Some Calculable Contributions to Holographic Entanglement Entropy
aa r X i v : . [ h e p - t h ] M a y Preprint typeset in JHEP style - HYPER VERSION arXiv:1105.nnnn [hep-th]
Some Calculable Contributions to HolographicEntanglement Entropy
Ling-Yan Hung, Robert C. Myers and Michael Smolkin
Perimeter Institute for Theoretical Physics,31 Caroline Street North, Waterloo, Ontario N2L 2Y5, Canada
Abstract:
Using the AdS/CFT correspondence, we examine entanglement entropyfor a boundary theory deformed by a relevant operator and establish two results. Thefirst is that if there is a contribution which is logarithmic in the UV cut-off, then thecoefficient of this term is independent of the state of the boundary theory. In fact,the same is true of all of the coefficients of contributions which diverge as some powerof the UV cut-off. Secondly, we show that the relevant deformation introduces newlogarithmic contributions to the entanglement entropy. The form of some of thesenew contributions is similar to that found recently in an investigation of entanglemententropy in a free massive scalar field theory [1]. ontents
1. Introduction 12. Universality with a Boundary CFT 4
3. New Universal Terms with a Deformed Boundary Theory 16 m = 2 213.1.2 m = 3 223.1.3 m = 4 223.2 Spherical entangling surfaces 223.2.1 m = 2 233.2.2 m = 3 253.2.3 m = 4 26
4. Curved boundaries 275. PBH transformations with matter 316. Discussion 36A. Universality and Odd d
1. Introduction
Entanglement entropy was first considered in the context of the AdS/CFT correspon-dence by Ryu and Takayanagi [2, 3]. They provided a simple conjecture for calculatingholographic entanglement entropy. In the d -dimensional boundary field theory, the en-tanglement entropy between a spatial region V and its complement ¯ V is given by thefollowing expression in the ( d +1)-dimensional bulk spacetime: S ( V ) = 2 πℓ d − P ext v ∼ V [ A ( v )] (1.1)– 1 –here v ∼ V indicates that v is a bulk surface that is homologous to the boundary region V [4, 5]. In particular, the boundary of v matches the ‘entangling surface’ ∂V in theboundary geometry. The symbol ‘ext’ indicates that one should extremize the area overall such surfaces v . Implicitly, eq. (1.1) assumes that the bulk physics is described by(classical) Einstein gravity and we have adopted the convention: ℓ d − P = 8 πG N Hencewe may observe the similarity between this expression (1.1) and that for black holeentropy. While this proposal passes a variety of consistency tests, e.g., see [4, 3, 6],there is no general derivation of this holographic formula (1.1). However, a derivationhas recently been provided for the special case of a spherical entangling surface in [7].One aspect of entanglement entropy (EE), which eq. (1.1) reproduces, is that thisquantity diverges and so it is only well-defined with the introduction of a short-distancecut-off δ in the boundary field theory. Generically, the leading term obeys an ‘area law,’being proportional to A d − /δ d − where A d − denotes the area of the entangling surfacein the boundary theory. While the coefficient of this divergent contribution is sensitiveto the details of the UV regulator, universal data characterizing the underlying fieldtheory can be found in subleading contributions. In particular, in a conformal fieldtheory (CFT) with even d , one typically finds a logarthmic term log( R/δ ) where R issome macroscopic scale characterizing the size of the region V . The coefficient of thislogarithmic contribution is a certain linear combination of the central charges appearingin the trace anomaly of the CFT. The precise details of the linear combination willdepend on the geometry of the background spacetime and of the entangling surface[6, 3, 8, 9].A similar class of universal contributions were recently identified in a calculationwith a free massive scalar field [1]. In particular, considering the scalar field theory (ina ‘waveguide’ geometry) with an even number of spacetime dimensions, a logarithmiccontribution to the EE appears with the form S univ = − γ d A d − µ d − log( µδ ) , (1.2)where µ is the mass of the scalar and γ d is a numerical factor depending on the spacetimedimension. While this calculation is simplified by having a free field theory, onewould still characterize the mass term as being a relevant operator. That is, the massdominates the physics of the scalar theory at low energies but it leaves the leading UVproperties unchanged, e.g., the area law contribution to the EE would not be affected. We are using ‘area’ to denote the ( d –1)-dimensional volume of v . If eq. (1.1) is calculated in aMinkowski signature background, the extremal area is only a saddle point. However, if one first Wickrotates to Euclidean signature, the extremal surface will yield the minimal area. See [10] for related results in d = 4. To be precise, γ d = ( − ) d − [6(4 π ) d − Γ( d/ − [1]. – 2 –ith this perspective, one is led to examine a natural strong coupling analog ofthis calculation using holography. In particular, with the standard AdS/CFT dictionary[11], we can introduce a relevant deformation of the boundary field theory by turning ona (tachyonic) scalar field in the bulk. Asymptotically the bulk geometry still approachesan AdS spacetime reflecting the fact that the boundary theory still behaves as a CFTin the far UV. However, the details of the bulk geometry are changed by the back-reaction of the scalar field and so this naturally introduces the possibility that applyingeq. (1.1) in this geometry will yield new universal contributions of the form given above.In fact, our holographic calculations reveal a general class of logarithmic contributionswhich can appear with a relevant deformation. Schematically, they take the form of anintegral over the entangling surface ∂V : S univ = X i,n γ i ( d, n ) Z ∂V d d − σ √ H [ R, K ] ni µ d − − n log µδ , (1.3)where n < ( d − / µ is the mass scale appearing in the coupling of the new relevantoperator, H ab is the induced metric on ∂V and [ R, K ] ni denotes various combinationsof the curvatures with a combined dimension 2 n . Both the curvature of the back-ground geometry or the extrinsic curvature of the entangling surface may enter theseexpressions. The coefficients γ i depend on the details of the underlying field theoryand provide universal information characterizing this theory. For an even-dimensionalCFT, only the contributions with n = ( d − / µ wouldbe replaced by a scale in the geometry) and the coefficients γ i are proportional to thecentral charges of the CFT [3, 8]. Of course, for n = 0, there is a single contributionwhich matches that appearing in eq. (1.2).One feature which is typically implicit in calculations of entanglement entropy isthat the QFT is in its vacuum state. However, it is further assumed that the coeffi-cients appearing in these logarithmic contributions are ‘universal,’ including that theyare independent of the state of the underlying QFT. For example, calculating the en-tanglement entropy in a thermal bath would yield precisely the same result as in thevacuum state. While this feature is relatively ‘obvious’ and can be confirmed withexplicit calculations, a rigorous proof is lacking. The basic idea is that the propertiesof the state will not modify the UV properties of the theory. One of our results in thefollowing will be to demonstrate that the logarithmic contributions are in fact indepen-dent of the state in a holographic setting. Further our analysis makes clear that thecoefficient of these contributions is a local functional of the geometry of the backgroundin which the boundary theory resides and of the entangling surface.An overview of the paper is as follows: We begin in section 2 by demonstratingthat the coefficient of any logarithmic contribution is independent of the state of the– 3 –oundary theory. Our first discussion in this section considers the boundary theorybeing a pure CFT but in section 2.1, we show that this result extends to the case wherethe boundary theory is deformed by a relevant operator. In fact our conclusion is that ingeneral any UV divergent terms involve local functionals of the geometry and couplingsof the boundary theory. An additional feature which our analysis reveals is that newuniversal contributions to the entanglement entropy can arise from the presence ofthe relevant deformation. Hence in the subsequent sections, we present some explicitexamples where such logarithmic contributions are calculated. We begin in section 3by considering flat and spherical entangling surfaces with the boundary theory in aflat background. This exercise demonstrates that, as well as terms of the form (1.2),there are also universal contributions where the mass scale of the relevant deformationcombines with the curvature of the entangling surface, as shown in eq. (1.3). In section4, we investigate the latter contributions further by considering examples where thebackground in which the boundary theory resides is curved, e.g., R × S d − and R × H d − .In section 5, we extend the approach of [12] to properly identify the precise structureof these contributions in the leading case. We conclude with a discussion of our briefresults, in section 6. Finally, in appendix A, we present a holographic calculation whichexplicitly shows that the constant terms, which are often interpreted as universal, inodd dimensions do indeed depend on the state of the boundary theory.
2. Universality with a Boundary CFT
In this section, we establish that the logarithmic contribution to the entanglemententropy (EE) is independent of the state of the boundary field theory in the AdS/CFTcorrespondence. To begin, we consider the case where the boundary theory is purelya conformal field theory. We must then also choose the boundary dimension to beeven, since for a CFT, it is only in this case that a logarithmic contribution arises inthe EE. It has been shown that the coefficient of this term is related to the centralcharges appearing in the trace anomaly [13, 3, 8] — see also the discussion in [9].Implicitly, the calculations establishing this connection are made in the vacuum of thecorresponding CFT and so here we are establishing that, at least in a holographicframework, the result is independent of the state of the CFT. Our present observationcomes as a simple extension of the holographic calculations in [6]. There our holographiccalculations were able to reproduce the precise expression for the logarithmic term in However, our discussion here will consider both odd and even d because both are germane whena relevant deformation is introduced in the next subsection. We comment further on the case of odd d in appendix A. – 4 –he EE for a general entangling surface in a four-dimensional CFT, matching that whichwas originally determined in [8].Let us begin by denoting the spacetime dimension of the boundary theory as d andhence the dual gravity theory has d + 1 dimensions. In the AdS/CFT correspondence,the bulk geometry asymptotically approaches anti-de Sitter space for any generic stateof the boundary theory. This asymptotic geometry can then be described with theFefferman-Graham expansion, as follows [14] — see also [15]: ds = L dρ ρ + 1 ρ g ij ( x, ρ ) dx i dx j , (2.1)where L is the AdS curvature scale. The asymptotic boundary is approached with ρ → g ij ( x, ρ ) admits a series expansion in the (dimensionless) radial coordinate ρ g ij ( x, ρ ) = (0) g ij ( x i ) + ρ (1) g ij ( x i ) + ρ (2) g ij ( x i ) (2.2)+ · · · + ρ d/ ( d/ g ij ( x i ) + ρ d/ log ρ ( d/ f ij ( x i ) + O ( ρ d +12 ) . The leading term (0) g ij corresponds to the metric on which the boundary CFT resides. Asshown in eq. (2.2), the first few terms fall into a Taylor series expansion but this simpleform breaks down at order ρ d/ . In particular, for even d , a logarithmic term appearsat this order while for odd d , non-integer powers of ρ begin to make an appearance— no logarithmic term appears for odd d . With this choice of coordinate system, theexpectation value of the boundary stress-energy tensor becomes [15, 16] h T ij i = d ℓ d − P L ( d/ g ij + e X ij [ ( n ) g ] . (2.3)where e X ij denotes the contribution coming from the conformal anomaly. Hence thisterm vanishes for odd d , while for even d , it is determined by coefficients ( n ) g ij with n < d/ (0) g ij and ( d/ g ij can be regarded as theindependent boundary data needed to determine the bulk spacetime. As noted above, Our notation is essentially the same as that established in [6]. Explicitly then, our index conven-tions are as follows: Directions in the full (AdS) geometry are labeled with letters from