Holographic Entanglement Entropy in Lovelock Gravities

We study entanglement entropies of simply connected surfaces in field theories dual to Lovelock gravities. We consider Gauss-Bonnet and cubic Lovelock gravities in detail. In the conformal case the logarithmic terms in the entanglement entropy are governed by the conformal anomalies of the CFT; we verify that the holographic calculations are consistent with this property. We also compute the holographic entanglement entropy of a slab in the Gauss-Bonnet examples dual to relativistic and non-relativistic CFTs and discuss its properties. Finally, we discuss features of the entanglement entropy in the backgrounds dual to renormalization group flows between fixed points and comment on the implications for a possible c-theorem in four spacetime dimensions.


Introduction and summary
Entanglement entropy is an interesting non-local observable which carries important information about field theory. Refs. [1][2] proposed a way of computing entanglement entropy in the strongly coupled conformal field theories dual to gravitational theories whose gravity sector is described by the Einstein-Hilbert lagrangian with the negative cosmological constant. The set of CFTs that admit duals of this type is strongly restricted. In particular in four spacetime dimensions, all such CFTs necessarily have their a and c central charges equal to each other.
It is known that some interesting phenomena in CFTs and, more generally, in quantum field theories, are associated with the regime where a = c. For example, there are unitarity constraints in CFTs [3] which restrict the ratio of a/c to lie within certain bounds. Another interesting and important question is whether one can formulate and prove the analog of Zamolodchikov's c-theorem [4] in three and more spacetime dimensions. It has been suggested that the value of the c-function in four dimensions is equal to a at fixed points [5]; recent work in field theory includes [6][7][8]. The conjecture is known by the name "atheorem".
Holographic theories with higher derivative terms provide a natural arena for investigating these phenomena. Recently Myers and Sinha [9][10] have shown that one can formulate an a-theorem in certain higher derivative theories of gravity, but it is not completely clear what the field theoretic counterpart of the corresponding a-function is. Their results have been generalized to Lovelock theories in [11]. (Work which uses higher derivative gravitational theories to study unitarity constraints in CFTs includes [12][13][14][15][16][17][18][19][20][21][22][23][24][25].) Interestingly, entanglement entropy provides an independent way of computing the a and c central charges in the CFTs. In particular, it has been noticed in [26] that depending on the shape of the surface which defines entanglement entropy, the logarithmic terms contain a linear combination of the a and c central charges. This provides an additional motivation to investigate the holographic entanglement entropy (EE) in the theories with higher derivative gravitational terms. In this paper we consider Lovelock gravities, paying special attention to the Gauss-Bonnet and cubic Lovelock cases. We make use of the prescription of [27] and generalize it to the Lovelock case to compute the logarithmic terms in the holographic entanglement entropy for a few simple geometries such as a ball, a cylinder and a slab.
The rest of the paper is organized as follows. In the next Section we review the results of [26] which imply that EE of a ball contains a logarithmic term proportional to the a central charge, while the EE of a cylinder in the similar manner encodes the c central charge. In Section 3 we give a brief review of Lovelock theories of gravity. Section 4 contains the description of holographic entropy proposal of [1][2] together with the generalization to the Gauss-Bonnet case [27]. There we holographically compute the logarithmic terms in the EE of a ball and a cylinder in CFTs dual to Gauss-Bonnet gravity in AdS 5 and confirm that they are proportional to the a and c central charges respectively. We also compute the entanglement entropy of a slab as a function of Gauss-Bonnet parameter. In Section 5 we make an educated guess for the holographic formula valid in all Lovelock theories. We show that the coefficient of the logarithmic term in the EE of a cylinder in six-dimensional CFT dual to cubic Lovelock gravity in AdS 7 is proportional to the linear combination of the B-type anomaly coefficients. In Section 6 we consider the solution in the bulk which is holographically dual to the non-relativistic field theory with Lifshitz symmetries. We compute the entanglement entropy of a slab and a cylinder and compare with earlier results. Section 7 is devoted to studying entanglement entropy in the bulk geometries dual to renormalization group flows between conformal fixed points, motivated by the search for a c-theorem in four spacetime dimensions.
Note added: As we were working on this project, we became aware of the forthcoming paper [28] which partially overlaps with our results (see also [29]).

Entanglement entropy and conformal field theories in four dimensions.
