Probing renormalization group flows using entanglement entropy

In this paper we continue the study of renormalized entanglement entropy introduced in [1]. In particular, we investigate its behavior near an IR fixed point using holographic duality. We develop techniques which, for any static holographic geometry, enable us to extract the large radius expansion of the entanglement entropy for a spherical region. We show that for both a sphere and a strip, the approach of the renormalized entanglement entropy to the IR fixed point value contains a contribution that depends on the whole RG trajectory. Such a contribution is dominant, when the leading irrelevant operator is sufficiently irrelevant. For a spherical region such terms can be anticipated from a geometric expansion, while for a strip whether these terms have geometric origins remains to be seen.


I. INTRODUCTION AND SUMMARY
In renormalizable field theories, the entanglement entropy (EE) for a spatial region is divergent in the continuum limit, with the leading divergence given by the so-called area law [2,3]: where δ is a short-distance cutoff, d is the number of spacetime dimensions, A Σ is the area of the entangling surface Σ, and the dots stand for less singular terms. Equation (1.1) can be interpreted as coming from degrees of freedom at the cutoff scale δ near Σ.
More generally, for a smooth Σ, one expects that local contributions (including all divergences) near Σ to the entanglement entropy can be written in terms of local geometric invariants of Σ [1,4] S (Σ) where σ denotes coordinates on Σ, F is a sum of all possible local geometric invariants formed from the induced metric h ab and extrinsic curvature K ab of Σ. For a scalable surface Σ of size R, 1 the local contribution (1.2) should then have the following geometric expansion S (Σ) local = a 1 R d−2 + a 2 R d−4 + · · · (1. 3) with the first term coinciding with (1.1). In (1.3) terms with positive exponents of R are expected to be divergent, 2 while a n is finite, when the corresponding exponent of R is negative.
The area law (1.1) and other subleading divergences in (1.3) indicate that EE is dominated by physics at the cutoff scale and thus is not a well defined observable in the continuum limit. This UV-sensitivity makes it difficult to extract long range correlations from EE. A standard practice is to subtract the divergent part by hand. This may not be sufficient to remove all the short-distance dependence, and is often ambiguous. For example, consider the entanglement entropy of a disk of radius R in the vacuum of a (2+1)-dimensional free massive scalar field theory. It was obtained in [5,6] that for mR 1, the entanglement entropy has the behavior Subtracting the divergent # R δ piece by hand, from the second term in (1.4) one finds that in the IR limit (R → ∞), the resulting expression approaches −∞. This result appears to be in conflict with the expectation that in the IR limit the system should have no correlations.
Ideally, we would have liked EE to go to zero. To understand what is going on, note that the second term in (1.4) also has the form of an area law R/δ withδ ∼ 1 m , and thus can be interpreted as coming from physics at scale 1 m , which is still short-scale physics compared to the IR scale R → ∞. A related observation is that the second term in (1.4) is in fact ambiguous in the continuum limit, as its coefficient can be modified by the following redefinition of δ δ → δ (1 + c mδ + · · ·) (1. 5) with c some constant. 1 A scalable surface can be specified by a size R and a number of dimensionless parameters characterizing the shape. 2 We assume a continuous regularization in which the size R can unambiguously defined.
In [1] we introduced the renormalized entanglement entropy (REE) (1.6) which was designed to remove all divergent terms in (1.3). It was shown there that the REE has the following desired properties: 3 1. It is unambiguously defined in the continuum limit.
2. For a CFT it is given by a R-independent constant s of the UV and IR fixed points as R is increased from zero to infinity. 4. It is most sensitive to degrees of freedom at scale R.
For example, applying (1.6) to (1.4) we find that the differential operator in (1.6) (for d = 3) removes the first two terms in (1.4) and changes the sign of the last term, resulting S 3 (R) = + π 120 1 mR + · · · , mR → ∞ (1.7) which monotonically decreases to zero at large distances as desired.
For a general quantum field theory the REE can be interpreted as characterizing entanglement at scale R. In particular, the R-dependence can be interpreted as describing the renormalization group (RG) flow of entanglement entropy with distance scale. In [1], it was conjectured that in three spacetime dimension the REE for a sphere S sphere 3 is monotonically decreasing and non-negative for the vacuum of Lorentz invariant, unitary QFTs, providing a central function for the F-theorem conjectured previously in [7,8]. The monotonic nature of S sphere 3 , and thus the F-theorem, was subsequently proved in [9]. In (1 + 1)-dimension, S 2 reduces to an expression previously considered in [10], where its monotonicity was also established. There are, however, some indications [1] that in four spacetime dimensions S sphere 4 is neither monotonic nor non-negative. 3 The differential operator (1.6) can be applied to the Rényi entropies and the following statements also apply to renormalized Rényi entropies.