the secondhalf of the Greek alphabet, i.e., µ, ν, ρ, · · · . Letters from the ‘second’ half of the Latin alphabet, i.e., i, j, k, · · · , correspond to directions in the background geometry of the boundary CFT. Frame indicesare denoted by a hat, i.e., ˆ ı, ˆ . Meanwhile, directions along the entangling surface in the boundary aredenoted with letters from the beginning of the Latin alphabet, i.e., a, b, c, · · · , and directions along thecorresponding bulk surface are denoted with letters from the beginning of the Greek alphabet, i.e., α, β, γ, · · · . – 5 –he first fixes the boundary metric while the second determines the boundary stresstensor. That is, ( d/ g ij ( x ) is related to the state of the boundary CFT. Further, wenote that the coefficients ( n ) g ij ( x ) with 0 < n < d/ (0) g ij . More precisely, by expanding the gravitational equations ofmotion near the boundary, one solves for each of these coefficients in terms of the lowerorder terms in the expansion (2.2) — see section 2.1 for more details. For example (aslong as d > (1) g ij = − L d − (cid:18) R ij [ (0) g ] − (0) g ij d − R [ (0) g ] (cid:19) , (2.4)where R ij and R are the Ricci tensor and Ricci scalar constructed with the bound-ary metric (0) g ij . An alternative approach was presented in [12]. There the authorsshowed that these coefficients are almost completely fixed by conformal symmetries atthe boundary. This method examines the effect of Penrose-Brown-Henneaux (PBH)transformations, the subgroup of bulk diffeomorphisms which generate Weyl trans-formations of the boundary metric. Consistency of the PBH transformations on theasymptotic expansion (2.2) essentially determines all of the coefficients up to order n < d/ ( n ) g ij ( x ) contains 2 n derivatives.As these coefficients in the asymptotic geometry (2.2) depend only on the boundarymetric (0) g ij and are completely independent of the state of the boundary CFT, we referto them as the ‘fixed boundary data.’ Hence, if the logarithmic contribution to the EEis independent of the state, we must demonstrate that in our holographic calculationsof the EE (for even d ), this contribution depends only on this fixed boundary data andis independent of any coefficients ( n ) g ij with n ≥ d/ i.e., the coefficient of the logarithmic contribution, depends on the geometryof the extremal surface v . Hence, as well as the bulk geometry (2.2), we must alsoconsider the embedding of the corresponding ( d − d + 1)-dimensional bulk geometry. This embedding may be described by X µ = X µ ( y a , τ ),where X µ = { x i , ρ } are the bulk coordinates and σ α = { y a , τ } are the coordinates onsurface m (with a = 1 , .., d −
2) – recall our conventions from footnote 5. The inducedmetric on the bulk surface is then given by h αβ = ∂ α X µ ∂ β X ν g µν [ X ] , (2.5) These calculations leave some small ambiguity that must still be fixed by the equations of motion. – 6 –here g µν denotes the full ( d + 1)-dimensional bulk metric.The calculations below are simplified somewhat if we fix reparameterizations of thecoordinates on v with the following gauge choices (as in [12]) τ = ρ and h aτ = 0 . (2.6)Now following [17], one finds that the remaining embedding functions X i ( y a , τ ) arethen described by the following series expansion for small ρ : X i ( y a , ρ ) = (0) X i ( y a ) + τ (1) X i ( y a ) + τ (2) X i ( y a ) (2.7)+ · · · + τ d/ ( d/ X i ( y a ) + τ d/ log τ ( d/ Y i ( y a ) + O ( τ d +12 ) . Essentially the form of this expansion matches that for the bulk metric in eq. (2.2). Thefirst term (0) X i ( y a ) describes the position of ∂v in the boundary of the asymptotically AdSspace. That is, this matches the position of the entangling surface ∂V in the boundarymetric (0) g ij ( x ). Here as in eq. (2.2), the simple Taylor series expansion appearing forthe first few terms breaks down at order ρ d/ . Again, for even d , a logarithmic termappears at this order while for odd d , non-integer powers of ρ begin to appear — nologarithmic term appears for odd d . This expansion (2.7) is constructed in detail bysolving the local equations of motion for X µ ( y a , τ ) which extremize the area of the bulksurface v [17] — see section 2.1 for more details. In solving these equations, the fullsurface is determined by independently fixing both (0) X i ( y a ) and ( d/ X i ( y a ). In particular,the latter data would be chosen to ensure that the surface v closes off smoothly in theinterior of the asymptotically AdS space. As the equations are solved iteratively orderby order in τ , the coefficients ( n ) X i ( y a ) with n < d/ (0) X i ( y a ) and (0) g ij . In particular then, these terms are independent of thestate of the boundary CFT. Hence in discussing the extremal surface m , we extend themeaning of the ‘fixed boundary data’ to include both ( n ) g ij and ( n ) X i ( y a ) with n < d/ k . In general, they found that the second set ofindependent coefficients entered the expansion of the embedding functions at order The gauge condition (2.6) is not the same as in [17], however, the general structure of the asymp-totic expansion (2.7) remains unaltered in both cases. – 7 – ( k +2) / . Hence it is only for k = d −
2, the case of present interest, that the form ofthe expansion (2.7) matches the metric expansion (2.2).Given the expansions of the bulk metric (2.2) and the embedding functions (2.7),the induced metric (2.5), compatible with the gauge choice (2.6), can also be expandedin the vicinity of the AdS boundary h ττ = L τ (cid:16) (1) h ττ τ + · · · (cid:17) , h ab = 1 τ (cid:18) (0) h ab + (1) h ab τ + · · · (cid:19) . (2.8)Note that (0) h ab = H ab , i.e., it is precisely the induced metric on the entangling surface ∂V as evaluated in the boundary CFT. A crucial feature emerging from this perturbativeconstruction is that the coefficients ( n ) h αβ depend only on the fixed boundary data for n < d/ i.e., they are completely determined by (0) X i ( y a ) and (0) g ij . Now in calculatingthe holographic EE (1.1), we must evaluate the area A ( v ) = Z v d d − σ √ h = Z v d d − y dτ L τ d/ q det (0) h (cid:20) (cid:18) (1) h ττ + (0) h ab (1) h ab (cid:19) τ · · · (cid:21) . (2.9)The integral over the radial direction τ extends down to an asymptotic regulator surfaceat τ min = ρ min = δ /L where δ is a short distance cut-off in the boundary theory. Weare interested in the appearance of a logarithmic contribution of the form log δ , hence wemust carry out the expansion in the bracketed expression to order τ d − , which producesthe term in the integral with an overall power 1 /τ . The explicit term appearing atthis order for general d would be quite complicated. However, for our purposes, itsuffices to know that this term will involve coefficients ( n ) h αβ with n ≤ ( d − /
2. Hencethis logarithmic contribution to the holographic EE is completely determined by thefixed boundary data. That is, we do not require the state dependent coefficient ( d/ g ij ,nor details of the shape of the extremal surface v beyond what is determined by theboundary geometry (0) g ij and (0) X i ( y a ). We also observe that, as expected, there is no termat the appropriate order in eq. (2.9) to produce a logarithmic contribution in the case ofodd d . Further, our analysis above shows that all of the divergent contributions to theholographic EE will only depend on this fixed boundary data, i.e., these contributionsare completely determined as local functionals of (0) g ij and (0) X i ( y a ). Motivated by the recent results in [1], we wish to consider how the holographic EEis modified by the introduction of a mass term in the boundary theory. For example,– 8 – scalar mass term would deform the boundary theory by an operator of dimension∆ = d − d −
1. Our analysis here will be more general and consider modifications ofholographic EE when the boundary theory is deformed by a general relevant (scalar)operator with ∆ < d . The dual of such a scalar operator will be a scalar field in thebulk. Hence our starting point is the following bulk action where we have Einsteingravity coupled to a scalar field I = 12 ℓ d − P Z d d +1 x √− G (cid:20) R −
12 ( ∂ Φ) − V (Φ) (cid:21) , (2.10)where V (Φ) = − d ( d − L + 12 m Φ + κ L Φ + O (Φ ) . (2.11)Now a relevant operator primarily affects the IR properties of the theory but has aninsignificant effect in the UV regime. In the present holographic framework then, thedual bulk geometry still approaches AdS space in the presence of the relevant operator.Hence we consider the background geometry which asymptotically approaches AdS d +1 in Graham-Fefferman coordinates [14], as in eq. (2.1), ds = L dρ ρ + 1 ρ g ij ( x, ρ ) dx i dx j . (2.12)Of course, with Φ = 0, the vacuum solution in the bulk will be precisely AdS d +1 , asdescribed in the previous section. In general, the boundary theory’s metric is stillgiven by g ij ( x, ρ = 0) = (0) g ij , however, as we will see below, the details of the small- ρ expansion will change with Φ = 0. If we turn on the scalar as a probe field in thisbackground, the scalar has two independent solutions asymptotically [11]Φ ≃ ρ ∆ − / φ (0) + ρ ∆ + / φ (∆ − d ) , (2.13)where ∆ ± = d ± r d m L . (2.14)The standard approach is to interprete the conformal dimension of the dual operatoras ∆ = ∆ + . Then, the leading coefficient of the more slowly decaying solution, φ (0) , isinterpreted as the coupling for the dual operator in the boundary theory, while φ (∆ − d ) yields the expectation value of this operator as [18] hO ( x ) i = (2∆ − d ) φ (∆ − d ) ( x ) . (2.15) In general, additional contributions may appear on the right-hand side involving φ ( n ) with n < ∆ − d . These terms are related to contact terms in correlation functions of O with itself and with thestress-energy tensor [15]. – 9 –ince we wish to consider a relevant operator in the boundary theory, eq. (2.14) requiresthat m < ρ → ≥ d/ < d . Hence even in the full nonlinear analysis of the equations ofmotion, Φ → ρ → ρ expansion of the scalar field is ρ α/ with α = ∆ − and 0 < α ≤ d . (2.16)Here, the upper limit on α is the BF bound while the lower limit is simple the require-ment that ∆ < d .Let us now turn to holography with the fully back-reacted solutions of the Einstein-scalar theory (2.10). This construction is discussed in some detail in [15] and we followtheir discussion below. The Einstein equations can be expressed as R µν = 12 ∂ µ Φ ∂ ν Φ + 1 d − G µν V (Φ) , (2.17)and the scalar wave equation is1 √− G ∂ µ (cid:16) √− GG µν ∂ ν Φ (cid:17) − δVδ Φ = 0 . (2.18)Now we write the scalar field as Φ( x, ρ ) = ρ α/ φ ( x, ρ ) (2.19)where we have extracted the leading asymptotic decay for this field. Combining this In the case of a marginal operator with ∆ = d , the boundary theory remains a CFT and so theresults regarding the holographic EE are unchanged from our previous discussion. – 10 –orm with the metric ansatz (2.12), the Einstein equations (2.17) yield [15] ρ (cid:0) g ′′ ij − g ′ ik g kℓ g ′ ℓj + g kℓ g ′ kℓ g ′ ij (cid:1) + L R ij [ g ] − ( d − g ′ ij − g kℓ g ′ kℓ g ij == ρ α (cid:18) L ∂ i φ ∂ j φ + g ij ( d − ρ (cid:16) m L φ + κ ρ α/ φ + O ( ρ α φ ) (cid:17)(cid:19) ∇ i (cid:0) g kℓ g ′ kℓ (cid:1) − ∇ k g ′ ki = ρ α L (cid:18) φ ′ ∂ i φ + α ρ φ ∂ i φ (cid:19) (2.20) g kℓ g ′′ kℓ − g ij g ′ jk g kℓ g ′ ℓ i = ρ α (cid:18) φ ′ + αρ φ φ ′ + α ρ φ + 14( d − ρ (cid:16) m L φ + κ ρ α/ φ + O ( ρ α φ ) (cid:17)(cid:19) where the primes denote differentiation with respect to ρ and ∇ i is the covariant deriva-tive constructed from the metric g ij ( x, ρ ). Further R ij [ g ] in the first line above denotesthe Ricci tensor calculated for the d -dimensional metric g ij ( x, ρ ), treating ρ as an extraparameter — in particular then, this is not just the boundary Ricci tensor calculatedwith (0) g ij . Meanwhile the scalar wave equation becomes0 = ρφ ′′ + (cid:18) α + 1 − d (cid:19) φ ′ + 12 ∂ ρ log( − g ) (cid:16) α φ + ρφ ′ (cid:17) + L (cid:3) g φ − κ ρ α − φ + O ( ρ α − φ ) . (2.21)Here we eliminated the leading term in this equation since vanishes for α = d − ∆or ∆, just as in the probe analysis leading to eq. (2.13). Further, we have defined (cid:3) g φ ≡ √− g ∂ i ( √− gg ij ∂ j φ ) where the full metric g ij ( x, ρ ) is again inserted in this waveoperator along the boundary directions.Now we are in a position to construct solutions with a small- ρ expansion near theasymptotic boundary. We should say that our objective here is primarily to understandthe form of these solutions. We will delay considering explicit solutions to the next twosections. Hence, to get a feeling for the expansion of g ij ( x, ρ ), we consider the equationsof motion (2.20) and (2.21) with g ij ( x, ρ ) ≃ (0) g ij ( x ) + ρ β ( β ) g ij ( x ) and φ ( x, ρ ) ≃ φ (0) ( x ) + ρ β ′ φ ( β ′ ) ( x ) , (2.22)where we assume both exponents β , β ′ are positive. We begin by examining the firstequation in eq. (2.20) to linear order in ( β ) g ij . First we find that there are two homo-geneous solutions, i.e., solving with all of the terms linear in g ′′ and g ′ . That is, wehave either β = 0 or β = d/ (0) g ij ( β ) g ij = 0. The first caseis simply deformation of the boundary metric (0) g ij where as the second is the secondindependent solution containing the state data about the stress-energy, as in eq. (2.3).