The entropy of entanglement (EE) in a d-dimensional quantum field theory on IR d−1 × IR is defined as the von Neumman entropy of the reduced density matrix associated with a subspace V of the total space IR d−1 where the field theory lives The EE can be ultraviolate (UV) divergent in the continuum limit and a cutoff ǫ needs to be introduced. The leading divergent term is usually proportional to the area of the boundary of V , (∂V ) where the proportionality coefficient γ depends on the regularization procedure. This result, known as the " area law" for EE, was first found numerically [30,31] and later derived analytically [32][33][34][35][36][37]. Note however that the area law is violated in the presence of a finite Fermi surface [38][39][40][41].
For d-dimensional conformal field theories (CFTs) the structure of the divergent terms usually takes the following form [42]  are not physical and depend on the regularization procedure. On the other hand, the coefficient of the logarithmic term is physical and universal in nature, not affected by cutoff redefinitions.
Here our primary focus will be on conformal field theories in d = 4 dimensions where the universal coefficient of EE was recently obtained [26]. In particular, using the replica trick and conformal invariance of a four dimensional CFT on a curved manifold, [26] proposed that the coefficient of the logarithmically divergent term in the entanglement entropy of a smooth and connected region V is given by where (c, a) are the central charges of the four dimensional CFT and g 0c , g 0a are defined as follows is parameterized by a set of coordinates x a with a = 1, 2 whereas the spacetime metric g µν where the CFT lives is spanned by coordinates X µ with µ = 0, 1, 2, 3. Then, n µ i in (2.5) denote two (i = 1, 2) vectors normal to the surface (∂V ) satisfying: (2.6) k i µν represents the extrinsic curvature tensor of (∂V ) associated to the normal n i and is given by where γ µν represents the induced metric equal to γ µν = g µν − n i µ n i ν . Note that µ, ν = 0, 1, 2, 3 are spacetime indices raised and lowered with the metric g µν . Finally, k i is the trace of the extrinsic curvature tensor k i = k i µν g µν . The coefficient µ in (2.5) cannot be determined by conformal invariance. In [26] it is fixed by requiring agreement between (2.4) and the holographic calculation. Using this result [26] concluded that the coefficient of the logarithmic term in the entanglement entropy of a ball B and a cylinder C in any four dimensional conformal field theory takes the form S(B) = · · · + a 90 ln ǫ + s(B) S(C) = · · · + c 720 where B, C denote a ball of radius R and an infinite cylinder of radius R and length l respectively 1 . This interesting result provides an additional characterization of the anomaly coefficients (c, a) through entanglement entropy.
Independent evidence in support of the work of [26] was first given in [43]. The authors of [43] used Srednicki's regularization method and numerically computed the coefficient of the logarithmic term in the entanglement entropy of a ball for a free bosonic CFT in d = 4 dimensions. Their result was in complete agreement with [26]. Recently, [44] analytically computed the entanglement entropy for the region of a ball in a massless scalar field theory in arbitrary dimensions, further verifying Solodukhin's formula for this case. General results for the entanglement entropy for a spherical region were established later on in [45][46][47][48]. 1 Note that for a region V with zero extrinsic curvature, e.g. a slab, the logarithmic term vanishes.
It is interesting to use eqs (2.4), (2.5) to compute the coefficients of the logarithmic term of the EE for spatial regions of different geometrical shape such as, ellipsoids, toroids e.t.c. It is important to stress here that the results of [26] are restricted to regions V of smooth geometrical shape. Otherwise, contributions from the non-smooth boundary are likely to modify the coefficients of the logarithmically divergent terms in the entropy [42].

Gauss-Bonnet gravity
Among all theories of gravity which contain higher derivative terms of the Riemann tensor in the their action there exists a special class of theories usually referred to, as Lovelock gravity. This class of gravitational theories stands out both for its simplicity and the several properties it shares with Einstein-Hilbert gravity. In particular, it is the most general theory of gravity whose equations of motion involve only second order derivatives of the metric. It is ghost free when expanded around a Minkowski spacetime background, while recently, the Palatini and metric formulations of Lovelock gravity have been shown to be equivalent [49].
The action for Lovelock gravity in d + 1-dimensions is denotes the integral part of d 2 , λ p is the p-th order Lovelock coefficient 2 and L p is the Euler density of a 2p-dimensional manifold. In d + 1 dimensions all L p terms with p ≥ [ d 2 ] are either total derivatives or vanish identically.
