cid:15)ρ cos θ (cid:17) + O ( (cid:15) ) , (118)where m i = − (cid:96) AdS ∂ i (cid:16) (cid:15)ρ cos θ (cid:17) , γ (0) ij = (cid:96) AdS cos θ ˜ g ij . (119) After preparing so much, we are ready to perform explicit calculations for the one-loop bulkentanglement entropy for deformed hemispheres. In the boundary, the entangling surfaceis described by [48] r (Ω d − ) /R = 1 + ˆ (cid:15) (cid:88) (cid:96) ,m ··· ,m d − a (cid:96) ,m ··· ,m d − Y (cid:96) ,m ··· ,m d − (Ω d − ) , (120)where Ω d − are the angular coordinates on S d − and Y (cid:96) ,m ··· ,m d − (Ω d − ) are hypersphericalharmonics . We follow [48] and normalize the hyperspherical harmonics as (cid:90) d Ω d − Y (cid:96) ,m ··· ,m d − Y (cid:96) (cid:48) ,m (cid:48) ··· ,m (cid:48) d − = δ (cid:96)(cid:96) (cid:48) δ m m (cid:48) · · · δ m d − m (cid:48) d − . (122)As in the four dimensional case, the deformed hemisphere can be most easily solved underthe ( ρ , θ ) coordinates, where the bulk metric reads ds = (cid:96) AdS ρ cos θ (cid:16) dt E + dρ + ρ (cid:0) dθ + sin θ d Ω d − (cid:1)(cid:17) . (123) They are eigenfunctions of the Laplacian on S d − :∆ Y (cid:96) ,m ··· ,m d − (Ω d − ) = − (cid:96) ( (cid:96) + d − Y (cid:96) ,m ··· ,m d − (Ω d − ) . (121) ρ/R = 1 + ˆ (cid:15) ρ ( θ , Ω d − ) + O (ˆ (cid:15) ) , (124)where again the higher order terms O (ˆ (cid:15) ) do not contribute to entanglement entropy at ˆ (cid:15) order. It follows that the deformation ρ ( θ , Ω d − ) can be solved analytically by minimizingthe area functional A Σ = (cid:96) dAdS (cid:90) π/ − ε dθ (cid:90) d Ω d − (sin θ ) d − ρ (cos θ ) d (cid:113) ρ + (cid:0) ρ θ + ρ (cid:1) sin θ , (125)where ρ θ ≡ ∂ θ ρ , ρ Ω = (cid:112) ˜ g ij ∂ j ρ , where ˜ g ij is the metric of unit S d − . Again the angular cutoff ε is related to the short distance cutoff δ at the asymptotic boundary as δ = R cos( π/ − ε ).In the D = 6 dimension, one finds [48] ρ ( θ , Ω ) = (cid:88) (cid:96) ,m ,m a (cid:96) ,m ,m Y (cid:96) ,m ,m (Ω ) f ( θ ) , (126)where f ( θ ) = tan (cid:96) (cid:0) θ (cid:1) (cid:96) + 1) cos θ + (cid:96) ( (cid:96) +2)3 cos θ θ . (127)Evaluation of the area functional yields S = R δ − Rδ + 4 π (cid:15) (cid:88) (cid:96) ,m ,m a (cid:96) ,m ,m (cid:96) ( (cid:96) − (cid:96) + 2)( (cid:96) + 3) . (128)To proceed, we present explicit formulas for the three dimensional hyperspherical har-monics [50] Y (cid:96) ,m ,m (Ω ) = P νµ , / (cos φ ) Y m ,m ( φ , φ ) , (129)where Y m ,m ( φ , φ ) is the usual spherical harmonics and P νµ , / (cos φ ) is a hyper-Legendrefunction, given by P νµ , / (cos φ ) = N (cid:96)m (cid:0) sin φ (cid:1) m F (cid:16) m − (cid:96) , m + (cid:96) + 22 , , (cid:0) cos φ (cid:1) (cid:17) , (130)where ν = ± (cid:112) m ( m + 1) , µ = − ± √ (cid:96) + 8 (cid:96) + 12 , (131)and N (cid:96)m is a normalization constant so that (cid:90) π dφ sin φ P νµ , / (cos φ ) P νµ (cid:48) , / (cos φ ) = δ (cid:96)(cid:96) (cid:48) . (132)According to (116), the key elements to derive the one-loop bulk entanglement entropyare surface integrals of extrinsic curvatures and their spatial derivatives evaluated on theminimal surface. By straightforward calculations, we find (cid:90) Σ (cid:96) − AdS K aij K aij = − ˆ (cid:15) (cid:88) (cid:96) ,m ,m a (cid:96) ,m ,m (cid:96) ( (cid:96) − (cid:96) + 2)( (cid:96) + 3) . (133)25gain as in the four dimensional case, this term depends on the shape of entangling surfacein the same manner as the leading entanglement entropy. However, for the derivative termsof extrinsic curvatures, the