– 11 –his structure matches precisely that found for the usual FG expansion (2.2) and de-pends only on the asymptotic geometry approaching AdS geometry. At this linearlevel, R ij ( (0) g ) introduces an inhomogeneous source term requiring β = 1. Similarly φ (0) introduces various source terms on the right-hand side. The leading source comes fromthe mass term which requires β = α , while the next source would be the cubic termwhich calls for β = 3 / α . Hence in the deformed background, the asymptotic expan-sion of g ij ( x, ρ ) involves terms with two powers of ρ , namely, integer powers ρ n as wellas powers ρ mα/ . To simplify the general expansion in a workable form, we consider thecase where α/ i.e., α/ N/M where N and M are relativelyprime. In this case, the general asymptotic expansion for the metric g ij ( x, ρ ) can bewritten as g ij ( x, ρ ) = N − X n =0 ρ n ( n ) g ij ( x ) + ∞ X m =2 ρ n + m α n + m α g ij ( x ) ! (2.23)+ ρ d/ N − X n =0 ρ n ( d n ) g ij ( x ) + ∞ X m =2 ρ n + m α d n + m α g ij ( x ) ! . If ρ d/ appears in the series in the first line, the expansion contains a logarithmic term g ij ( x, ρ ) = N − X n =0 ρ n ( n ) g ij ( x ) + ∞ X m =2 ρ n + m α n + m α g ij ( x ) ! (2.24)+ ρ d/ log ρ N − X n =0 ρ n ( d n ) f ij ( x ) + ∞ X m =2 ρ n + m α d n + m α f ij ( x ) ! . Of course, the latter expansion with the logarithmic contribution always arises if d iseven, as in the usual FG expansion (2.2). However, we note that this form can also arisein odd dimensions if the relevant operator has an appropriate dimension. For example,if d = 3, eq. (2.24) arises for α = 3 /m with m = 2 , , , .., which would correspond to∆ = 3( m − /m . In particular here, m = 3 yields ∆ = 2 which corresponds to thefermion mass term in d = 3.We might also note some simplifications that can occur in the above expansions.In particular, if the boundary curvature vanishes, the integer powers are not required, i.e., all of the coefficients with n > → − Φ, only (integer) powers of ρ α will appear, i.e., all of the coefficients with m being a odd integer vanish.Now using the test expansion (2.22), we can also examine the scalar wave equa-tion (2.21) to linear order in φ ( β ′ ) . Considering only the first two linear terms, we– 12 –nd two homogeneous solutions, namely β ′ = 0 and β ′ = d/ − α . The first case issimply a shift of the boundary coupling φ (0) , whereas the second power correspondsto the second independent solution. The overall power of this contribution is then ρ α + β ′ = ρ d − α = ρ ∆ + / . Hence we see again that this structure matches precisely thatfound with the probe analysis in eq. (2.13), which depends only on the fact that theasymptotic geometry approaches the AdS geometry. Further, as before, the second so-lution contains the expectation value of the corresponding boundary theory operator,as in eq. (2.15). Continuing with the linearized analysis, φ (0) introduces two inhomoge-neous source terms in eq. (2.21) from the derivative term (cid:3) g φ and from the higher ordercontributions from the scalar potential. The former requires β ′ = 1 while the latterrequires β ′ = α/
2. Hence in the full nonlinear solution, we see that the asymptoticexpansion of φ ( x, ρ ) also involves two powers of ρ , namely integer powers of ρ and ρ α/ separately, as in the metric expansion above. Given the expansions (2.23) and (2.24),we see the metric also feeds in source terms with both kinds of powers from the termwith ∂ ρ log( − g ). If we simplify the general expansion with the choice α/ N/M asabove, we findΦ( x, ρ ) = ρ α/ φ ( x, ρ ) = ρ α/ N − X n =0 ∞ X m =0 ρ n + m α φ ( n + m α ) ( x ) (2.25)+ ρ ( d − α ) / N − X n =0 ∞ X m =0 ρ n + m α φ ( d − α + n + m α ) ( x ) . In this case when d is even or when d is odd and α = ( d − n ) / ( m + 2) (subject to thecondition α > ρ , which gives rise to the following expansionΦ( x, ρ ) = ρ α/ φ ( x, ρ ) = ρ α/ N − X n =0 ∞ X m =0 ρ n + m α φ ( n + m α ) ( x ) (2.26)+ ρ ( d − α ) / log ρ N − X n =0 ∞ X m =0 ρ n + m α ψ ( d − α + n + m α ) ( x ) . Further the leading coefficient ψ ( d − α ) ( x ) can be related to matter conformal anomalies[15, 20].Now before leaving our discussion of the back-reacted bulk solution, we wish tocomment on the fixed boundary data. While the details of the small- ρ expansion in themetric (2.12) have changed, the second independent solution still arises at order ρ d/ with the coefficient ( d/ g ij . As before, this coefficient carries information about the state– 13 –f the boundary field theory through the relation in eq. (2.3). Similarly, the secondindependent solution appears in the expansion of the scalar field at order ρ ( d − α ) / , whichagain is determined by the state of the boundary theory through eq. (2.15). Hence wemay ask at what order the state data for the scalar, i.e., φ ( d − α ) , begins to contributeto the expansion of the metric. Examining the Einstein equation (2.20), we see theleading contribution will come from the mass term on the right-hand side with a crossterm φ (0) φ ( d − α ) . However, this contribution enters with a factor ρ d/ − and so we cansee that φ ( d − α ) will only effect the coefficients in the metric expansion ( n ) g ij with n ≥ d/ n < d/ φ ( n ) with n < d/ − α , since these coefficients are all independent ofthe state of the boundary theory and are completely fixed by the boundary metric (0) g ij and the coupling φ (0) .We note that the calculation of the holographic EE is a geometric one which relieson the details of the metric expansion (2.23) or (2.24), i.e., the scalar field does notdirectly enter into the extremal area (1.1). Hence, as in the previous section, we wouldlike to show that the logarithmic contribution to the holographic EE only dependson the fixed boundary data, i.e., ( n ) g ij with n < d/
2. Hence, we must next examinethe embedding functions X µ ( y a , τ ) to show that this universal contribution also onlydepends on the geometry of the entangling surface in the boundary and is independentof the details of the bulk surface, e.g., ensuring that it has a regular geometry.The embedding functions are determined by extremizing the area of the bulk surface v . It is a straightforward exercise to show that this leads to the following (local)equation of motion 1 √ h ∂ α (cid:16) √ hh αβ ∂ β X µ (cid:17) + h αβ Γ µνσ ∂ α X ν ∂ β X σ = 0 , (2.27)where the induced metric is given by eq. (2.5) and Γ µνσ denote the usual Christoffelsymbols constructed with the bulk metric g µν . As before, we make the gauge choicesgiven in eq. (2.6) and in this case, setting µ = ρ = τ in eq. (2.27) yields a spuriousconstraint, which is automatically satisfied upon solving the remaining equations for X i ( y a , τ ). Of course, the leading terms in a small- τ expansion of the latter are just theconstant terms describing the position of the entangling surface in the boundary, i.e., X i ( y a , τ ) ≃ (0) X i ( y a ). To determine the order at which a second independent solutionappears, we follow the analysis in [17]. We begin by assuming the equations (2.27) havebeen solved perturbatively to o ( τ s ), such that the coefficients ( s ) X i ( y a ) are to be solvedin terms of the previous terms. One can see that the leading contribution of these newcoefficients always comes from the terms in eq. (2.27) with two τ derivatives. If we– 14 –ubstitute in the leading form of the induced metric (2.8), this term is given by0 ≃ L τ d/ √ h (0) ∂ τ q (0) h τ − d ∂ τ (cid:18) τ s ( s ) X i ( y a ) (cid:19)! + 4 τ L g ik ∂ τ g kj ∂ τ (cid:18) τ s ( s ) X j ( y a ) (cid:19) + · · ·≃ s (cid:18) s − d (cid:19) τ s − ( s ) X i ( y a ) + · · · , (2.28)Now the latter result implies that ( s ) X i becomes undetermined for s = d/ e.g., to ensure that the extremal surface has a regular geometry.Note that this is precisely the same order at which this additional information entered inthe previous section (without the relevant deformation). Further, this is also preciselythe order at which the second independent set of coefficients appear in the small- τ expansion of the bulk metric. If we examine the full equations of motion (2.27) forthe embedding functions in more detail, we also find that, just as in the expansionsfor the metric and the scalar, there are two powers of τ (= ρ ) appear, namely, powersof τ and τ α/ . Further, it is straightforward to show that the small- τ expansion for X i ( y, τ ) takes an analogous form as that presented for the bulk metric in eq. (2.23)(or eq. (2.24), depending on the precise values of d and α ). Of course, the leadingcoefficients ( n ) X i with n < d/ (0) X i ( y a ), (0) g ij and φ (0) .Combining the asymptotic boundary expansions for the bulk metric and the em-bedding functions, one produces a similar expansion for the induced metric (2.5), e.g., h ab ( y, τ ) = 1 τ " N − X n =0 τ n ( n ) h ab ( y ) + ∞ X m =2 τ n + m α n + m α h ab ( y ) ! (2.29)+ τ d/ N − X n =0 τ n ( d n ) h ab ( y ) + ∞ X m =2 τ n + m α d n + m α h ab ( y ) ! . Of course, if τ d/ appears in the series in the first line above, then a logarithmic factorwill appear in the second line, as in eq. (2.24). Recall that the leading coefficient (0) h ab is the induced metric H ab on the entangling surface ∂V in the background for theboundary theory. The component h ττ ( y, τ ) has an analogous expansion with a pre-factor L / (4 τ ) and (0) h ττ = 1, as in eq. (2.8). For the present purposes, an essentialfeature of the induced metric is that all of the coefficients ( n ) h αβ depend only on the fixedboundary data for n < d/
2. That is, these leading coefficients are again completelydetermined by (0) X i ( y a ), (0) g ij and φ (0) . – 15 –ow turning to the holographic EE (1.1), we must evaluate the area of the extremalsurface. The area integral has the same basic structure as given in eq. (2.9) in theabsence of a relevant deformation. In particular, the leading expression provides afactor of τ − d/ and the radial integral ends at the regulator surface with τ min = δ /L where δ is the UV cut-off in the boundary theory. We are again primarily interested inthe contribution proportional to log δ and so we must expand the rest of the integrandto order τ d − . While the details of this expansion are now modified by the presenceof the relevant deformation, as before, it suffices to observe that a term at the desiredorder will only contain the coefficients ( n ) h αβ with n ≤ ( d − /
2. Hence this logarithmiccontribution to the holographic EE is completely determined by the fixed boundarydata. That is, this contribution is completely determined by (0) X i ( y a ), (0) g ij and φ (0) . Infact, the same result also applies for all of the divergent contributions to the holographicEE. While this conclusion has been unchanged by the introduction of a relevant operatorin the boundary theory, the appearance (or not) of a universal contribution in theholographic EE, proportional to log δ , depends very much on the details, i.e., on thedimension of the operator, as well as the spacetime dimension. In particular, in theexpansion of the integrand in eq. (2.9), we must identify a higher order term with τ n + m α = τ d − . Of course, as in the previous section, one finds such terms with m =0 for any even d . However, there can now be new terms for odd or even d if theoperator dimension of the relevant deformation is appropriate. For example, choosingan operator with ∆ = d + 1 yields α = ( d − / τ with n = 0 and m = 2. Similarly, for a scalar mass term with ∆ = d −
2, one finds α = d − ∆ = 2 and hence a logarithmic term appears in even dimensions with d ≥ One can compare this to the results of [1], where an analogous contribution (1.2) wasfound for a free massive scalar field. As a final note here, we observe that in certaininstances (with appropriate α and d ) the logarithmic term will be produced with both n and m nonvanishing. In the following sections, we present some explicit calculationsof these new universal contributions to the holographic EE.