In this work we will primarily focus on five dimensional gravitational theories and limit ourselves to the Gauss-Bonnet action. This is the simplest example of a Lovelock action, with only the 4-dimensional Euler density included Note that in eq. (3.2) we introduced a cosmological constant term Λ = − 12 L 2 and denoted the dimensionless Gauss-Bonnet parameter by λ instead of λ 2 since the other Lovelock terms vanish in this case. In what follows we will retain this notation except for section 5, where we discuss generic Lovelock theories. The Gauss-Bonnet term L (2) in (3.2) is Equations of motion derived from (3.2) are expressed in the following way where the curvature scale of the AdS space is related to the cosmological constant via 3 There are two ways to relate the gravitational parameters, the Gauss-Bonnet coupling λ, Newton's five dimensional coupling constant G N and the cosmological constant L, to the CFT parameters (c, a). One is via a holographic calculation of the three point function of the stress energy tensor and the other through the holographic computation of the Weyl anomaly [52][53] . Both calculations yield the same result, which is a good consistency 3 To be specific, Gauss-Bonnet gravity admits another AdS solution with α = 2 1− √ 1−4λ but this solution is unstable and contains ghosts [50].
check. The holographic calculation of the Weyl anomaly in Gauss-Bonnet gravity was performed in [54]. Here we simply quote the results where L AdS is given from (3.7). In our conventions the CFT central charges (c, a) are defined through the Weyl anomaly in the following way It will be helpful for the calculations in the next section to have the ratio (3.10) Finally, we should note that the correspondence between the positivity of the energy flux in a CFT [3] and causality of the boundary theory in Gauss-Bonnet gravity discussed in [12][13][14][15] , restricts the values of the Gauss-Bonnet parameter λ to lie within the region − 7 36 ≤ λ ≤ 9 100 . Similar results were obtained for generic Lovelock theories of gravity in arbitrary dimensions [16][17][18][19][20][21]25].

Holographic Entanglement Entropy Proposal
In the context of holography, entanglement entropy received a lot of attention after the work of Ryu and Takayanagi where a concrete proposal for evaluating the entanglement entropy was set forth [1,2]. In particular, the authors of [1,2] conjectured that the entanglement entropy of a spatial 4 region V in a d-dimensional CFT admitting a dual description in terms of Einstein-Hilbert gravity is given by where Σ is defined as the minimal area surface which asymptotes to the boundary of the spatial region V , (∂V ). For more details the reader is encouraged to consult [56].
This proposal has by now passed several tests. When for instance, the spatial region V extends to the whole of space, entanglement entropy should coincide with statistical entropy. Indeed, at finite temperature eq. (4.1) naturally reduces to the Bekenstein-Hawking entropy formula whereas for vanishing temperature, the dual gravitational description contains no horizon and the entropy vanishes as it should. Other properties of the entanglement entropy like strong subadditivity or the fact that V and its complement V c have the same entropy, are also satisfied by the holographic EE formula [57]. Moreover, precise agreement between the holographic computation and the field theoretic one has been shown in the cases where explicit results are available (mostly for two dimensional CFTs) [58][59][60].
It is interesting to generalize (4.1) to include higher derivative gravitational theories.
The most natural idea is to replace eq. (4.1) with Wald's entropy formula. In fact, for Gauss-Bonnet gravity, such a proposal already exists in the literature [27] 5 . To be specific, the author of [27] suggested that the entanglement entropy of a connected region V of the dual CFT, can be computed in the case of Gauss-Bonnet gravity through the following formula Here the integral is evaluated on Σ, the three dimensional surface which at the boundary of the holographic space reduces to the two dimensional boundary (∂V ) of the region whose entropy we want to compute and which is determined by minimizing (4.2). σ in the same expression, corresponds to the determinant of the induced metric on Σ whereas R Σ is the induced scalar curvature of Σ 6 .
To summarize the main reasoning of [27] recall that to compute the entanglement entropy on the CFT side, one starts by evaluating the partition function on a d-dimensional n-sheeted space -formed by gluing the n-copies of IR d along the boundary (∂V ). This 5 For related work on holographic entanglement entropy and higher curvature corrections see also [61]. 6 To make the variational problem well-defined a boundary term should in principle be added in (4.2). This term does not affect the solution of the embedding surface but it changes the value of the action evaluated on the solution and thus of the entanglement entropy. It turns out however that the boundary term only modifies the leading UV-divergent term in the entanglement entropy.
procedure produces a space R n with conical singularities on the surface (∂V ). To evaluate the partition function on R n holographically, it is necessary to identify the dual d + 1 dimensional geometry, S n . The latter should be a solution of the gravitational action with non-zero cosmological constant, which asymptotes to R n at the boundary. Finding such a solution is a difficult task. Instead, [27] assumed that S n is given by a n-sheeted AdS d+1 formed by gluing together n-copies of AdS d+1 along a surface of codimension two. Then, the problem essentially reduced to that of evaluating the gravitational action functional on a space with conical singularities. A method for performing this calculation (at least in some cases) has been developed in [62] (see also [63][64][65][66][67][68][69][70] Recently [71] questioned some of the assumptions that were used in [27] to derive (4.2). Still, (4.2) remains a reasonable generalization of (4.1) to Gauss-Bonnet gravity.