situation turns out to be much more complicated. To clarifythis, we may set (cid:90) Σ D (cid:96) K aij D (cid:96) K aij ≡ ˆ (cid:15) (cid:88) (cid:96) ,m ,m q (cid:96) ,m ,m a (cid:96) ,m ,m (cid:96) ( (cid:96) − (cid:96) + 2)( (cid:96) + 3) , (134)where the dependence on the shape of entangling surface is encoded in the functional relation q (cid:96) ,m ,m . However, it is of great difficult to calculate this term for general eigenvaluesanalytically. Nevertheless, we can check some special cases to see whether it is a sameconstant for general eigenvalues. This is enough for our purpose. We present some lowlying examples as follows q (cid:96) ,m , = − ,q , , = − (cid:39) − . ,q , , = − (cid:39) . ,q , , = − (cid:39) − . ,q , , = − (cid:39) − . . (135)It is clear that q (cid:96) ,m ,m is no longer a same constant for general eigenvalues. Mathematicallythis is not hard to explain since the spatial derivatives of the hyperspherical harmonicsgenerally mixes different Fourier modes of the entangling surface. It implies that in the D = 6 dimension, the bulk entanglement entropy at O (ˆ (cid:15) ) order depends on the shape ofentangling surface more strongly than the area of the RT surface. It may encode moreuniversal information about the boundary theories at this order. We expect that in generalthis will also be the case for higher dimensions, since more higher order derivative terms ofRiemann curvatures will appear in the heat kernel coefficient a [ D/ and hence more spatialderivative terms of extrinsic curvatures appear in the bulk entanglement entropy. In this paper, we adopt the heat kernel method to study one-loop bulk entanglement entropyin diverse dimensions. A shortcoming of the method is it does not capture the cut-offindependence piece of bulk entanglement entropy in odd dimensions. As a consequence, wefocus on even dimensions in this paper. 26e perform explicit calculations in the D = 4 dimension for several different shapesof subregions in the boundary. In particular, for a cusp subregion, we find that the bulkentanglement entropy encodes the same universal information about the boundary theoriesas the leading entanglement entropy, up to a fixed proportional constant. Furthermore,we study the shape dependence of bulk entanglement by considering a smoothly deformedcircle. We find that at leading order of the deformations, the bulk entanglement entropyshares the same shape dependence with the leading entanglement entropy in the boundary.This is interesting since the former just captures local information about O (1) degreesof freedoms in the boundary whilst the latter encodes nonlocal information about O ( N )degrees of freedoms. The result establishes a local/nonlocal duality for shape dependenceof entanglement entropy for holographic CFT .To see whether the same results hold for higher dimensions, we extend our investigationsto the D = 6 dimension. We find that the answer is no. The reason is in D ≥ a [ D/ , which captures the physical information about the under-lying theories, contains covariant derivatives of Riemann curvatures. As a consequence, theone-loop bulk entanglement entropy will depend on spatial derivatives of extrinsic curva-tures along the RT surfaces and hence depends on the shape of boundary subregions morestrongly. This implies that in general the nice result in the D = 4 dimension is not valid togeneral D ≥ Acknowledgments
Z.Y. Fan was supported in part by the National Natural Science Foundations of China withGrant No. 11805041 and No. 11873025.