3. New Universal Terms with a Deformed Boundary Theory
While our general discussion above indicated that a relevant deformation of the bound-ary theory may lead to new universal contributions in the holographic EE, we would Recall that m ≥ m = 0) because the stress tensor for the Einstein-scalar theory in the bulkis at least quadratic in the scalar field, e.g., all of the contributions on the right-hand side of eq. (2.20)are quadratic or higher order in the scalar. – 16 –ike to present some explicit examples where such logarithmic contributions appear. Tomake the problem of explicitly calculating the new universal terms simpler, we begin byconsidering here a background with a flat boundary metric. For convenience, we alsodepart from the conventions of the previous section by choosing a new radial coordinate z where ρ = z /L . We take the Einstein-scalar theory in eq. (2.10) with V (Φ) = − d ( d − L + 12 m Φ + κ L Φ , (3.1) i.e., we choose the potential to include only terms up to cubic in the scalar. Now witha flat boundary, our asymptotically AdS d +1 bulk metric becomes ds = L z (cid:0) dz + f ( z ) η ij dx i dx j (cid:1) , (3.2)and following eq. (2.19), we write the scalar profile as Φ( z ) = ( z/L ) α φ ( z ). Asymptoti-cally, as z → f ( z ) → φ ( z ) → φ (0) .Examining the Einstein equations of motion (2.17), there are two nontrivial equa-tions which may be written: d ( d − "(cid:18) f ′ f (cid:19) − z f ′ f − Φ ′ + ( mL ) z Φ + κ z Φ = 0 , (3.3)2( d − (cid:20) f ′′ − d − z f ′ + d − f ′ f (cid:21) + f (cid:18) Φ ′ + ( mL ) z Φ + κ z Φ (cid:19) = 0 . (3.4)However, the Bianchi identity ensures that these equations (combined with the scalarfield equation) are redundant. For simplicity we focus on eq. (3.3) in the following.The scalar field equation (2.18) reduces toΦ ′′ − d − z Φ ′ + d f ′ f Φ ′ − ( mL ) z Φ − κ z Φ = 0 . (3.5)Now constructing power series solutions for f and φ around z = 0, one finds f ( z ) = 1 + X k =2 a k (cid:0) φ (0) ( z/L ) α (cid:1) k , (3.6) φ ( z ) = φ (0) ( z/L ) α + X k =2 b k (cid:0) φ (0) ( z/L ) α (cid:1) k . Note that we are being somewhat cavalier in both of these expansions since neither in-cludes the second independent solution shown in, e.g., eqs. (2.23) and (2.25). However,as we showed in the previous section, none of the terms which have been neglected will– 17 –ontribute to the logarithmic contributions in the holographic EE. In comparing theabove expression for f ( z ) with eq. (2.23), we see that the terms involving n = 0 do notappear above. This simplification occurs because the boundary curvature vanishes.The first few coefficients in the above expansions are explicitly determined to be a = − d − , b = − κ α ( d − α ) , (3.7) a = 2 κ α ( d − d − α ) ,b = − d α d − d − α ) + κ α ( d − α )( d − α ) ,a = (3 d − α + 2 d d − ( d − α ) − κ (5 d − α )32 α ( d − d − α ) ( d − α ) ,b = κ d (17 d − α )72( d − d − α )( d − α )( d − α ) − κ (3 d − α )24 α ( d − α ) ( d − α )( d − α ) ,a = κ ((79 d − α − (19 d − dα − d )180 α ( d − ( d − α )( d − α )( d − α ) + κ (3 d − α )30 α ( d − d − α ) ( d − α )( d − α ) ,b = 3 d α d − ( d − α )( d − α ) − κ d (2655 α − αd + 163 d )576 α ( d − d − α ) ( d − α )( d − α )( d − α )+ κ (6 d − α )96 α ( d − α ) ( d − α )( d − α )( d − α ) . Here, we have used eqs. (3.3) and (3.5) to determine these coefficients. As an extracheck, we also explicitly checked that the above coefficients also solve eq. (3.4) to order (cid:0) φ (0) ( z/L ) α (cid:1) . Note that if we set κ = 0, the only nonvanishing coefficients are a k witheven k and b k with odd k .Recall that the calculation of the holographic EE (1.1) is purely a geometric oneand the scalar field only effects the result through its back-reaction on the bulk metric.Hence to evaluate eq. (1.1), we need only focus on the expansion of the metric, i.e., the expansion of f ( z ) in powers of z α , as seen in eq. (3.6). As shown in section 2.1, theuniversal part of holographic EE involves only the terms in the expansion up to thepower just proceeding z d ( ≃ ρ d/ ). Hence the expansion (3.6) of f ( z ) must be carriedout to a maximum value of k : d − α ≤ α k max < d . (3.8)Recall from eq. (2.16), we also have 0 ≤ α ≤ d/
2. Combining this inequality (3.8)with the above expansion (3.6), we see that k max = 2 , , d/ ≤ α ≤ d/ , d/ ≤ α ≤ d/ / ≤ α ≤ d/ d/ ≤ α ≤ d/
5, respectively. In general, a given k max is sufficientfor d/ (1 + k max ) ≤ α ≤ d/k max . To proceed further, we choose explicit values of d and∆, as well as the geometry of the entangling surface. In particular for the latter, weconsider 1) two flat planes bounding a slab and 2) a spherical surface S d − .Before proceeding with explicit calculations, we address a question about interpret-ing the results in terms of the boundary theory. Note that with the present conventions, φ (0) is a dimensionless parameter. However, following the standard AdS/CFT dictio-nary, we wish to relate this parameter in terms of the coupling to an operator withconformal dimension ∆ in the boundary theory. As such, this coupling should be di-mensionful defining some mass scale with φ (0) ∼ µ d − ∆ . The question is then how tomake this relation more precise, e.g., what scale enters on the bulk side of this equa-tion? Of course, in the AdS/CFT correspondence, the natural scale emerging from thebulk theory is simply the AdS curvature scale yielding φ (0) L d − ∆ = λ µ d − ∆ . (3.9)Here it is convenient to introduce a dimensionless parameter λ , which would controlthe strength of the deformation in the boundary theory. Note that given the presentframework, we can not provide a more specific relation than eq. (3.9) above. Forexample, if we consider a mass deformation like m φ in the boundary field theory,we could always redefine the operator by numerical factors, e.g., the dual operatorcould equally well be φ or φ / √ πφ and then accordingly the coupling wouldbe m or 2 m or m / ( √ π ). This example illustrates that distinguishing the operatorfrom the coupling part will not be well-defined without some additional informationabout the boundary theory. In fact, in certain cases, the required information may beprovided by supersymmetry and knowing more details of the duality between the bulkand boundary theories. One such example would be N = 2 ∗ theories [21], where moreprecise results can be obtained [22]. In this case, we introduce two flat parallel planes as the entangling surface. Hencesubsystem of interest in the boundary theory is the following slab: V F = { ≤ x ≤ ℓ, t = 0 } . The holographic EE has been calculated for this geometry in the case wherethe boundary theory is simply a d -dimensional CFT [2, 3]: S CFT ( V F ) = 4 πd − L d − ℓ d − P (cid:20) R d − δ d − − γ d R d − ℓ d − (cid:21) , (3.10) Upon converting our results below to parameters of the boundary theory, the power of λ keepstrack of the number factors of φ (0) appearing in the holographic calculation. – 19 –here γ d is a numerical factor: γ d = (cid:16) √ π Γ (cid:0) d d − (cid:1) / Γ (cid:0) d − (cid:1)(cid:17) d − . Here R is a regu-lator length along the x , , ··· ,d − directions, which was introduced so that the entanglingsurfaces have a finite area, i.e., R d − . Hence for a CFT in this geometry, no logarithmiccontribution appears in the EE for either even or odd d . Note that the pre-factor in theabove expression can be interpreted as a central charge in the boundary theory, e.g., the leading singularity in the two-point of the stress tensor is controlled by the centralcharge, i.e., C T ≃ L d − /ℓ d − P .Returning to the holographic EE in the presence of a relevant deformation, wedescribe the bulk surface v with the profile z = z ( x ) with the boundary conditions z ( x = 0) = 0 = z ( x = ℓ ). The induced metric h αβ on this surface embedded into thebackground (3.2) is given by h αβ dx α dx β = L z " ( f ( z ) + ˙ z ) ( dx ) + f ( z ) d − X i =2 ( dx i ) , (3.11)where ‘dot’ denotes a derivative with respect to x . Evaluating the area of v in thebulk then yields A ( v ) = Z d − Y i =1 dx i p det h αβ = R d − L d − Z ℓ dx f d/ − z d − p f + ˙ z . (3.12)We treat this expression as an action for z ( x ) to determine the profile which extremizesthis area. As the action contains no explicit x dependence, the conjugate ‘energy’ isconserved. This conserved energy functional then yields the following equation:˙ z = (cid:16) z ∗ z (cid:17) d − f d f d − ∗ − f , (3.13)where z ∗ corresponds to the (maximum) value of z where ˙ z = 0 and f ∗ = f ( z ∗ ). Fromthe inversion symmetry, x → ℓ − x , we must have ˙ z = 0 at x = ℓ/
2. To determine z ∗ , we can integrate the above equation ℓ Z z ∗ dz (cid:16) zz ∗ (cid:17) d − (cid:18) f d − f d − ∗ − (cid:16) zz ∗ (cid:17) d − (cid:19) − / f − / . (3.14)With these results, we can evaluate the holographic EE as follows S ( V F ) = 2 πℓ d − P A ( v ) = 4 π L d − ℓ d − P R d − Z z ∗ δ dz f d/ − z d − (cid:20) − (cid:16) f ∗ z f z ∗ (cid:17) d − (cid:21) , (3.15)where we have introduced the UV regulator surface at z = δ . We are interested inextracting a universal (logarithmic) contribution from the above expression, in the limit– 20 – →
0. Therefore we expand the integrand in powers of z and evaluate only the termwith 1 /z . In fact, the expression within the square brackets can be set to 1, since thehigher order terms which it contributes in this expansion begin at z d − . Therefore thedesired universal coefficient is independent of z ∗ . In the present notation, z ∗ representsthe undetermined data, appearing at higher order in the embedding functions, whichis specified to produce a smooth surface v in the bulk. Hence, as discussed aroundeq. (2.28), this data will not contribute to the universal terms in the holographic EE.Further, this observation allows us to consider the limit ℓ → ∞ in which case z ∗ → ∞ and the expression inside the square brackets above simply reduces to 1. In this limit, weare simply calculating the entanglement entropy upon dividing the boundary theoryinto two (semi-infinite) regions with a single wall at x = 0. In the case where theboundary theory is conformal, this limit leaves only the regulator dependent term, asshown in eq. (3.10). However, the limit leaves a more interesting result in the presentcase because the relevant deformation in the boundary theory has introduced a finitecorrelation length, ξ = 1 /µ . In particular, as we now show explicitly, the result caninclude a universal logarithmic contribution, of the form found in [1].As noted above, a logarithmic contribution will only appear from a term in thesmall- z expansion of the factor f d/ − in eq. (3.15). Further, we note that given theform (3.6) of f , this expansion only produce powers z mα with m ≥
2. Hence to producea 1 /z term in the integrand, we must have α = ( d − /m . Note that all such valuesappear in the allowed range given in eq. (2.16) and the conformal dimension of the dualoperator would be ∆ = d − d − m . We consider explicit examples for specific values of m below. m = 2In this case, we have α = ( d − / f . Examining eq. (3.15), we find S ( V F ) = π d − d − L d − ℓ d − P λ R d − µ d − log µδ + · · · (3.16)which applies for any odd or even d ≥
3. Above, we have used eq. (3.9) to write( φ (0) ) /L d − = λ µ d − . We have also introduced a factor of µ to make the argumentof the logarithm dimensionless, since it is the only natural scale to appear there. Weare implicitly assuming an operator arises with a specific conformal dimension which isdependent on d . However, we might note that for d = 4, ∆ = 3 which corresponds toa fermion mass term while for d = 6, ∆ = 4 which corresponds to a scalar mass term.– 21 – .1.2 m = 3In this case, α = ( d − / k max = 3. Eq. (3.15)then yields S ( V F ) = − πκ d − L d − ℓ d − P λ R d − µ d − log µδ + · · · (3.17)using ( φ (0) ) /L d − = λ µ d − . This result again applies for any odd or even d ≥
3. Notethat this contribution vanishes for κ = 0, e.g., for a free bulk scalar. m = 4With m = 4, α = ( d − / k max = 4. Then fromeq. (3.15), we obtain S ( V F ) = (cid:20) d + 34)( d − d + 6) κ − (3 d + 8)( d − d − (cid:21) π ( d − L d − ℓ d − P λ R d − µ d − log µδ + · · · (3.18)using ( φ (0) ) /L d − = λ µ d − . Again, we are implicitly assuming a specific operator di-mension which is dependent on d but with this assumption, the corresponding universalcontribution will appear for any odd or even d ≥ In this case, we wish to calculate the EE across a spherical surface in the boundarytheory. If we define the radial coordinate as usual, i.e., r = P i ( x i ) , in the flatboundary geometry, then the relevant subsystem is the ball: V S = { r ≤ R, t = 0 } .Again, the holographic EE has been calculated in this case with a conformal boundarytheory and a logarithmic contribution arises for even d [2, 3]: S CFT ( V S ) = ( − ) d − π d/ Γ( d/ L d − ℓ d − P log (2 R/δ ) + · · · . (3.19)In fact, this result can be calculated for any CFT without any reference to holographyand it is known that the pre-factor is precisely ( − ) d − A [8, 23, 9, 7] where the A is thecentral charge appearing in the A -type trace anomaly [24]. This contribution (3.19)will also appear in the calculation of the holographic EE when the boundary theoryis deformed by a relevant operator. However, in the following, we will focus on newcontributions related to the relevant deformation.To begin, we introduce polar coordinates P i ( dx i ) = dr + r d Ω d − for the bound-ary directions in the bulk metric (3.2). We describe the bulk surface v with a profile– 22 – = r ( z ) with the boundary condition r ( z = 0) = R . Then induced metric on v is givenby h αβ dx α dx β = L z (cid:2) ( f ( z ) r ′ + 1) dz + f ( z ) r d Ω d − (cid:3) , (3.20)where the ‘prime’ denotes a derivative with respect to z . The desired profile is chosento minimize the area A ( v ) = Z dz d Ω d − p det h αβ = L d − Ω d − Z δ dz f d/ − r d − z d − p f r ′ + 1 , (3.21)where Ω d − denotes the area of a ( d − i.e., Ω d − = 2 π d − / Γ (cid:0) d − (cid:1) .As before, we introduce a UV regulator surface at z = δ .In a pure AdS background, i.e., f ( z ) = 1, the profile which extremizes the area(3.21) has a simple form [2, 3] r ( z ) = √ R − z ≡ r . (3.22)Unfortunately, we could not find a closed form solution in the background with ageneric relevant deformation. Hence to extract the universal contribution, we canproceed by solving the corresponding Euler-Lagrange equation order by order in z andthen substitute the results back into the area functional (3.21). However, to leadingorder in this expansion f ( z ) = 1, for which r = r ( z ) is an exact solution. Hence itwill be convenient to organize our calculations by expanding around this profile, i.e., to evaluate corrections, δr = r − r , induced by the higher order terms in f ( z ).In the discussion towards the end of section 2.1, we found that the small- τ expansionof the extremal area produced a series involving powers τ n + m α . In the present notationthen, we expect the small- z expansion to produce terms with powers z n + mα . Further,from eq. (3.21), we see that the leading term in the integrand begins with 1 /z d − andso to produce a logarithmic contribution the expansion must contain a term where2 n + mα = d −
2. In fact, for even d , one finds a term where m = 0 and n =( d − / m is nonvanishing — as usual,this requires m ≥
2. Hence let us consider some explicit examples for specific values of m . m = 2In this case, we only keep the first correction k = 2 in the expansion of f , givenin eq. (3.6). Note then that the cubic, as well as any higher order interactions in– 23 –he potential of the bulk scalar play no role. Expanding eq. (3.21) to linear order in δf = f − δr = r − r yields A ( v ) = L d − Ω d − Z δ dz r d − z d − q r ′ + 1 (cid:18) d − d − r ′ r ′ + 1) δf + . . . (cid:19) (3.23)Above the term linear in δr vanishes, as it must since it is proportional to the equationsof motion for r . Focusing our attention on the δf term above, we substitute the leadingterm from eq. (3.6), as well as a small- z expansion of r , which combine to yield A ( v ) ≃ − d − d −