First, the proposal agrees with Wald's entropy formula for AdS-Schwartzchild black holes in Gauss-Bonnet gravity [72][73][74][75]. Therefore, whenever the spatial region V coincides with the total space, the entanglement entropy is guaranteed to be equal to the thermal entropy.
Moreover, the strong subadditivity property of EE is satisfied [57]. Finally, the difference between (4.2) and (4.1) is the integral of a topological quantity i.e., the euler density in two dimensions, just like the difference between the Einstein-Hilbert and Gauss-Bonnet lagrangian is the Euler density term of four dimensions.
In the following we will use the proposal of [27] to compute the entanglement entropy for a region bounded by ball, a cylinder and a slab. Comparison with (2.8) will provide yet another check of (4.2).

The entanglement entropy of a ball.
To compute the entanglement entropy of a ball of radius R, it is useful to parameterize the AdS space in the following form The first step is to identify a three dimensional surface in the bulk of AdS which reduces to a sphere of radius R at the boundary. Taking into account the symmetries of the problem we see that the surface in question is determined by a single function r(ρ). With this ansatz the induced metric of the surface can be written as follows Using (4.4) to compute the induced curvature R Σ and substituting into (4.2) yields where α is given in (3.7) and R, the induced scalar curvature in units of the AdS radius, is Eq. (4.5) gives the equations of motion which determine r(ρ). To specify the coefficient of the logarithmic term it suffices to solve for r(ρ) to the next to leading order in the neighborhood of the boundary ρ = 0. We find that The solution is identical in this order to the case λ = 0. Substituting (4.7) into (4.5) yields Using (3.10) and the definition of α from (3.7) we arrive at which is in complete agreement with (2.8).

4.2.
The entanglement entropy of a cylinder.
Consider a three dimensional surface in AdS which reduces to a two dimensional cylindrical surface of radius R and length l on the boundary of the AdS space. It is then natural to parameterize the AdS metric as follows The symmetries of the problem lead us to consider a surface described by a single function r(ρ). The induced metric is then Plugging this ansatz into (4.2) yields where now R is the induced curvature of the surface in units of the AdS radius is (4.13) The equations of motions in the vicinity of ρ = 0 are satisfied by 7 (4.14) Substituting (4.14) into (4.12) yields With the help of (3.10) and (4.13) we arrive at which agrees with (2.8).

4.3.
The entanglement entropy of a slab.
The slab geometry corresponds to the region of space bounded by − y 2 ≤ x 1 ≤ y 2 and infinitely extended along the x 2 , x 3 directions. This is the simplest configuration to consider because of the large amount of symmetry. Here, it is convenient to write the AdS metric as ds 2 = L 2 AdS r 2 η ij dx i dx j + dr 2 (4.17) 7 Notice that the solution is again identical in this order to the case λ = 0.
with η ij the four dimensional Minkowski metric. The three dimensional induced surface can be parametrized by a single function x 1 (r) as follows The induced curvature of the surface (in units of the AdS radius) is non-vanishing and equal to where the primes indicate differentiation with respect to the radial coordinate. The Lagrangian of the system is independent of x 1 (r) so there is a constant of motion which allows us to solve for x ′ 1 (r) exactly. Since the theory is conformal, one can rescale the coordinate by denoting τ = r/r * . Then eq. (4.20) can be written as where h(τ ) = x ′ 1 (r) 2 . It is easy to see that there are three solutions for h(τ ) but only one of them is continuously connected with the solution of the λ = 0 case. In the following we restrict our attention to this solution 8 . It would be interesting to examine the other two solutions which at first glance appear to be complex valued. We leave this analysis to future work.
We proceed to relate the constant of motion r ⋆ with the width y of the slab Evaluating the action on the solution of (4.21) yields Recall from (3.10) that λ can be expressed as a function of the ratio a c whereas . This allows us to express the final result as with γ (a, c) = 3c−a 4π×90 γ(λ). The result of numerical integration in eq. (4.24) is shown in Fig. 1.