A Curvatures for orbifold geometry
Let us calculate curvatures for a product metric as ds = G ab dx a dx b + G ij dy i dy j = e φ ˆ g ab dx a dx b + G ij dy i dy j , (136)where G ab = G ab ( x ) , G ij = G ij ( x , y ) , G ai = G ai = 0 . (137)27e are interested in evaluating the curvature tensors at a given codimension − y directions. One has the metric expansion G ij ( x , y ) = γ ij ( y ) + 2 K aij x a + Q abij x a x b + · · · , (138)where K aij are extrinsic curvatures of Σ. Notice that under the above coordinates, K aij = ∂ a G ij and Q abij ≡ ∂ a K bij . We use the metric ˆ g ab and ˆ g ab to raise and lower indices a , b , · · · for curvature tensors, for example K aij = ˆ g ab K bij . The inverse metric G ij is given by G ij = γ ij − K ija x a − (cid:0) Q ijab − K i(cid:96)a K jb(cid:96) (cid:1) x a x b + · · · , (139)where K ija = − ∂ a G ij , where the minus sign emerges owing to the requirement G ik G kj = δ ij . However, we are interested in deriving Riemann curvatures on the surface Σ. In manycases, we can take the approximation G ij ( x , y ) = γ ij ( y ) but it is not always true. We needkeep in mind that all the relevant terms around the surface Σ should be included whennecessary. Without confusion, we will raise and lower the indices i , j , · · · by using theinduced metric γ ij on the surface Σ.The Christoffel connection and the Riemann tensor can be calculated from their standarddefinitions. One has R abcd = r abcd , R iabc = 0 , R aijk = D k K aij − D j K aik ,R ijab = γ kl ( K bik K aj(cid:96) − K aik K bj(cid:96) ) ,R iajb = Γ cab K cij + γ k(cid:96) K ajk K bi(cid:96) − Q abij ,R ijk(cid:96) = R ijk(cid:96) + e − φ ( K ai(cid:96) K ajk − K aik K aj(cid:96) ) , (140)where Γ abc = 2 δ a ( b ∂ c ) φ − ∂ a φ ˆ g bc . (141)The Ricci tensor and scalar can be derived as R ai = D j K aji − ∂ i K a ,R ab = r ab + Γ cab K c + K aij K ijb − γ ij Q abij ,R ij = R ij + e − φ (cid:0) K a(cid:96)i K a(cid:96)j − K a K aij − Q aa ij (cid:1) , (142)and R = r + R − e − φ (cid:0) K a K a − K aij K aij + 2 γ ij Q aa ij (cid:1) . (143)Note that the above relation is nothing else but an equivalent expression for Gauss-Codazziidentity. The Riemann tensor in the cone directions up to linear order in (cid:15) is given by R abcd = r abcd = e φ (cid:0) ˆ g ad ∂ b ∂ c φ + ˆ g bc ∂ a ∂ d φ − ˆ g ac ∂ b ∂ d φ − ˆ g bd ∂ a ∂ c φ (cid:1) + O ( (cid:15) ) , (144)28here φ = − (cid:15) log ρ (without introducing a cut-off away from the cone ρ = 0). This impliesthat r ab = − ˆ g ab ˆ ∇ φ and r = − e − φ ˆ ∇ φ , where ˆ ∇ is defined with respect to the twodimensional metric ˆ g . For later convenience, the metric ˆ g will be expressed frequently inthree types of coordinates: the Cartesian coordinates ( x , x ), the cylindrical coordinates( ρ , τ ) and the complex coordinates ( z , ¯ z ), which are defined as x = ρ cos τ , x = ρ sin τ ,z = ρe iτ = x + ix , ¯ z = ρe − iτ = x − ix . (145)One has d ˆ s = δ ab dx a dx b = dρ + ρ dτ = dzd ¯ z . (146)To exclude contributions from the conical singularity of the orbifold geometry ˆ M n , thefunction φ should be properly regularized, for