1) Ω d − L d − R d − Z δ dzz d − (cid:18) φ (0) z α L α (cid:19) (3.24) × (cid:16) − ( d − d − d − z R + ( d − d − d − d − z R + . . . (cid:17) Now if α = ( d − / S ( V S ) = π d − d − L d − ℓ d − P λ Ω d − R d − µ d − log µδ + · · · , (3.25)where we have used eq. (3.9) to write ( φ (0) ) /L d − = λ µ d − . This result (3.25) isessentially the same as that in eq. (3.16). In our holographic construction, the similarityof the results reflects the fact that to leading order r ( z ) ≃ R is constant and there isno distinction between a flat or a spherical entangling surface. Comparing eqs. (3.16)and (3.25), it appears this universal contribution can be written in the general form: S univ = π d − d − L d − ℓ d − P λ A d − µ d − log µδ , (3.26)where A d − is the area of the entangling surface. Again this logarithmic term willarise for any odd or even d ≥
3, for a relevant deformation with conformal dimension∆ = d + 1.Given eq. (3.24), we can also begin to consider contributions arising from termsin the expansion where n is also nonvanishing. If we consider the second term inthe parenthesis in eq. (3.24), we see a new logarithmic contribution will appear if α = ( d − /
2. In this case, S ( V S ) = − π d − L d − ℓ d − P λ Ω d − ( R µ ) d − log µδ + · · · (3.27)where we use ( φ (0) ) /L d − = µ d − , as implied by the present choice of α . Of course, thisresult only applies for d ≥
5. As should be evident from our construction above, the– 24 –erms with nonvanishing n appear in the expansion of the area integrand because of thecurvature of the sphere. That is, the background geometry has vanishing curvature,both the intrinsic and extrinsic curvatures of the entangling surface are non-vanishinghere. For example, the Ricci scalar of the intrinsic geometry on the entangling surface S d − is given by R = ( d − d − /R . This suggests that we might express the resultin eq. (3.27) as an integral over the sphere (contributing a factor of Ω d − R d − ) butthe integrand would be µ d − multiplying some appropriate combination of curvatures(contributing a factor of 1 /R ). However, given the large amount of symmetry in thepresent geometry, it is not possible to precisely fix that latter curvature expression. Wecontinue to investigate this question in sections 4 and 5.Of course, it is also possible to continue with examining higher order terms inthe expansion in eq. (3.24). This would in turn lead to logarithmic contributionsproportional to higher powers of curvature. Schematically, these terms would take theform S univ ≃ L d − ℓ d − P λ Ω d − ( R µ ) d − − n log µδ , (3.28)for α = ( d − − n ) /
2. Following the discussion above, it appears that these contribu-tions take the form of an integral over the entangling surface with a factor of µ d − − n multiplying some combination of curvatures contributing a factor of 1 /R n . m = 3In this case, we are focusing on new contributions which might come from the k = 3term in the expansion of f , given in eq. (3.6). Again we consider the linear expansiongiven in eq. (3.23) but substitute the k = 3 term for δf . A new logarithmic contributionarises when we assume that α = ( d − /
3. Of course, this is the same exponent thatappeared in section 3.1.2 and the contribution identified here has essentially the sameform as in eq. (3.17). We combine these results to write a general expression, S univ = − πκ d − L d − ℓ d − P λ A d − µ d − log µδ , (3.29)where A d − is again the area of the entangling surface. Such a logarithmic term gener-ically apppears for any odd or even d ≥ ( d + 1). Comparing eqs. (3.26) and (3.29), we see that thesetwo expressions have essentially the same structure, however, the details of the overallfactors differ. In particular, the present contribution depends on the cubic coupling inthe potential for the bulk scalar. Hence it will vanish for a free scalar field or moregenerally where the bulk theory is symmetric under Φ → − Φ.– 25 –s above, we can also consider higher order terms in the expansion with nonvanish-ing n which introduce additional factors of the curvature of the sphere ( i.e., factors of1 /R n ) in the logarithmic contributions. Schematically these terms again take a formvery similar to that found in the previous analysis. In particular, for α = ( d − − n ) / S univ ≃ κ L d − ℓ d − P λ Ω d − ( R µ ) d − − n log µδ , (3.30)similar to those in eq. (3.28). We might also note that in particular cases the universalterm receives contributions from more than one of the expressions outlined above. Forexample, consider the special case where α = 2 and d = 8 — note that this correspondsto ∆ = 6, which is the dimension of a scalar mass term in eight dimensions. With thischoice of parameters, we satisfy both α = ( d − / α = ( d − / S ( V S ) = − π L ℓ P Ω R (cid:20) κ λ µ + λ µ R (cid:21) log µδ + · · · . (3.31) m = 4In this case, we extend the expansion (3.6) of f to the order k = 4 and hence forconsistency, we must also expand the area (3.21) to quadratic order in δf = f −
1. Infact, we extend the latter expansion to quadratic order in both δf and δr = r − r toproduce A ( v ) = L d − Ω d − Z a dz r d − z d − q r ′ + 1 (cid:16) d − d − r ′ r ′ + 1) δf + g ( z ) δf (3.32)+ g ( z ) δf δr + g ( z ) δr + g ( z )( δr ′ ) + g ( z ) δf δr ′ + g ( z ) δrδr ′ + · · · (cid:17) , with g ( z ) = ( d − d −
4) + 2( d − d − r ′ + ( d − d − r ′ r ′ + 1) ,g ( z ) = ( d − d − r ′ + ( d − r ( r ′ + 1) ,g ( z ) = ( d − d − r , g ( z ) = 12( r ′ + 1) ,g ( z ) = ( d − r ′ + dr ′ r ′ + 1) , g ( z ) = ( d − r ′ r ( r ′ + 1) . (3.33)– 26 –ext we must solve to the extremal profile by varying the above ‘action’ with respectto δr . The solution of the resulting equation of motion must also satisfy the boundarycondition δr ( z = 0) = 0. To leading order in z , we find δr = d ( α −
1) + 28( d − α )( d − − α ) (cid:18) φ (0) z α L α (cid:19) z R + · · · . (3.34)Hence we have δr ∼ z δf . Therefore if we choose α = ( d − / δf, δf in eq. (3.32) contribute to the logarithmic divergence. Assuming further thatneither ( d − / d − / S univ = (cid:20) (3 d + 34)( d − d + 6) κ − (3 d + 8)( d − d − (cid:21) π ( d − L d − ℓ d − P λ A d − µ d − log µδ , (3.35)where as before A d − is the area of the entangling surface. Such a logarithmic termappears for any odd or even d ≥ d + 2) / d − / d − / δf in eq. (3.32). These terms are associatedwith the effect of intrinsic curvature of the sphere and in general will be of the formgiven by eqs. (3.28) and (3.30). Furthermore, by suitably changing the value of α , onecan also consider possible scenarios where terms involving δr start contributing to theuniversal divergence.As a specific example, let us consider the case of a ten-dimensional CFT with α = 2,which corresponds to the deformation of the CFT with a scalar mass term. Then thelogarithmic divergence is given by the expression in eq. (3.35) supplemented with δS ( V S ) = π L ℓ P R Ω (cid:20) λ µ R + κλ µ R (cid:21) log µδ + . . . (3.36)We have used eqs. (3.6), (3.7) and (3.23) to evaluate this term.
4. Curved boundaries
In section 3.2, we examined the holographic EE for a spherical entangling surface. Ourcalculations there began to illustrate an interesting interplay in the coefficient of theuniversal contributions between the curvature of the entangling surface and the massscale introduced by the relevant deformation. In particular, our results suggest thatvarious new universal contributions to the holographic EE appear where the coefficientis given by an integral over the entangling surface with a factor of µ d − − n multiplying– 27 –ome combination of curvatures contributing a factor of 1 /R n . However, with the re-sults of the previous section alone, the details of these contributions remain incomplete.That is, the precise form of the appropriate curvature factor remains unclear. Here weexamine these issues further by calculating the holographic EE for various entanglingsurfaces when the background in which the boundary theory resides is also curved.In particular, we consider the boundary theory on certain simple backgrounds ofthe form R × Σ k where Σ k is a maximally symmetric space, where k ∈ {± , } indicatesthe sign of the curvature. That is, Σ + = S d − , Σ = R d − and Σ − = H d − . Further, weintroduce R as the background curvature scale so that the Ricci scalar takes the form R [Σ k ] = k ( d − d − R . (4.1)Of course, R can be scaled away in the case of k = 0 — the simplest choice is to set R = L in this case. The corresponding bulk metric can be written as ds = L z (cid:0) dz − f t ( z ) dt + R f k ( z ) d Σ k (cid:1) , (4.2)where d Σ k = dθ + F k ( θ ) d Ω d − , F k = sin θ , k = 1sinh θ , k = − θ , k = 0 (4.3)and d Ω d − is the metric on a ( d − f t,k ( z ) are given by: f t = (cid:18) k z R (cid:19) ≡ f t, , f k = (cid:18) − k z R (cid:19) ≡ f k, . (4.4)We wish to consider the Einstein-scalar theory (2.10) with the cubic potential (3.1),as in the previous section. The Einstein equations (2.17) now yield three nontrivialcomponents but only two of these are independent. We chose to consider the followingtwo equations:( d − d − (cid:20) (cid:18) f ′ k f k (cid:19) − z (cid:18) f ′ k f k (cid:19) (cid:21) +( d − f ′ k f ′ t f k f t − d − f ′ t zf t − R [Σ k ] f k − Φ ′ + ( mL ) z Φ + κ z Φ = 02( d − (cid:20) f ′′ k − d − z f ′ k + ( d − f ′ k f k (cid:21) − R [Σ k ] (4.5)+ f k (cid:18) Φ ′ + ( mL ) z Φ + κ z Φ (cid:19) = 0– 28 –here R [Σ k ] is the Ricci scalar (4.1) of the boundary geometry. The scalar field equation(2.18) becomesΦ ′′ − d − z Φ ′ + Φ ′ (cid:18) ( d − f ′ k f k + f ′ t f t (cid:19) − ( mL ) z Φ − κ z Φ = 0 . (4.6)In order for the bulk metric (4.2) to be an asymptotically AdS solution, f k and f t must approach a constant ( i.e.,
1) at the boundary z →
0. Substituting the asymptoticform Φ ∼ z α into the scalar equation again yields the expected indicial equation, α ( α − d ) = ( mL ) , which has the two solutions ∆ ± given in eq. (2.14). As before, weintroduce a profile for Φ beginning with z α , where α = ∆ − = d/ − p d / mL ) , todescribe a dual operator of dimension ∆ + = d − α . The back-reaction on the metric, i.e., in f k,t , again begins at order z α . However, the boundary curvature now alsoappears as an explicit source in the Einstein equations (4.5) and its effect begins toappear at order z . The asymptotic expansion of the metric thus generally take theform given in eq. (2.23).Now we would like to compute the EE in the case where the entangling surface isan S d − at θ = θ in the metric (4.3) for the spatial geometry Σ k . Hence we specifythe bulk surface with a profile θ ( z ) satisfying the boundary condition θ ( z = 0) = θ .This calculation then requires a generalization of eq. (3.21), S = 2 πℓ d − P L d − R d − Ω d − Z δ dz f d/ − k F d − k z d − p f k R θ ′ ( z ) . (4.7)In a pure AdS background (4.4), we can find an exact solution for θ ( z ): θ ( z ) ≡ θ k, ( z ) = cos − (cos θ (4 R + z ) / (4 R − z )) , k = 1cosh − (cosh θ (4 R − z ) / (4 R + z )) , k = − p θ − z /R , k = 0 (4.8)We were able to find these solutions because these profiles all specify essentially thesame surface in different coordinate systems of the AdS geometry. Following ref. [7],this surface corresponds to the bifurcation surface of a topological AdS black hole.Now we follow the same procedure as in section 3.2 expanding around the pureAdS solutions. That is, we expand eq. (4.7) in powers of δf and δθ , which are definedas δf = f k − f k, , δθ = θ − θ k, . (4.9)Note that f t ( z ) does not appear in our integral (4.7) and so we need not considerperturbations of this metric function. To obtain the leading contribution in φ (0) , we– 29 –xpand eq. (4.7) to leading order in δf and δθ , which gives δS ≃ π L d − ℓ d − P R d − Ω d − Z δ dz " f d/ − k, F d − k ( θ k, ) z d − q R f k, θ ′ k, × d − f k, + θ ′ k, R f k, θ ′ k, ! δf ≃ π ( d − L d − ℓ d − P R d − Ω d − F k ( θ ) d − Z δ dzz d − δf × (cid:20) −
12 ( d − (cid:18) d − d − c k + k (cid:19) z R (cid:21) (4.10)where c k = cot θ , k = 1coth θ , k = − θ − , k = 0 . (4.11)Note that the term linear in δθ vanishes in eq. (4.10) by the equations of motion (forthe extremal profile in AdS space).Following the discussion in section 2, the correction to the metric δf may be writtenas δf = N − X n =0 ∞ X m =2 a ( m,n ) (cid:0) φ (0) ( z/L ) α (cid:1) m ( z/R ) n , (4.12)where we have assumed the exponent α has the form α/ N/M , as in the previousanalysis. Our convention to normalize the factors of z n with powers of R , rather than L , is convenient in the following but it is also a natural choice because R [Σ k ] appears asa source in the Einstein equations 4.5. The leading coefficient a (2 , in this expansion isunaffected by the boundary curvature and takes precisely the same value as in eq. (3.7), i.e., a (2 , = a = − / [4( d − δS arising when α = ( d − /
2, asappeared in sections 3.1.1 and 3.2.1. In this case, the leading contribution in δf is oforder z d − and the new logarithmic term is identical to that in eq. (3.26). In particularthen, this result is unaffected by the background curvature.Next we turn to α = ( d − /
2, as was considered in sections 3.1.2 and 3.2.2. In thiscase, δf begins at z d − but must be expanded up to z d − to identify the logarithmiccontribution to δS . The equations of motion give δf = ( φ (0) ) (cid:16) zL (cid:17) α (cid:20) − d −
1) + k ( d − d − d + 8)32( d − d − z R (cid:21) + · · · . (4.13)– 30 –he logarithmic contribution in eq. (4.10) then becomes S univ = π ( d − L d − ℓ d − P R d − Ω d − F k ( θ ) d − λ µ d − log µδ × (cid:20)
12 ( d − (cid:18) d − d − c k + k (cid:19) a (2 , − a (2 , (cid:21) = − π L d − ℓ d − P λ µ d − log µδ Z S d − d d − σ √ H (4.14) × (cid:20) ( d − d − ( K ˆ θ aa ) + ( d − d − d + 4)32( d − ( d − R [Σ k ] (cid:21) . In the second expression, we have tentatively expressed the result as an integral over theentangling surface, to illustrate the kind of general expression that we anticipate. Weare denoting the induced metric on this boundary surface as H ab . Note that implicitlythe result contains two curvature scales, 1 /R and c k /R and hence the integrandincludes two independent curvature terms. The last term involves the Ricci scalar (4.1)of the background geometry in which the boundary theory resides. The first terminvolves the extrinsic curvature of the entangling surface which is given by K ˆ θab = − t ia t jb ∇ i n ˆ θj = − c k R (0) h ab , (4.15)where n ˆ ıj and t ia are respectively the normal and tangent vectors, to the entanglingsurface ∂V — see [6] for further details and a full discussion of our conventions. Notethat in principle, there is also an extrinsic curvature associated with the normal vectorin the time direction however K ˆ tab = 0 in the present case.Hence our present calculation demonstrates that S univ takes a form slightly morecomplicated than anticipated in the discussion in section 3.2.1. In particular, thereare two independent curvature contributions, whereas we could only detect one inour calculations in the previous section. We should note however that the curvatureswhich we have written in eq. (4.14) are only representative. For example, we easilycould replace ( K ˆ θ aa ) by ( d − K ˆ θ ab K ˆ θ ba . Alternatively we could use the fact that theintrinsic curvature of our entangling surface has R ∝ (1 /R + c k /R ). Of course,when we set k = 0 in eq. (4.14), the result agrees with this previous calculation in aflat background. While they are informative, unfortunately these simple examples arestill too symmetric to give us enough insight to properly fix the covariant expressionthat describes this universal term for a general entangling surface.