Generalization to Lovelock
So far we used Fursaev's proposal to compute the leading divergent terms in the entanglement entropy of a ball, an infinite cylinder and a slab in a four dimensional CFT defined from Gauss-Bonnet gravity via gauge-gravity duality. Gauss-Bonnet gravity allows for the holographic description of CFTs with unequal central charges (c, a) giving an opportunity to check the proposal of [27] against the results of [26]. According to [26], the coefficient of the logarithmic term in the entropy should be proportional to a for a ball, c for an infinite cylinder and vanish for a slab, which is precisely what we found by using the prescription of [27]. This result led us to generalize the proposal of [27] to any Lovelock theory of gravity Expression (5.1) coincides with the expression for the entropy of black holes in Lovelock gravity as established in [72][73], which in turn agrees with Wald's entropy formula [74][75].
As a result (5.1) satisfies by construction several properties of the entanglement entropy.
An interesting check of the proposal (5.1) is the computation of the entanglement entropy of the cylinder in the case of cubic Lovelock gravity dual to the six-dimensional CFT. In this case, we expect the log term in the EE to be proportional to the coefficients of the B-type anomaly (just as in the four-dimensional case, the coefficient was proportional to c) 9 . In fact, there is a non-trivial check of this statement. In Lovelock gravity, the three coefficients of the B-type anomaly, b i , i = 1, . . . , 3 are not linearly independent. Using the results of [19] one can write them as where we used to eliminate terms linear in λ 3 . The overall coefficient c is related to the definition of the invariants I i , i = 1, . . . , 3 and is therefore not important. One can fix its numerical value to be c = −1/2304. We can now use (5.1) to compute the EE of the cylinder. In particular, (4.7) becomes while the result for the EE Using (5.3) we can eliminate the term proportional to λ 3 from the square brackets. The resulting expression has three terms, but turns out to be proportional to the linear combination of the two central charges (5.2), which is a highly nontrivial check of (5.1). Note that the third central charge, b 3 , is a linear combination of b 1 and b 2 (since Lovelock gravity was shown to satisfy the supersymmetric constraint [77]). Finally, the results of this section can be helpful in extending the proposal of [26] to CFTs of higher dimensionality. We hope to investigate this issue further in the future.

Entanglement entropy in Lifshitz backgrounds.
Lifshitz spacetimes are geometries of the form which preserve the scaling symmetry w is usually referred to as the dynamical critical exponent. It is clear that any metric of the form (6.1) with w = 1 breaks Lorentz invariance.

Lifshitz spacetimes received considerable attention in the literature (together with
Schrodinger geometries) because they provide a natural playground for the study of strongly coupled systems near non-relativistic critical points [78]. Recently, Lifshitz geometries have been embedded in string theory [79][80][81][82]. These solutions typically require non-trivial profile for fields other than the metric such as e.g. the dilaton.
It is an interesting fact, that Lovelock gravity admits Lifshitz solutions of arbitrary critical exponent w for special values of the Lovelock parameters [83][84]. This is to be contrasted with the Einstein-Hilbert case, where additional matter fields are required for the theory to support such solutions 10 . Restricting our attention to Gauss-Bonnet gravity, we find that (6.1) solves the equations of motion (3.4) as long as λ = 1 4 and L 2 = L 2 2 11 .
Note that when λ = 1 4 symmetries preserved by the gravitational action functional are enhanced to the SO(4, 2) group. The Lagrangian of (3.2) then coincides with the Chern-Simons Lagrangian for the AdS group and admits a natural supersymmetrization [85]. It is perhaps worth mentioning that the AdS solution of Gauss-Bonnet gravity for λ = 1 4 is difficult to interpret in the context of the AdS/CFT correspondence since the dual conformal field theory has vanishing central charge c.