example we may set φ = − (cid:15) log ( ρ + a )and take the limit a → ∂ z φ = − (cid:15)/ z , ∂ ¯ z φ = − (cid:15)/ z and ∂ z ∂ ¯ z φ = π (cid:15) δ ( x ). These relations determine the singular terms in the Riemann curvatures R z ¯ zz ¯ z = e φ ∂ z ∂ ¯ z φ + · · · ,R izjz = 2 ∂ z φ K zij + · · · ,R i ¯ zj ¯ z = 2 ∂ ¯ z φ K ¯ zij + · · · , (147)as well as those in the Ricci tensors R zz = 2 ∂ z φK z + · · · ,R ¯ z ¯ z = 2 ∂ ¯ z φK ¯ z + · · · ,R z ¯ z = r z ¯ z + · · · = − π(cid:15) δ ( x ) + · · · . (148)In the above, the first two equalities imply that the trace of the extrinsic curvatures shouldvanish for Einstein’s gravity. This proves the RT formula. Finally, for Ricci scalar R = r + · · · = e − φ (cid:104) − π(cid:15) δ ( x ) (cid:105) + · · · . (149)These results are sufficient to derive holographic entanglement entropy for general Rieman-nian gravities [12].As a simple application, we take Σ as a bifurcate event horizon, which has vanishingextrinsic curvatures K aij = 0. In this case, all the singular contributions to the curvature29ensors come from the conical two dimensions. The results can be expressed compactly as R µλνρ = ¯ R µλνρ − π(cid:15) ε µλ ε νρ δ Σ ,R µν = ¯ R µν − π(cid:15) n µν δ Σ ,R = ¯ R − π(cid:15) δ ( x ) , (150)where ¯ R stands for the curvatures in the smooth region of ˆ M n ; n µi are two normal vectorsof Σ (here the normal vectors are normalized to unity). ε µν = n µ n ν − n µ n ν is binormalvector of Σ and n µν = Σ i =1 n iµ n iν . Some useful relations are n µ n µ = 2 = n µν n µν and ε µλ ε νρ = n µν n λρ − n µρ n νλ , ε µσ ε σν = n µν . (151)In the n → n = ( dz + d ¯ z ) ,n = i ( dz − d ¯ z ) . (152)This leads to ε z ¯ z = i/ n zz = 0 = n ¯ z ¯ z , n z ¯ z = . (153)These results are particularly useful to transform the entropy formula for higher derivativegravities into covariant forms. B Derivation details about (106)
We list our results for each term in (103) as follows R : E µνρσ = 3 R g ρ [ µ g ν ] σ ,RR µν : E µνρσ = g ρ [ µ g ν ] σ R αβ + R (cid:0) g ρ [ µ R ν ] σ − g σ [ µ R ν ] ρ (cid:1) ,RR µνρσ : E µνρσ = 2 RR µνρσ + g ρ [ µ g ν ] σ R αβγδ ,R µσ R σν R µν : E µνρσ = 32 (cid:0) g ρ [ µ R ν ] α R ασ − g σ [ µ R ν ] α R αρ (cid:1) ,R µρ R νσ R µνρσ : E µνρσ = R ρ [ µ R ν ] σ + R αβ (cid:0) g ρ [ µ R ν ] ασβ − g σ [ µ R ν ] αρβ (cid:1) ,R µλ R µνρσ R λνρσ : E µνρσ = − R [ µα R ν ] αρσ + 12 (cid:0) g ρ [ µ R ν ] αβγ R σαβγ − g σ [ µ R ν ] αβγ R ρ αβγ (cid:1) ,R αβµν R ρσαβ R µνρσ : E µνρσ = 3 R µναβ R ρσαβ ,R λ ρµ ν R α βλ ρ R µ να β : E µνρσ = 32 (cid:0) R αρβ [ µ R ν ] σβ α − R ασβ [ µ R ν ] ρβ α (cid:1) . (154)30t is straightforward to evaluate the Wald entropy terms R : Wald = − π (cid:90) Σ R ,RR µν : Wald = − π (cid:90) Σ (cid:0) R µν + RR µν n µν (cid:1) ,RR µνρσ : Wald = − π (cid:90) Σ (cid:0) R µνρσ + RR µνρσ ε µν ε ρσ (cid:1) ,R µσ R σν R µν : Wald = − π (cid:90) Σ R µσ R σν n µν ,R µρ R νσ R µνρσ : Wald = − π (cid:90) Σ (cid:0) R µρ R νσ ε µν ε ρσ + 2 R µλνρ R µν n λρ (cid:1) ,R µλ R µνρσ R λνρσ : Wald = − π (cid:90) Σ (cid:0) R µαβγ R αβγν n µν + 2 R λµ R λνρσ ε µν ε ρσ (cid:1) ,R αβµν R ρσαβ R µνρσ : Wald = − π (cid:90) Σ R αβµν R αβρσ ε µν ε ρσ ,R λ ρµ ν R α βλ ρ R µ να β : Wald = − π (cid:90) Σ R µαρβ R α βν σ ε µν ε ρσ . (155)For RT surfaces in AdS vacuum, the results can be even more simplified. We shall not listthem here. 31or the anomaly terms, after straightforward but lengthy derivations, we obtain R : ∂ L∂R izjz ∂R k ¯ zl ¯ z = 0 = ⇒ Anomaly = 0 ,RR µν : ∂ L∂R izjz ∂R k ¯ zl ¯ z = 12 g ij g k(cid:96) R = ⇒ Anomaly = 2 π (cid:90) Σ R K a K a ,RR µνρσ : ∂ L∂R izjz ∂R k ¯ zl ¯ z = 2 g ik g j(cid:96) R = ⇒ Anomaly = 8 π (cid:90) Σ R K aij K aij ,R µσ R σν R µν : ∂ L∂R izjz ∂R k ¯ zl ¯ z = 34 g ij g k(cid:96) R a a = ⇒ Anomaly = 3 π (cid:90) Σ R a a K b K b ,R µρ R νσ R µνρσ : ∂ L∂R izjz ∂R k ¯ zl ¯ z = 12 (cid:0) g ij R k(cid:96) + g k(cid:96) R ij + 4 g ij g k(cid:96) R z ¯ zz ¯ z (cid:1) Anomaly = 2 π (cid:90) Σ (cid:16) K a K aij R ij − R ab ab K c K c (cid:17) , (156) R µλ R µνρσ R λνρσ : ∂ L∂R izjz ∂R k ¯ zl ¯ z = 12 (cid:0) g ij R kz(cid:96) ¯ z + g k(cid:96) R i ¯ zjz + g ik g j(cid:96) R z ¯ z (cid:1) + g j(cid:96) R ik Anomaly = 2 π (cid:90) Σ (cid:16) K aij K aij R b b + K a K aij R i jbb + 23 K a(cid:96)i K a(cid:96)j R ij (cid:17) ,R αβµν R ρσαβ R µνρσ : ∂ L∂R izjz ∂R k ¯ zl ¯ z = 3 (cid:0) g ik R jz(cid:96) ¯ z + g j(cid:96) R izk ¯ z (cid:1) Anomaly = 2 π (cid:90) Σ K ai(cid:96) K (cid:96)b j R i jc d (cid:0) n ab n cd + ε ab ε cd (cid:1) ,R λ ρµ ν R α βλ ρ R µ να β : ∂ L∂R izjz ∂R k ¯ zl ¯ z = 32 R ikj(cid:96) + 38 g ik g j(cid:96) R z ¯ zz ¯ z + 34 (cid:0) g ik R j(cid:96)z ¯ z + g j(cid:96) R ikz ¯ z (cid:1) Anomaly = 2 π (cid:90) Σ (cid:16) − K aij K aij R cd cd + 35 K aij K ak(cid:96) R i jk (cid:96) + 2 K a(cid:96)i K (cid:96)b j R ijab (cid:17) . The results can be greatly simplified by using the fact that Σ is extremal and hence K a = 0.Moreover, the spacetime is pure AdS so that R a a = − D − (cid:96) − AdS , R ab ab = − (cid:96) − AdS . Itfollows that R , RR µν , R µσ R σν R µν , R µρ R νσ R µνρσ : Anomaly = 0 ,RR µνρσ : Anomaly = − π (cid:96) − AdS (cid:90) Σ D ( D − K aij K aij ,R µλ R µνρσ R λνρσ : Anomaly = − π (cid:96) − AdS (cid:90) Σ ( D − K aij K aij ,R αβµν R ρσαβ R µνρσ : Anomaly = − π(cid:96) − AdS (cid:90) Σ K aij K aij ,R λ ρµ ν R α βλ ρ R µ να β : Anomaly = π (cid:96) − AdS (cid:90) Σ K aij K aij . Combing all the results above, one finally arrives at (106).32 eferences [1] S. Ryu and T. Takayanagi,
Holographic derivation of entanglement entropy fromAdS/CFT,
Phys. Rev. Lett. , 181602 (2006) [arXiv:hep-th/0603001 [hep-th]].[2] V. E. Hubeny, M. Rangamani and T. Takayanagi, A Covariant holographic entangle-ment entropy proposal,
JHEP , 062 (2007) [arXiv:0705.0016 [hep-th]].[3] G. Penington, Entanglement Wedge Reconstruction and the Information Paradox,
JHEP , 002 (2020) [arXiv:1905.08255 [hep-th]].[4] A. Almheiri, N. Engelhardt, D. Marolf and H. Maxfield, The entropy of bulk quantumfields and the entanglement wedge of an evaporating black hole,
JHEP , 063 (2019)[arXiv:1905.08762 [hep-th]].[5] A. Almheiri, R. Mahajan, J. Maldacena and Y. Zhao, The Page curve of Hawkingradiation from semiclassical geometry,