5. PBH transformations with matter
In this section, we revisit the powerful approach developed in [12] to get a more precise– 31 –nderstanding of the new universal contributions to the holographic EE. Here oneis able to determine essentially all of the fixed boundary data by examining theirbehaviour under PBH transformations, the subgroup of bulk diffeomorphisms whichgenerate Weyl transformations in the boundary. In [12] however only pure gravitytheories in the bulk are considered and so we must extend their analysis to include abulk scalar. An essential feature of our analysis is that we must not just consider φ (0) to be a coupling constant in the boundary theory, rather we must elevate it to a field.That is, we consider φ (0) ( x ) to take full advantage of this approach. Just as in the puregravity case, these calculations leave some undetermined constants that must be fixedby the equations of motion. While there are no immediate obstacles to performing ageneral analysis, in the following, we only work out a specific example which includesa scalar field in the bulk to illustrate the general approach.In particular, we will focus our attention on α = ( d − / g ij ( φ (0) ) = g ij + ∆ g ij ( φ (0) ) , (5.1)where g ij is the asymptotically AdS solution without the relevant deformation turnedon, i.e., before the back-reaction of the scalar field is considered. Our goal is to solve forthe leading terms in the expansion of φ (0) in ∆ g ij . The leading terms in the expansionof the metric and the scalar field Φ are g ij ( x, ρ ) = (0) g ij + ρ (1) g ij + · · · , ∆ g ij = ρ α (cid:18) ( α ) g ij + ρ ( α +1) g ij (cid:19) + · · · , Φ( x, ρ ) = ρ α (cid:0) φ (0) + ρφ (1) (cid:1) + · · · . (5.2)Now the coordinate transformations which preserve the FG gauge take the form [12], ρ = ρ ′ (1 − σ ( x ′ )) , x i = x ′ i + a i ( x ′ , ρ ′ ) , (5.3)to leading order in some function σ , where a i ( x, ρ ) = L Z ρ dρ ′ g ij ( x, ρ ′ ) ∂ j σ ( x ) . (5.4)The form of a i is independent of the form of the series expansion of g ij ( x, ρ ) in ρ , whichis modified from that in eq. (2.2) to the more general form in eq. (2.23) in the presenceof matter back-reaction. Now following the approach of [12], we substitute the metricexpansion (2.23) into eq. (5.4) and use δG ij = δg ij ( x, ρ ) ρ = ξ µ ∂ µ G ij + 2 ∂ ( i ξ µ G j ) µ , (5.5)– 32 –here ξ ρ = − σ ( x ) ρ , ξ i = a i ( x, ρ ) . (5.6)Our notation is such that the ‘symmetrization bracket’ is defined as A ( i B j ) = 1 / A i B j + B i A j ). This allows one to determine how each coefficient in the general expansion (2.23)transforms under a general PBH transformation.One can tell immediately from ξ ρ ∂ ρ G ij that there is a homogeneous scaling termfor each coefficient of the form δ ( n ) g = − σ ( x )( n − ( n ) g + · · · . (5.7)Since the PBH transformations reduce to Weyl rescalings in the boundary, the aboveimplies that ( n ) g ij has conformal dimension 2( n − ρ multiplying the coefficient of interest.Particularly, (0) g ij always carries conformal dimension −
2, as expected.We are interested in how ( α ) g ij and ( α +1) g ij transform. Expanding a i ( x, ρ ) in ρ , a i ( x, ρ ) = a i (1) ρ + a i (2) ρ + a i ( α +1) ρ α +1 + · · · , (5.8)and substituting into eq. (5.5), we have δ (0) g ij = 2 σ (0) g ij , δ (1) g ij = a k (1) ∂ k (0) g ij + 2 ∂ ( i a k (1) (0) g j ) k ,δ ( α ) g ij = − σ ( α − ( α ) g ij , (5.9) δ ( α +1) g ij = − σα ( α +1) g ij + a k (1) ∂ k ( α ) g ij + a k ( α +1) ∂ k (0) g ij + 2 ∂ ( i a k (1) ( α ) g j ) l + 2 ∂ ( i a k ( α +1) (0) g j ) k , where a i (1) = L (0) g − ) ij ∂ j σ , a i ( α +1) = − L (0) g − ( α ) g (0) g − ) ij ∂ j σ . (5.10)These give the Weyl transformation properties of the coefficients, with which one couldin principle reconstruct the series. The building blocks in the boundary theory con-sidered in [12] include the boundary metric, its curvature tensors and their covariantderivatives. The only extra component that we have at our disposal here is the non-trivial boundary source φ (0) ( x ) of conformal dimension α = d − ∆ and its covariantderivatives.The solution for (1) g ij is unaffected by the scalar profile and is again given by eq. (2.4).Meanwhile ( α ) g ij transforms homogenously with conformal dimension 2 α −
2. Including φ (0) amongst our building blocks, the solution is uniquely determined as ( α ) g ij = c ( φ (0) ) (0) g ij , (5.11)– 33 –here the constant c is fixed by the bulk equations of motion.Substituting ( α ) g back into a i ( α +1) and hence the transformation of ( α +1) g , we find thelatter must have the form ( α +1) g ij = c L (cid:18) d ( φ (0) ) R ij + d (0) g ij ( φ (0) ) R + d ∂ i φ (0) ∂ j φ (0) (5.12)+ d ∇ i ∇ j ( φ (0) ) + d (0) g ij (cid:3) ( φ (0) ) (cid:19) . Here we have more degrees of freedom than equations, and we obtain d = − ( d − d + 2 d ( d − d + 12))2( d − , d = ( d − d ( d − d − d − ( d − ,d = − d − d + 4 + 2 d ( d − d + 36 d − d − d − , d = 12 − d ( d − , (5.13)leaving d to be determined by equations of motion.The transformation of the scalar field gives δφ (0) = − σαφ (0) ,δφ (1) = − σ ( α + 2) φ (1) + L (0) g − ) ij ∂ i σ ∂ j φ (0) , (5.14)implying that φ (1) = L d − α + 1)) (cid:18) (cid:3) φ (0) − d − φ (0) R (cid:19) . (5.15)In general if nα = 2 for some integer n then an extra homogenous term ( φ (0) ) n +1 couldappear in φ (1) and the coefficient of this term would have to be determined from theequations of motion.As a check, our results above were compared with those obtained from directlysolving the equations of motion with d = 6 , α = ( d − / d is over-determined by the equations of motion but may be consistentlysolved. Hence this serves as a non-trivial check.With these results, we can compute the leading φ (0) contribution to the universallogarithmic term in the entanglement entropy for arbitrary boundary entangling sur-face. The procedure is similar to the previous section. We begin by assuming that thebulk surface in the absence of relevant perturbation is given by X µ ( x α , τ ) — wherewe are again working with the gauge (2.6). The back-reaction of the scalar field thenintroduces changes in the background metric ∆ g and also in the minimal surface δX .– 34 –he former has been solved to leading order in φ (0) above. The latter however, doesnot contribute to the entanglement entropy to leading order because X i ( x α , τ = ρ )extremizes the action at φ (0) = 0, a fact we have made used of already in the previoussections. Since ∆ g begins at τ α = τ ( d − / , together with the measure of the minimalsurface √ h ∼ τ − d/ , the leading correction to the entanglement entropy goes like τ − .To extract the log-term, one needs to expand the remaining integrand to linear order in τ . While we do not know the complete solution of X ( x, τ ) for arbitrary asymptoticallyAdS background and boundary entangling surface, the linear τ term in its asymptoticexpansion is universal, completely dictated by fixed boundary data, independent of thegravity theory concerned. That is, it can also be fixed by the PBH transformationswhich yield [12] X i ( x α , τ ) = (0) X i ( x α ) + τ (1) X i ( x α ) + · · · , (1) X i = L K i d − , (5.16)where K i is the trace of the extrinsic curvature of the entangling surface — seeeq. (4.15).The leading φ (0) dependence of the area of the minimal surface is then given by δA = Z d d − ydτ δ (cid:18)q h ττ ( φ (0) ) det h ij ( φ (0) ) (cid:19) (5.17)= L Z d d − y dτ τ d/ q ˜ h ττ det ˜ h ab τ ∂ τ X µ ∂ τ X ν ˜ h ττ + ∂ a X µ ∂ b X ν ˜ h ab ! (cid:12)(cid:12)(cid:12)(cid:12) φ =0 ∆ g µν , where we have defined ˜ h ττ = 4 τ h ττ , ˜ h ab = τ h ab , such that these quantities beginat O ( τ ). Further recall that the radial integral ends at the UV regulator surface τ min = δ /L . With φ (0) = 0, the expansion of ˜ h ττ and det ˜ h ij are given by [12]˜ h ττ = L + 4 τ ( (1) X i ) + · · · , det ˜ h ij = det (0) h ab (1 + τ (0) h ab (1) h ab + · · · ) , (5.18)where (1) h ab = (1) g ab − L d − K i K jab (0) g ij , (5.19)and (1) g ab is as defined in eq. (2.4), but projected on to the boundary entangling surfaceby contracting with tangent vectors ∂ a (0) X i . We finally have S univ = − π L d − ℓ d − P λ Z d d − y q (0) h ab (cid:18) − ( d − d − d − c K i K j (0) g ij + ( d − L c (1) g aa + 12 L φ (0)2 ( α +1) g aa (cid:19) µ d − log µ δ . (5.20)– 35 –his expression is now completely fixed when we apply c = − d − , d = 18 , (5.21)which were determined by solving the equations of motion.We can combine the preceding results to explicitly write out this universal contri-bution for the case that φ (0) is a constant. With this simplification, the result (5.20)reduces to S univ = − ( d − π d − L d − ℓ d − P λ µ d − log µδ (5.22) × Z d d − y q (0) h ab (cid:18) K i K j (0) g ij + d + 4 d − R aa − d d − d − R (cid:19) . where R and R aa are, respectively, the background Ricci scalar and the backgroundRicci curvature contracted with H ij , the induced metric expressed as a d -dimensionaltensor: H ij = (0) g ij − n ˆ ıi n ˆ ıj . Evaluating this expression (5.22) for the geometries consid-ered in the previous section, we find complete agreement with eq. (4.14). However, thepresent result is completely general and can be applied for any background geometryand any (smooth) entangling surface.