Here we will focus on computing the entanglement entropy holographically in d = 4 dimensional field theories with Lifshitz scaling. When the time coordinate is the only coordinate with a different scaling, the computation of entanglement entropy is identical to the one in AdS spacetime. Therefore the results of the previous sections go through unmodified as long as we make the substitution λ → 1 4 and L AdS → L √ 2 . In the following we will consider the more interesting case where rotational invariance is broken and instead of the time coordinate, one of the spatial coordinates e.g. x 1 , scales like x 1 → κ w x 1 . In particular, we will compute the entanglement entropy of an infinitely extended slab and a cylinder. Generalization to higher dimensions is straightforward. 10 Asymptotically Lifshitz black hole solutions however do not exist in pure Lovelock gravity [83]. 11 For completeness, we remark that Gauss-Bonnet gravity (for particular values of the parameter λ) admits Lifshitz solutions of the most generic form where several boundary coordinates scale like x i → κ w x i . However, the anisotropic scaling is a relative notion and thus, in d = 4 there are effectively only two physically distinct cases. The first case, with a single coordinate of anisotropic scaling, is discussed above. The second case, of a Lifshitz spacetime with two coordinates scaling anisotropically, also satisfies the equations of motion of Gauss-Bonnet gravity as long as λ = 1+w+w 2 12w and L 2 = 1+w+w 2 6 L 2 . We will not address the latter case in the following which can be studied in a similar manner.

Entanglement entropy of an infinite belt.
Consider double Wick rotation of (6.1), Now there are two distinct orientations for the belt geometry, depending on whether its width (smallest size) extends along the direction with w-scaling or not. Here we concentrate on the case where the slab is infinitely extended along the direction with anisotropic scaling.
So let us consider a stripe, infinitely extended along directions x 1 , x 3 and width y along x 2 . It is convenient to parametrize the bulk surface by a single function x 2 (r). The induced metric and curvature of the surface are Substituting (6.4) into (4.2) results in a Lagrangian independent of x 2 (r) which leads to the following equation of motion Note that r ⋆ is related to the width of the slab through where the integral I 0 is expressed in terms of the dimensionless variable τ = r r ⋆ . and the function h(τ ) = ∂x 2 ∂r 2 .
Expressed in terms of h(τ ) eq. (6.5) has three solutions. Only one of them is real and continuously connected to the w = 1 case. Again we will restrict our attention to this case. Evaluating (4.2) on the particular solution of (6.5) yields the entanglement entropy where γ is a numerical constant equal to γ = 4I 2 0 × −w 2 −w w+1 − I . I 0 is defined in (6.6) while I denotes the following integral Let us discuss eq. (6.7). First of all, we find that the entanglement entropy is proportional to the boundary area l 1 × l 3 of the belt. This is in accordance with expectations from field theory considerations [86]. We see that the logarithmically divergent term is absent just like in the relativistic case. The power of the leading divergent term depends on the critical exponent w and is fixed by dimensional analysis. The coefficient of the second term in (6.7) which is physical and assumed to measure the total degrees of freedom of the system, scales in the same way the cutoff scales. It is interesting to note that the sign of the leading divergent term is negative for w > 0 in contrast to the relativistic case. This is not in contradiction with the results of the previous sections for Gauss-Bonnet gravity, because for w = 1, i.e., AdS space, the central charge c of the dual theory vanishes and a becomes negative (recall that Lifshitz solutions exist for λ = 1 4 ). For this reason the case w = 1 appears to be unphysical. Nevertheless, the leading divergent term in the entropy does not have a physical meaning and its overall sign is immaterial. We expect that a proper treatment of the boundary term would give a positive result but do not pursue this issue further here. Similar results for the entanglement entropy of a slab in a Lifshitz background were obtained in [87].

Entanglement entropy of an infinitely long cylinder.
Another interesting case to consider is that of an infinite cylinder extended along a direction with anisotropic scaling. To make contact with sections 4.2 and 4.3 we express the Lifshitz metric in the following form The induced metric and scalar curvature (in units L √ 2 = 1) of the surface are where primes denote differentiation with respect to ρ. Combining (4.2) and (6.10) determines the equation of motion for r(ρ) which takes a rather complicated form but remains second order in derivatives. Studying the equation of motion in the vicinity of the boundary ρ = 0 we were able to eventually determine the exact solution. We find that Note that (6.11) is independent of the critical exponent w 12 . Evaluating the action (4.2) for λ = 1 4 on the solution (6.11) leads to where l z regularizes the length of the cylinder. Performing the integral in (6.12) determines the entanglement entropy to be The computation of the entanglement entropy makes sense only for w > 1 13 . Recall that Lifshitz geometries are unstable for w < 1 (see for example [88]). We see a manifestation of this fact here through the computation of entanglement entropy. As expected, the leading divergence is proportional to the area 2πl z R whereas the scaling of the cutoff is fixed by dimensional analysis. Note that the logarithmic divergence, characteristic of the entanglement entropy on the cylinder in relativistic theories, is absent here. It would be interesting to examine this point further from the field theory point of view. 12 It is interesting to expand the solution close to the boundary ρ = 0 as r(ρ) = R − ρ 2R − · · ·. Note that the solution is not equal to next leading order to the embedding function for the case of a cylinder in Gauss-Bonnet gravity (section 4.2 eq. (4.14)). This shows once more that the order of limits λ → 1 4 and w → 1 cannot be interchanged. 13 We do not consider the case w = 1 since its physical meaning is ambiguous as explained earlier.