JHEP , 149 (2020) [arXiv:1908.10996 [hep-th]].[6] S. W. Hawking, Black hole explosions,
Nature , 30-31 (1974).[7] S. W. Hawking,
Particle Creation by Black Holes,
Commun. Math. Phys. , 199-220(1975) [erratum: Commun. Math. Phys. , 206 (1976)].[8] G. W. Gibbons and S. W. Hawking, Action Integrals and Partition Functions inQuantum Gravity,
Phys. Rev. D , 2752-2756 (1977).[9] A. Lewkowycz and J. Maldacena, Generalized gravitational entropy,
JHEP , 090(2013) [arXiv:1304.4926 [hep-th]].[10] B. Chen and J. j. Zhang, Note on generalized gravitational entropy in Lovelock gravity,
JHEP , 185 (2013) [arXiv:1305.6767 [hep-th]].[11] A. Bhattacharyya, A. Kaviraj and A. Sinha, Entanglement entropy in higher deriva-tive holography,
JHEP , 012 (2013) [arXiv:1305.6694 [hep-th]].[12] X. Dong, Holographic Entanglement Entropy for General Higher Derivative Gravity,
JHEP , 044 (2014) [arXiv:1310.5713 [hep-th]].[13] J. Camps, Generalized entropy and higher derivative Gravity,
JHEP , 070 (2014)[arXiv:1310.6659 [hep-th]]. 3314] A. Bhattacharyya and M. Sharma, On entanglement entropy functionals in higherderivative gravity theories,
JHEP , 130 (2014) [arXiv:1405.3511 [hep-th]].[15] R. X. Miao and W. z. Guo, Holographic Entanglement Entropy for the Most GeneralHigher Derivative Gravity,
JHEP , 031 (2015) [arXiv:1411.5579 [hep-th]].[16] H. Casini, M. Huerta and R. C. Myers, Towards a derivation of holographic entangle-ment entropy,
JHEP , 036 (2011) [arXiv:1102.0440 [hep-th]].[17] T. Hartman, Entanglement Entropy at Large Central Charge, [arXiv:1303.6955 [hep-th]].[18] T. Faulkner,
The Entanglement Renyi Entropies of Disjoint Intervals in AdS/CFT, [arXiv:1303.7221 [hep-th]].[19] X. Dong, A. Lewkowycz and M. Rangamani,
Deriving covariant holographic entan-glement,
JHEP , 028 (2016) [arXiv:1607.07506 [hep-th]].[20] T. Faulkner, A. Lewkowycz and J. Maldacena, Quantum corrections to holographicentanglement entropy,
JHEP , 074 (2013) [arXiv:1307.2892 [hep-th]].[21] N. Engelhardt and A. C. Wall, Quantum Extremal Surfaces: Holographic Entangle-ment Entropy beyond the Classical Regime,
JHEP , 073 (2015) [arXiv:1408.3203[hep-th]].[22] T. Barrella, X. Dong, S. A. Hartnoll and V. L. Martin, Holographic entanglementbeyond classical gravity,
JHEP , 109 (2013) [arXiv:1306.4682 [hep-th]].[23] A. Belin, N. Iqbal and J. Kruthoff, Bulk entanglement entropy for photons and gravi-tons in AdS , SciPost Phys. , no.5, 075 (2020) [arXiv:1912.00024 [hep-th]].[24] C. A. Ag´on, S. F. Lokhande and J. F. Pedraza, Local quenches, bulk entanglemententropy and a unitary Page curve,
JHEP , 152 (2020) [arXiv:2004.15010 [hep-th]].[25] B. Chen, P. X. Hao and W. Song, R´enyi mutual information in holographic warpedCFTs,
JHEP , 037 (2019) [arXiv:1904.01876 [hep-th]].[26] C. Ag´on and T. Faulkner, Quantum Corrections to Holographic Mutual Information,
JHEP , 118 (2016) [arXiv:1511.07462 [hep-th]].[27] B. Chen, Z. Y. Fan, W. M. Li and C. Y. Zhang, Holographic Mutual Information ofTwo Disjoint Spheres,
JHEP , 113 (2018) [arXiv:1712.05131 [hep-th]].3428] S. Ryu and T. Takayanagi, Aspects of Holographic Entanglement Entropy,