6. Discussion
Our calculations have demonstrated two interesting properties about holographic en-tanglement entropy. First of all, the coefficient of any universal contribution which islogarithmic in the short-distance cut-off is independent of the state of the boundarytheory. Secondly, when the boundary theory is deformed by turning on a relevantoperator, new universal contributions appear including a class of the form found in [1].Let us begin here with some discussion of the first result. The observation thatthese universal coefficients are independent of the state of the underlying field theorymay seem trivial. As previously noted for an even dimensional CFT, the universalcoefficients will be given by some linear combination of central charges in a generalsetting, even without holography. However, while our result is implicitly regarded as‘obvious’ in discussions of EE, a rigorous proof has not been provided. In the AdS/CFTframework, we were able to make the separation of data depending on the state versusdata depending on the action very explicit, even when the boundary theory is deformedby a relevant operator, and it is clear only the latter data contributes to determiningthe universal terms in the holographic EE.– 36 –et us point out that there is the potential to produce a contradiction with rele-vant deformations with low conformal dimensions. Recall that the standard approach,described in section 2, allows us to study ∆ ≥ d/
2. The lower bound arises with m = − d / L , which corresponds to the Breitenlohner-Freedman bound in d + 1 di-mensions [19]. However, the unitarity bound for a scalar operator in a d -dimensionalCFT allows for ∆ ≥ ( d − /
2. To study operators in the range d/ ≥ ∆ ≥ ( d − / − d L ≤ m ≤ − d L + 1 L . (6.1)Hence in this regime then, we can choose the dimension of the dual operator as ∆ = ∆ − and in this case, the roles of φ (0) and φ (∆ − d ) are interchanged. We note, however, thatthis alternate quantization does not change the powers of ρ appearing in eq. (2.13). Inparticular, the leading power is still given by ρ ∆ − / . However, the key difference (forour purposes) is that the leading coefficients appearing in this asymptotic solution arenow related to the state of the boundary field theory.Hence it seems that in this situation any new universal term appearing in the holo-graphic EE must depend on the state. However, as we now show, there is no problembecause deformations in this regime do not produce any such universal contributions.Recall from our discussion in section 2.1 that a universal contribution appears in theholographic EE when, in the expansion of the integrand in eq. (2.9), a term appearswith τ n + m α = τ d − . Further, recall that apart from m = 0, the minimum value of m is2 because of the structure of the Einstein-scalar theory in the bulk. This means thatthere is a maximum value which α can have in order to produce a logarithmic contri-bution in the holographic EE. In particular, a logarithm will only arise for α ≤ d − ≥ d + 1 or in terms of the mass of the bulk scalar, m ≥ − d L + 1 L . (6.2)However, the lower limit here is interesting because comparing to eq. (6.1), we seethat it precisely excludes the range of allowed masses where it is possible to makean alternate quantization of the bulk scalar. Hence the potential problem, arising The case of ∆ = ( d − / m = − d / (4 L ) + 1 /L . However, this is precisely the unitarity bound for which the dual operator isexpected to be a free scalar field. However, such a CFT would be beyond the scope of the holographicmodels which we are considering here. – 37 –rom the interchange of the roles of the different terms in the asymptotic scalar inthe alternate quantization, is cleanly avoided because the deformation will simply notgenerate a log δ contribution in the holographic EE.While the focus of our discussion has been the possible logarithmic contributions tothe holographic EE, these are only the least divergent terms as δ →
0. The expansionof the area (2.9) will generally produce a series of terms diverging as 1 /δ d − − n − mα . Ofcourse, the first term ( i.e., with n = 0 = m ) yields the expected area law. Further, ouranalysis shows that the coefficients of all of these divergent terms are determined bythe fixed boundary data in the asymptotic expansion, i.e., they are all independent ofthe state of the boundary theory. In the present holographic framework, the generalcoefficient will contain m factors of the coupling φ (0) and an integral of n curvatures overthe entangling surface. This observation then guarantees that the mutual information I ( A, B ) = S ( A ) + S ( B ) − S ( A ∪ B ) (6.3)for two disjoint regions A and B is free of any UV divergences in our holographiccalculations. The finiteness of the mutual information is another generally acceptedfeature which is believed to be true in general but never rigorously proven.We should add that implicitly we are considering a constrained class of states inthis discussion, e.g., the energy density of the states being studied must be kept finite.This constraint becomes evident with the following thought experiment: Consider theboundary theory with a finite cut-off δ , in which case it contains a finite number ofdegrees of freedom (if the total volume is also kept fixed). In this case, one can easilyimagine choosing a state in which there is simply no entanglement between a particularregion V and its complement ¯ V . That is, we seem to have removed the potentiallydivergent contributions to the entanglement entropy with a particular choice of state.However, the price to be paid for this lack of correlations would be that the energydensity of such a state will be of order 1 /δ d . Hence if we wish to maintain this vanishingentanglement in the limit δ →
0, we would require an infinite entanglement entropy.Holographically, such a state would not be dual to an asymptotic AdS geometry andso it lies outside of the class of states considered here.In the discussion above eq. (2.8), we noted that the analysis in refs. [17, 12] provideda general analysis for bulk submanifolds with an arbitrary dimension k +1. In this case,the second set of independent coefficients appear in the expansion of the embeddingfunctions at order τ ( k +2) / . Our analysis focussed on k = d −
2, however, for smallervalues of k , the second set of free coefficients would appear at a lower order than in theexpansion given in eq. (2.7). Despite appearing at a lower order, this state data does RCM thanks Mark van Raamsdonk for an interesting conversation on this point. – 38 –ot contribute to the coefficients of any UV divergences, in particular a logarithmicdivergence, appearing in the calculation of the area of the corresponding surfaces. Thisoccurs precisely because the dimension of the submanifold is also reduced. Hence whenwe evaluate the analog of eq. (2.9), the leading power becomes precisely τ − ( k +2) / andso the state dependent coefficients will only produce finite contributions to the areafor general k . Hence our results extend beyond the calculation of the entanglemententropy. For example, this analysis would apply to the calculation of the expectationvalues of Wilson lines and shows that the coefficients of any divergent terms appearingin such a calculation are also independent of the state of the boundary theory.In certain cases, no logarithmic contribution appears in the EE, e.g., with a CFTin an odd number of spacetime dimensions. However, the constant term independent ofthe short-distance cut-off may then still be a universal contribution to the EE [2, 3]. Theuniversality of this constant contribution is established for a variety of d = 3 conformalquantum critical systems [25], as well as certain three-dimensional (gapped) topologicalphases [26]. However, from the discussion of the present paper, it is natural to expectthat such a finite contribution will in fact depend on the details of the state in whichthe EE is calculated. Certainly in the holographic framework, this finite term shoulddepend on ( d/ g ij and higher order terms in the FG expansion. We have confirmed thisexpectation with an explicit calculation in appendix A. Hence in general, while sucha constant contribution to the EE certainly contains information with which we maycharacterize the underlying field theory, it will not be completely universal in the samesense as the coefficient of a logarithmic contribution. In particular then, in order toproperly compare or distinguish theories with a constant contribution to EE, we mustspecify that this term was calculated in the vacuum state of the underlying theory. Of course, one of the interesting results arising from our holographic investigationswas that relevant deformations of the boundary theory will produce new universalcontributions to the EE, which are logarithmic in the cut-off. Schematically, the generalform of the logarithmic contribution is an integral over the entangling surface ∂V : S univ = X i,n γ i ( d, n ) Z ∂V d d − σ √ H [ R, K ] ni µ d − − n log µδ , (6.4)where n < ( d − / µ is the mass scale appearing in the coupling of the relevantoperator, H ab is the induced metric on ∂V and [ R, K ] ni denotes various combinations ofthe curvatures with a combined dimension 2 n . Both the curvature of the background In general, we would have to refine further our characterization of these constant contributions tothe entanglement entropy. In particular, different regulators will modify the details of the expansionthe EE in terms of the cut-off and hence ambiguities should be expected to appear in the definitionof the constant contribution. – 39 –eometry or the extrinsic curvature of the entangling surface may enter these expres-sions. The universal information which would distinguish different theories is carriedin the pre-factors γ i ( d, n ).As noted previously, for n = 0, we have simply [ R, K ] = 1 and the integral simplyyields the area of the entangling surface ∂V . In this case, this contribution matches theform of the universal terms (1.2) recently found for a massive free scalar [1]. Furtherfor n = ( d − /
2, the above integral involves only curvatures ( i.e., the µ factor reducesto one) and our expression will match the form found for an even-dimensional CFT[3, 8] — see below. More generally, the presence of these new universal terms with n > R, K ] ni cannot bedetermined in these calculations. However, one feature that is already evident there isthat the combination [ R, K ] ni µ d − − n appearing in the integrand has dimension d − R, K ] ni can in principle bedetermined but this is a somewhat tedious exercise. Hence we have only examined theparticular case of α = ( d − / φ (0) to be a field which various over the boundarygeometry. This approach also highlights the connection of the entanglement entropyto a Graham-Witten anomaly [17] for the entangling surface, as noted previously forpure CFT’s in [12, 6]. We should note, however, that the spacetime dimension d andthe conformal dimension of the relevant operator ∆ must satisfy a particular constraintbefore the various terms in eq. (6.4) can appear. These constraints simply reflect therelations discussed at the end of section 2 where the logarithmic contribution appearsif a term in the expansion of the area (2.9) with τ n + m α = τ d − . Hence, for a term with n to appear in eq. (6.4), we require∆ = d − d − − nm with integer m ≥ . (6.5)We might note that the universal contributions from relevant deformations typicallyappear in higher dimensions. The leading term with n = 0 appears for d ≥
3. However,the terms mixing the curvatures with a power of µ require larger values of d . For exam-ple, we require d ≥ d ≥ n = 1 and 2, respectively. Ofcourse, as we illustrated in section 3.2.2, that a single deformation may produce morethan one of these universal terms in higher dimensions, i.e., with d = 8 and ∆ = 6,eq. (6.5) can be satisfied with n = 0 , m = 3 and n = 1 , m = 2.Recall the integer m cannot be 1 above in eq. (6.5) because the stress tensor in– 40 –he bulk Einstein equations (2.17) is at least quadratic in the scalar field. We mightcompare this feature of our calculations to a similar result in [37]. There, a relevantoperator λ O is introduced perturbatively in a two-dimensional CFT and it is noted thatthis deformation only begins to have effect at order λ because the one-point function hOi vanishes in the CFT vacuum. Of course, this observation extends to CFT’s in anynumber of dimensions and then agrees with our result that the new universal termsappear with a factor of λ or a higher power of the coupling. However, we mightalso contrast the differences between the two situations. First, in our holographiccalculations, we are not working perturbatively, i.e., we are not assuming that λ issmall in any sense. Further, one of our key observations is that the results for theuniversal coefficients is independent of the state of the boundary CFT and so does notrely on calculating in the vacuum state. It would be interesting to see if in fact thesefeatures also extend beyond our holographic setting to more general CFT’s.We must emphasize that eq. (6.4) is schematic. In particular, for a given value of n , there may be several independent combinations of curvatures that appear, includingboth background curvatures and the extrinsic curvature of the entangling surface. Ourresult in eq. (5.22) explicitly illustrates the possible complications. Further we must addthat even when eq. (6.5) is satisfied, the coefficient of the universal term may still vanish,depending on further details of the underlying theory. For example, if the bulk scalartheory, respects a discrete symmetry Φ → − Φ, the coefficient will vanish unless m is aneven integer. It is interesting to consider how these results would change if there weretwo or more relevant deformations with different conformal dimensions. We expectthat in fact there would be no essential changes. The asymptotic expansions wouldhave to be extended to allow separate factors ρ α i / from each of the deformations andthe nonlinearities of the bulk theory would mix these terms. However, the schematicstructure of the universal terms would remain as given in eq. (6.4) and the constrainton the conformal dimensions to produce a particular term would become X m i ( d − ∆ i ) = d − − n . (6.6)Of course, one contribution, which appears irrespective of the precise conformaldimension(s) of the relevant deformation(s), is the term with n = ( d − /
2. Asnoted above, such contributions are known to appear for any even-dimensional CFTand the universal coefficients γ correspond to the central charges of the CFT [8, 3].Even with the relevant deformation the present case, it is precisely these terms thatappear with the central charges of the CFT that emerges in the UV regime. As wasdemonstrated in [6], the precise structure of these terms and their dependence on thegeometry of the boundary metric and of the entangling surface can be derived usingthe PBH approach discussed in section 5. The latter was originally derived in the case– 41 –here the boundary theory was a pure CFT, however, all of the same contributionsstill appear in the asymptotic expansion when the relevant deformation is turned on.Hence the structure of this term does not change in the case where the boundary theoryis deformed by a relevant deformation.It is interesting that our holographic calculations indicate that for even d , the samecentral charges for the CFT emerging in the UV actually appear in the coefficients ofall of the logarithmic contributions. This can be seen from the pre-factor of ( L/ℓ P ) d − which appears in all of our results. Since our bulk theory corresponds to Einsteingravity, all of the central charges are equal and we can not distinguish precisely whichcentral charges appear in the various contributions. It may be interesting to repeat ouranalysis in the case where the bulk gravity theory includes higher curvature interactionssince in principle, this would allow us to distinguish the different central charges [38]— see below.The appearance of central charges in these new universal contributions hints atthe close relation of these new terms with the trace anomaly. As is well known, witheven d , EE in a CFT can be directly calculated using the trace anomaly, at leastfor geometries with sufficient symmetry [3, 8, 6, 9]. Typically, we consider the traceanomaly in a curved background, where it is usually related to various conformallyinvariant combinations of the curvature [24]. However, deforming the CFT with arelevant operator will also introduce additional terms in the trace anomaly related tothe coupling to the new operators [15, 20]. While this situation has not been studied indetail, it is already evident that terms involving both the curvature and the couplingof the relevant operator appear in the trace anomaly in this situation. For example,the simplest such term, which arises for ∆ = ( d + 2) /
2, takes the form [20] h T ii i = 12 φ (0) (cid:18) (cid:3) + d − d − R (cid:19) φ (0) + · · · . (6.7)Given such a result, we can apply the approach, alluded to above, to calculate theentanglement entropy using the trace anomaly [3, 8, 6, 9] and we have confirmed thateq. (6.7) does indeed yield a universal contribution to the EE of precisely the form givenin eq. (3.26). More generally, we expect that the new universal terms in eq. (6.4) aresimilarly related to new terms which the relevant operator induces in the trace anomaly.We hope to return to these issues elsewhere. Note that the calculations which we havesketched above apply to any general CFT with a relevant deformation and does notrefer to holography. This would demonstrate that our results apply more broadly thanto holographic field theories.Another framework where these aspects of entanglement entropy are easily studiedis with free field theories. In particular, ref. [1] considered a free massive scalar in– 42 – flat background with a flat entangling surface. They found the logarithmic termsin eq. (1.2) appear for any even d ≥ n = 0 terms inthe general expression (6.4). It is amusing to compare this result to our holographicresults which would correspond to a strongly coupled field theory. The natural relevantoperator to consider would be one with ∆ = d −
2, as for a scalar mass term. In thiscase, the holographic contribution appears again for any even d but for d ≥
6. One caneasily extend the free scalar calculation to examples where the geometry is curved [39]and logarithmic terms mixing curvatures and powers of the mass also appear in higherdimensions, similar to the results of our holographic study. Another simple extensionof the free field calculations in [1] is to consider massive fermions [39]. In this case,it appears that various logarithmic contributions as in eq. (6.4) again arise in evendimensions. On the holographic side, it would be natural to compare to a relevantoperator with ∆ = d −
1, as for a fermionic mass term. In this case, the holographiccalculations yield a logarithmic contribution for any odd or even dimension with d ≥ S ( V ) = 2 πℓ d − P ext v ∼ V Z v d d − y √ h (cid:2) λ L R (cid:3) , (6.8)where R denotes the Ricci scalar for the intrinsic geometry on v and λ is the (di-mensionless) coupling which controls the strength of the curvature-squared interaction.While the appropriate entropy functional is not known for general higher curvaturetheories, we expect that it will have the form S EE = 2 πℓ d − P ext v ∼ V Z v d d − y √ h (cid:2) f ( R, K i , Φ) (cid:3) , (6.9)where f is some local scalar constructed from the bulk curvature R , the extrinsic cur-vature of the surface K i , the bulk scalar Φ (when a relevant deformation is introduced)– 43 –nd derivatives of these building blocks. Note that eq. (6.8) can be re-expressed in thisform using the Gauss-Codazzi equations [6]. Now the key observation is that in theFG gauge, any such scalar will admit an expansion in τ beginning at order τ . Hencethe expansion of the full integrand begins with τ − d , just as in section 2. Further, thePBH transformations will continue to fix the asymptotic form of the asymptotic metric(and scalar), as well as the embedding functions, as discussed in section 5. The onlychange is that the constants appearing at various orders may take on new values asthe equations of motion will have changed. In any event, as in the main text, anylogarithmic contribution will only depend on the fixed boundary data and so we expectthat the corresponding coefficient remains state independent when the bulk gravitytheory is extended to include higher curvature interactions. Hence our previous resultfor the universal logarithmic contribution to the holographic EE is not changed in sucha generalized holographic framework. Further we do not expect that the basic form ofthe universal terms will change in this scenario. However, it may be useful to examinethese expressions in, e.g., Lovelock gravity, as it may allow one to identify the specificcentral charge associated with the pre-factor (
L/ℓ P ) d − , as discussed above. Acknowledgments
We would like to thank Alex Buchel, Joao Penedones, Mark van Raamsdonk, SimonRoss, Brian Swingle and especially Horacio Casini for useful conversations and corre-spondence. Research at Perimeter Institute is supported by the Government of Canadathrough Industry Canada and by the Province of Ontario through the Ministry of Re-search & Innovation. RCM also acknowledges support from an NSERC Discovery grantand funding from the Canadian Institute for Advanced Research. RCM would also liketo thank the Aspen Center for Physics in the final stages of preparing this paper.