7.1. The entanglement entropy of a ball along the RG flow.
To compute the entanglement entropy of a ball of radius R we will take steps similar to the conformal case. Writing the domain wall metric as we see that the ansatz r(ρ) is still natural. The induced metric is then and substituting into (4.2) we arrive at where R = R(r(ρ), U (ρ)) is the scalar curvature of the induced three dimensional surface in units of the asymptotic AdS radius in the UV, L. The equation of motion for r(ρ) derived from (7.6) is rather complicated. In the vicinity of the boundary however it is solved by r(ρ) = R − ρ 2R + · · · exactly like the case λ = 0, U (ρ) = 0. Knowledge of the near boundary behavior of r(ρ) together with (7.3) allows us to determine the divergent terms in the entanglement entropy of a ball (7.7) Here α = L 2 L 2 . To leading order in the vicinity of ρ = 0 it is given by (3.7). The coefficient of the logarithmically divergent term is modified compared to the pure AdS case by the overall factor 1 − 2β 1 R 2 1−2αλ 1−6αλ and α given in (3.7). Since 1−2αλ 1−6αλ is positive, the behavior of the coefficient of the logarithmic term under rescalings of the radius of the ball R → ΛR depends on the sign of β 1 .

The entanglement entropy of a cylinder along the RG flow.
The case of a cylindrical surface is dealt with in a similar manner. We write the domain wall solution as and select an ansatz of the form r(ρ) which leads to the following induced metric We subsequently substitute into (4.2) and derive an equation of motion for r(ρ) which we solve near the boundary ρ = 0. The solution r(ρ) = R − ρ 4R + O(ρ 2 ), together with (7.3), help us obtain the divergent terms for the entanglement entropy of a cylinder (7.10) Notice that the coefficient of the logarithmic term differs by a factor equal to 1 − 8β 1 R 2 compared to the conformal case. Depending on the sign of β 1 , the overall coefficient behaves in exactly the same way for both the ball and the cylinder. It will be interesting to understand the implications of this statement.
Here we will examine the weak (or null) energy condition and investigate whether it is possible to constrain the sign of β 1 without further specifying the geometry (7.2). The nullenergy condition implies that the matter stress energy tensor T M N satisfies T M N ζ M ζ N ≥ 0 for any arbitrary null vector ζ M . Consider the most general null vector ζ M where η ij represents the four dimensional Minkowski metric. It directly follows from (7.11) that η ij ζ i ζ j ≤ 0. Substituting eq. (7.11) into the null energy condition yields Here 15 we used T x i x i = −T tt which follows from the symmetries of the metric (7.2). As we saw earlier −η ij ζ i ζ j ≥ 0 which results in T tt + 4ρe 2U(ρ) T ρρ ≥ 0.
Recall that the equations of motion (3.4) for d = 4 in the presence of matter reduce Eq. (7.13) allows the determination of T tt + 4ρe 2U(ρ) T ρρ for arbitrary matter sector 14) The energy condition implies that the left hand side is positive for any ρ, therefore for ρ = 0 as well. The Fefferman-Graham expansion (7.3) helps us evaluate (7.14) at ρ = 0 and combined with the null-energy condition leads to To leading order close to the boundary α is defined by (3.7). For the stable AdS solution of Gauss-Bonnet gravity (α − 2) > 0 fixes the sign of β 1 to be negative.
This is exactly what happens in Einstein-Hilbert gravity with a cosmological constant. It is interesting that had we chosen the unstable AdS solution we would have a positive β 1 .