JHEP ,045 (2006) [arXiv:hep-th/0605073 [hep-th]].[29] T. Nishioka, S. Ryu and T. Takayanagi, Holographic Entanglement Entropy: AnOverview,
J. Phys. A , 504008 (2009) [arXiv:0905.0932 [hep-th]].[30] S. Sugishita, Entanglement entropy for free scalar fields in AdS,
JHEP , 128 (2016)[arXiv:1608.00305 [hep-th]].[31] D. V. Vassilevich, Heat kernel expansion: User’s manual,
Phys. Rept. , 279-360(2003) [arXiv:hep-th/0306138 [hep-th]].[32] S. N. Solodukhin,
The Conical singularity and quantum corrections to entropy of blackhole,
Phys. Rev. D , 609-617 (1995) [arXiv:hep-th/9407001 [hep-th]].[33] D. V. Fursaev and S. N. Solodukhin, On the description of the Riemannian geometryin the presence of conical defects,
Phys. Rev. D , 2133-2143 (1995) [arXiv:hep-th/9501127 [hep-th]].[34] S. N. Solodukhin, Entanglement entropy of black holes,
Living Rev. Rel. , 8 (2011)[arXiv:1104.3712 [hep-th]].[35] D. V. Fursaev, A. Patrushev and S. N. Solodukhin, Distributional Geometry ofSquashed Cones,
Phys. Rev. D , no.4, 044054 (2013) [arXiv:1306.4000 [hep-th]].[36] R. M. Wald, Black hole entropy is the Noether charge,
Phys. Rev. D , no.8, 3427-3431 (1993) [arXiv:gr-qc/9307038 [gr-qc]].[37] V. Iyer and R. M. Wald, Some properties of Noether charge and a proposal for dy-namical black hole entropy,
Phys. Rev. D , 846-864 (1994) [arXiv:gr-qc/9403028[gr-qc]].[38] X. Dong and R. X. Miao, Generalized Gravitational Entropy from Total DerivativeAction,
JHEP , 100 (2015) [arXiv:1510.04273 [hep-th]].[39] T. Albash and C. V. Johnson, Holographic Studies of Entanglement Entropy in Su-perconductors,
JHEP , 079 (2012) [arXiv:1202.2605 [hep-th]].[40] R. G. Cai, S. He, L. Li and Y. L. Zhang, Holographic Entanglement Entropy inInsulator/Superconductor Transition,
JHEP , 088 (2012) [arXiv:1203.6620 [hep-th]]. 3541] X. M. Kuang, E. Papantonopoulos and B. Wang, Entanglement Entropy as aProbe of the Proximity Effect in Holographic Superconductors,
JHEP , 130 (2014)[arXiv:1401.5720 [hep-th]].[42] Y. Ling, P. Liu, C. Niu, J. P. Wu and Z. Y. Xian, Holographic Entanglement EntropyClose to Quantum Phase Transitions,
JHEP , 114 (2016) [arXiv:1502.03661 [hep-th]].[43] Y. Ling, P. Liu and J. P. Wu, Characterization of Quantum Phase Transition us-ing Holographic Entanglement Entropy,
Phys. Rev. D , no.12, 126004 (2016)[arXiv:1604.04857 [hep-th]].[44] T. Hirata and T. Takayanagi, AdS/CFT and strong subadditivity of entanglemententropy,
JHEP , 042 (2007) [arXiv:hep-th/0608213 [hep-th]].[45] P. Bueno, R. C. Myers and W. Witczak-Krempa, Universality of corner entanglementin conformal field theories,
Phys. Rev. Lett. , 021602 (2015) [arXiv:1505.04804[hep-th]].[46] P. Bueno and R. C. Myers,
Corner contributions to holographic entanglement entropy,
JHEP , 068 (2015) [arXiv:1505.07842 [hep-th]].[47] A. Allais and M. Mezei, Some results on the shape dependence of entanglement andR´enyi entropies,
Phys. Rev. D , no.4, 046002 (2015) [arXiv:1407.7249 [hep-th]].[48] M. Mezei, Entanglement entropy across a deformed sphere,
Phys. Rev. D , no.4,045038 (2015) [arXiv:1411.7011 [hep-th]].[49] P. Bueno and R. C. Myers, Universal entanglement for higher dimensional cones,