A. Universality and Odd d In many cases, no logarithmic contribution appears in the EE, e.g., with a CFT inan odd number of spacetime dimensions. However, the constant term appearing inusual expansion in powers of the short-distance cut-off may still be regarded as auniversal contribution to the EE [3]. The universality of this constant contributionis well-established for a variety of d = 3 conformal quantum critical systems [25], aswell as certain three-dimensional (gapped) topological phases [26]. However, from thediscussion of the present paper, it is natural to expect that such a finite contributionwill in fact depend on the details of the state in which the EE is calculated. Certainlyin the holographic framework, this finite term should depend on ( d/ g ij and higher order– 44 –erms in the FG expansion. Hence in general, if we are to interpret this constantcontribution to the EE as characteristic of the underlying field theory, we must alsospecify that the calculations were performed in the vacuum state of the theory.In the following, we verify that the finite term in the EE does in fact depend onthe state of the underlying theory with a simple holographic calculation. We considera boundary CFT at finite temperature T and calculate the holographic EE across aspherical entangling surface of radius R . Working at low temperature, i.e., RT ≪ d -dimensional spacetime and so atfinite temperature, the holographic dual is a ( d + 1)-dimensional planar AdS black hole.The metric for this bulk solution can be written as ds = L z (cid:0) − f ( z ) dt + dr + r d Ω d − (cid:1) + L f ( z ) dz z , (A.1)where we have introduced polar coordinates in the boundary directions and f ( z ) isgiven by f ( z ) = 1 − (cid:16) zz + (cid:17) d . (A.2)Note that in this solution, the horizon appears at z = z + . Further, in the limit z + → ∞ ,we recover the AdS vacuum metric in Poincare coordinates. The Hawking temperatureof this black hole solution is given by T = d πz + . (A.3)To evaluate the holographic EE (1.1) for a spherical entangling surface, we mustdetermine the extremal bulk surface described by a profile r = r ( z ) with the boundarycondition r ( z = 0) = R , as in section 3.2 except that the bulk space is now given byeq. (A.1). The induced metric on such a surface is given by h αβ dx α dx β = L z h ( r ′ + 1 /f ) dz + r d Ω d − i , (A.4)where the ‘prime’ denotes a derivative with respect to z . As a result the EE is givenby S = 2 π L d − ℓ d − P Ω d − Z dz r d − z d − p r ′ + 1 /f . (A.5)– 45 –n the case of pure AdS ( z + = ∞ ) the shape of the extremal surface can be obtainedin the closed form given in eq. (3.22) [2, 3]. However, for general z + we did not succeedto find a closed analytic expression and thus we proceed perturbatively in R/z + << T R << d ). In this regime, the extremal surface will be closeto that in eq. (3.22) and so we have z . R . Therefore we expand the integrand ofeq. (A.5) in powers of ǫ ( z ) = ( z/z + ) d S = 2 π L d − ℓ d − P Ω d − Z z max δ dz r d − z d − √ r ′ + 1 (cid:16) ǫ ( z )2( r ′ + 1) + O ( ǫ ) (cid:17) . (A.6)Recall that with z + = ∞ , ǫ ( z ) = 0 and, according to eq. (3.22) (3.22), the profileof the extremal surface is given by r ( z ) = √ R − z and z max = R . However, with ǫ ( z ) = 0, both r ( z ) and z max acquire corrections r ( z ) = r ( z ) + δr ( z ) , z max = R + δz max , (A.7)where these corrections are at least of order ǫ . To solve for δr ( z ), we may substituteeq. (A.7) into the action (A.6) and consider extremizing with respect to δr ( z ). However,in doing so, we find that to leading order there are two contributions, one of order δr and the other of order ǫ δr . Hence upon substituting the solution back into eq. (A.6),we would find that to leading order δr only makes contributions of O ( ǫ ) and so wecan ignore this change in the profile. Similarly, the contribution to eq. (A.6) from thechange δz max involves evaluating the integrand (with r = r ) at z max = R but thisvanishes to leading order since r ( R ) = 0. Therefore if we work only to linear order in ǫ ( z ), we need only evaluate the second term in eq. (A.6) with the profile r ( z ): δS = 2 π L d − ℓ d − P Ω d − Z Rδ dz r d − z d − ǫ ( z )2 p r ′ + 1 + O ( ǫ ) = 2 π L d − ℓ d − P Ω d − d + 1) (cid:16) Rz + (cid:17) d + O ( δ d , ǫ ) . (A.8)Hence combining this result with eq. (A.3), we find that δS ∼ ( RT ) d and so we havefound a finite contribution to the EE which depends on the temperature (state) of theboundary CFT. Let us also note that for two-dimensional CFT’s, one can get a closedexpression for the EE at finite temperature [30] and expanding the latter in the limitof low temperature reproduces precisely our correction (A.8).Let us consider extending the above calculation to a more general entangling surfaceto provide a general estimate for the contribution δS calculated above for a sphericalsurface. To make progress here, it is simplest to adopt the general framework andnotation introduced in section 2. In particular, we begin by adopting the usual radial– 46 –oordinate of the FG expansion (2.1) z = Lρ / (cid:18) Lz + (cid:19) d ρ d/ ! − /d ≃ L ρ / − d (cid:18) Lz + (cid:19) d ρ d/ + · · · ! . (A.9)With this choice, the asymptotic expansion of the planar black hole metric (A.1) be-comes ds ≃ L ρ dρ + 1 ρ " − − d − d (cid:18) Lz + (cid:19) d ρ d/ ! dt + d (cid:18) Lz + (cid:19) d ρ d/ ! X ( dx i ) . (A.10)Hence as expected, we see that the leading effect of the temperature appears in ( d/ g ij inthe FG expansion (2.2). Now we choose some entangling surface ∂V in the flat boundarymetric which is described by (0) X i ( y a ) and there will be some bulk surface described bythe profile X i ( y a , τ ) satisfying the boundary condition X i ( y a , τ = 0) = (0) X i ( y a ). Wemake the same gauge choice as in eq. (2.6) and then the induced metric (2.5) becomes h ττ = L τ (cid:18) τL ∂ τ X i ∂ τ X j g ij (cid:19) ≡ L τ ˜ h ττ ,h ab = 1 τ ∂ a X i ∂ b X j g ij ≡ τ ˜ h ab . (A.11)As only the spatial coordinates are relevant here, we may write g ij ≃ δ ij (1 + ˜ ǫ ( ρ )) where ˜ ǫ ( ρ ) = 1 d (cid:18) Lz + (cid:19) d ρ d/ . (A.12)Now following the analysis above for the spherical entangling surface, we will ex-pand the holographic EE in powers of ˜ ǫ ( ρ ) and only keep the contribution that is linearin this term. As above, the deformation of the background metric will produce pertur-bations of the profile and the maximum value of ρ , both of which begin at linear orderin ˜ ǫ ( ρ ): X i ( y a , τ ) = X i ( y a , τ ) + δX i ( y a , τ ) , ρ max = ρ ,max + δρ max . (A.13)Again, to solve for δX i ( y a , τ ), we would extremize the area functional with respect tothese functions. However, also as above, we find that to leading order there are two– 47 –ontributions, one of order δX i and the other of order ǫ δX i . Hence upon substitut-ing the solution back into area, we would find that to leading order δX i only makescontributions of O ( ǫ ) and so we can ignore these changes in the profile. Similarly,the contribution from the change δρ max involves evaluating the integrand with X i at ρ ,max but this vanishes (to leading order) since by definition ρ ,max is the point wherethe bulk surface (smoothly) closes off. Hence at this point, we have √ h | ρ = ρ ,max = 0.Therefore if we work only to linear order in ǫ ( z ), we need only evaluate the variation tothe area coming from the change of the background metric, given in eq. (A.12), withthe profile X i : δS = 2 πℓ d − P I ∂V d d − y Z ρ max δ dτ L τ d/ q ˜ h (cid:20)
12 ˜ h αβ δ ˜ ǫ ˜ h αβ (cid:21) = 2 πℓ d − P I ∂V d d − y Z ρ max δ dτ L d q ˜ h (cid:20) d − − h ) ττ (cid:21) (cid:18) Lz + (cid:19) d . (A.14)An essential feature of this result is that it is finite, i.e., the leading factor of τ − d/ has been canceled by the τ -dependence of ˜ ǫ . In the vicinity of τ = 0, the integrandreduces to essential the q det (0) h ab , i.e., the area measure on ∂V in the boundary metric.Hence we would argue that the y integration essentially contributes a factor of A d − , thearea of the entangling surface. Certainly such a factor appears for entangling surfaceswith sufficient symmetry, such as the spherical surface in the previous calculation.Similarly, the contribution of the τ integral can be estimated to be roughly ρ max ≃ ℓ /L , where ℓ is some characteristic scale of the geometry of ∂V which controls howfar the extremal surface extends into the bulk. Again, for surfaces with sufficientsymmetry, we can readily identify ℓ . For example, in the above calculations, ℓ is theradius R of the spherical entangling surface or in section 3.1, ℓ would be the width ofthe slab with flat parallel boundaries. Therefore up to overall numerical factors, ourestimate of this contribution to the holographic EE becomes δS ≃ L d − ℓ d − P A d − ℓ T d . (A.15)Thus, our holographic calculations explicitly demonstrate that the constant contri-bution to the EE depends on the state in which the latter is calculated. Hence, whilesuch a contribution certainly contains information that characterizes the underlyingfield theory, we must be careful in comparing various results to specify the state ( e.g., the vacuum) for which the calculations were performed.Let us consider our holographic result (A.15) further. Given that the boundaryCFT is at finite temperature T , the thermal bath will produce a uniform entropy density– 48 – ∼ ( L d − /ℓ d − P ) T d − and so for a general region with volume V d − , the correspondingthermal entropy would be δS therm = ( L d − /ℓ d − P ) V d − T d − . Hence comparing thisresult to eq. (A.15), we see that the finite temperature dependent contribution to theholographic EE which we have identified does not correspond to this thermal entropy. Itmay seem that we have found a discrepancy since we should expect that δS therm shouldappear as a finite contribution in the EE [2, 3] and in fact, it seems that this contributionwould dominate in the low temperature limit (given that δS therm is proportional to asmaller power of T ). However, the latter limit provides the resolution of this apparentdiscrepancy. Since we are working in the limit ℓT ≪
1, the typical wavelength of thethermal excitations is much larger than the size of the entangling geometry and so itshould not be a surprise that δS therm has not appeared in our calculations. Insteadthis contribution would be expected to appear in the opposite limit ℓT ≫
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