We conclude that the coefficient of the logarithmically divergent term in the entropy of both the ball and the infinite cylinder is monotonically increasing with dilatations (as long as β 1 is non-vanishing). Unfortunately, this does not mean much. To apply the reasoning of [93][94] the full result, not just the logarithmic term, is required; mainly because away from the conformal fixed point the EE depends on two, rather than one length scale, i.e., the radius R of the ball and the scale of the theory µ. In the previous section we saw that the coefficient of the logarithmically divergent term in the entanglement entropy depends only on the UV data of the theory. It is therefore clear that this coefficient is not a good candidate for a function decreasing along the RG flow and being equal to a at the fixed points. This result, together with the work of Casini and Huerta [42] in two-dimensional field theories, lead us to consider instead the following quantity Q ≡ −R ∂S EE,reg ∂R → a CF T . (7.16) Here S EE,reg contains only the finite part of the entanglement entropy of a ball of radius R.
We would like to examine the monotonicity properties of Q(R) holographically. A priori, this does not seem an easy task since we cannot compute (7.16) exactly in an arbitrary domain wall geometry. However, it is possible that the null energy condition and general characteristics of the spacetime (7.2) will determine whether (7.16) behaves monotonically along renormalization group trajectories.
In the following we will attack this problem in the context of Einstein-Hilbert gravity.
This has the advantage of being technically simpler while the relevant features of the problem remain the same. To simplify the analysis we will parametrize the surface in the bulk by ρ(x i ) where x i for i = 1, 2, 3 are cartesian coordinates at the boundary. This choice yields a lagrangian independent of x i where ρ(x i ) determines the profile of the embedding surface. Translational invariance implies that the system has a conserved "stress-energy" tensor T j i T j i = ∂L ∂(∂ j ρ) (∂ i ρ) − Lδ j i (7.18) obeying ∂ i T j i = 0. Due to spherical symmetry T ij takes the form T ij = A(r) r i r j + δ ij B(r) (7.19) where A, B can be explicitly found to be A(r) = e 2U(ρ) 2ρ 2 ρ ′ (r) 2 ρ ′ (r) 2 + 4e 2U(ρ) ρ = e 2U(ρ) 2ρ 2 1 |r ′ (ρ)| 1 + 4e 2U(ρ) ρr ′ (ρ) 2 B(r) = − e 2U(ρ) 2ρ 2 ρ ′ (r) 2 + 4e 2U(ρ) ρr ′ (ρ) 2 = − e 2U(ρ) 2ρ 2 |r ′ (ρ)| 1 + 4e 2U(ρ) ρr ′ (ρ) 2 (7.20) In the first equality we have replaced the dependence of ρ on the cartesian coordinates x i with the spherical coordinate r. In the second equality we give A(r) , B(r) in terms of r(ρ) instead of ρ(r). Expressing the geometry of the embedding surface with r(ρ) will be more convenient in the following. Note that r(ρ) satisfies the conservation equation of T ij , in other words the equations of motion derived from (7.17). The conservation equation can be written in a simple form in terms of the functions A(r) and B(r) 2A(r) + r d dr (A(r) + B(r)) = 0 (7.21) Solving for r(ρ) in the vicinity of the UV with the help of (7.3) and (7.20) yields where σ 1 is a function of the parameters (β 1 , β 2 , β 3 ) which specify the domain wall geometry but also of σ 3 , the coefficient of a higher order term in the near boundary expansion of r(ρ) Unfortunately, σ 1 cannot be determined without knowledge of the exact form of the profile function r(ρ). This will turn out to be the main difficulty in determining the behavior of Q(R) under rescalings of R.
Now suppose we vary the boundary conditions in the entanglement entropy computation as follows ρ(x i + δx i ) = ǫ + δǫ (7.24) where the original boundary condition is ρ(x i ) = ǫ for x i ∈ D (here D is a sphere of radius R). The change in the on-shell action would then be equal to with dΣ i the volume element on D. Substituting (7.19) into (7.25) leads to Although we have obtained a simple expression for Q(R), we cannot determine whether it has a monotonic behavior under rescalings of the radius R. It appears that one would need to know the exact solution for the profile r(ρ). Smoothness and other generic characteristics of the geometry of the induced surface did not suffice to prove the monotonicity of Q(R).
Moreover, it is not clear how the null energy condition, pertinent to the ambient spacetime, will affect the behavior of Q(r) which depends on the details of the embedding function r(ρ). It is likely that even if Q(R) is a c-function in the sense of Zamolodchikov, it is most likely a different c-function than the standard holographic one -which in the coordinates used here is expressed as c hol ∼ 1 (−2U ′ (ρ)ρ+1) 3 . Finally, the two main ingredients in the proof of [93][94] in two dimensions, are Lorentz invariance and the strong subadditivity property of the entanglement entropy. To make use of the latter, it might be more appropriate to consider the geometry of an annulus.
We leave this issue to future investigation.