More generally, regardless of whether it is monotonic, REE provides a new set of observables to probe RG flows. 4 From REE, one can introduce an "entropic function" defined in the space of couplings (or in other words the space of theories) C (Σ) (g a (Λ)) ≡ S (Σ) (RΛ, g a (Λ)) R= 1 Λ = S (Σ) (1, g a (Λ)) (1.8) where g a (Λ) denotes collectively all couplings and Λ is the RG energy scale. Given that S (Σ) is a measurable quantity, it should satisfy the Callan-Symanzik equation which leads to Λ dC (Σ) (g a (Λ)) dΛ = −R dS (Σ) (RΛ, g a (Λ)) dR (1.10) The R-dependence of S (Σ) is translated into the running of C (Σ) (g(Λ)) in the space of couplings, with R → 0 and R → ∞ limits correspond to approaching UV and IR fixed points of RG flows. At a fixed point g * , C (Σ) (g * ) = s (Σ) d and the monotonicity of S (Σ) with respect to R translates to the monotonicity of C (Σ) with respect to Λ.
For Σ being a sphere, some partial results were obtained earlier in [1,11] for the small and large R behavior of REE (or equivalently for C near a UV and IR fixed point) in holographic theories. From now on we will focus on a spherical region and suppress the superscript (Σ) on S and C. For a (UV) fixed point perturbed by a relevant operator of dimension ∆ < d, it was found that where µ is a mass scale with the relevant (dimensional) coupling given by g = µ d−∆ , and A(∆) is some positive constant. The above equation leads to an entropic function given by where g ef f (Λ) = gΛ ∆−d is the effective dimensionless coupling at scale Λ. Equation (1.12) has a simple interpretation that the leading UV behavior of the entropic function is controlled by the two-point correlation function of the corresponding relevant operator. We expect this 4 See [12,13] for other ideas for probing RG flows using entanglement entropy.
result to be valid also outside holographic systems. This appears to be also consistent with general arguments from conformal perturbation theory [14]. It is curious, however, that low dimensional free theories defy this expectation. For example in d = 2, as R → 0 [15] free scalar : Dirac fermion : while for a d = 3 free massive scalar [16] ruled out the m 4 R 4 short distance behavior based on numerics. 5 Near an IR fixed point, it was argued in [1] that the large R behavior of S(R) should have the form  In addition to domain wall geometries, we also consider a class of geometries, which are singular in the IR. These correspond to either gapped systems, or systems whose IR fixed point does not have a gravity description (or has degrees of freedom smaller than O(N 2 )).
We will see that for these geometries the asymptotic behavior of REE provides a simple diagnostic of IR gapless degrees of freedom.
While in this paper we focus on the vacuum flows, the techniques we develop can be used to obtain the large R expansion of the entanglement entropy for generic static holographic geometries, including nonzero temperature and chemical potential. As an illustration we study the behavior of extremal surfaces in a general black hole geometry in the large size limit. We also show that, in this limit, for any shape of the entangling surface the leading behavior of the EE is the thermal entropy. While this result is anticipated, a general holographic proof appears to be lacking so far.
For d = 2, 3, the monotonicity of S d in R leads to a monotonic C d in coupling space, 6 Since here we consider the R → ∞ limit s n should thus depend on the full RG trajectory from δ to ∞.
i.e. C d is a c-function. Equations (1.13)-(1.14) show that for a free massive field, C 2 is not stationary near the UV fixed point, and neither is C 3 for a free massive scalar field, as pointed out in [16]. From (1. 16) we see that C d is in fact generically non-stationary near an IR fixed point for∆ − d > 1 2 (∆ − d > 1) for odd (even) dimensions. The physical reason behind the non-stationarity is simple: while the contribution from degrees of freedom at short length scales are suppressed in S d , they are only suppressed as a fixed inverse power of R, and are the dominant subleading contribution, when the leading irrelevant operator is sufficiently irrelevant. The non-stationarity of S (or C) is independent of the monotonic nature of S (or C) and should not affect the validity of c-or F-theorems. In contrast to the Zamolodchikov c-function [17], which is stationary, in our opinion, the non-stationarity of C should be considered as an advantage, as it provides a more sensitive probe of RG flows.
For example, from (1.16) by merely examining the leading approach to an IR fixed point, one could put constraints on the dimension of the leading irrelevant operator.
While in this paper we will be mainly interested in taking the entangling surface to be a sphere of radius R, for comparison we also examine the IR behavior for a strip. Since the boundary of a strip is not scalable, the definition (1.6) has to be modified. Consider a strip where for convenience we have put other spatial directions to have a finite size → ∞. Note that due to translational symmetries of the entangled region in x i directions, the EE should have an extensive dependence on , i.e. it should be proportional to d−2 . Furthermore, for the boundary of a strip the extrinsic curvature and all tangential derivatives vanish. Hence we conclude that the only divergence is the area term (1. 19) In particular, the divergent term should be R-independent. This thus motivates us to consider R dS dR , which should be finite and devoid of any cutoff dependent ambiguities. Given that all the dependence in S on comes from the over factor d−2 , it is convenient to introduce dimensionless quantity R d defined by This quantity was considered earlier in [18,19]. For a CFT there is no scale other than R, hence R d should be a R-independent constant, which can be readily extracted from expressions in [19,20]. For a general QFT, R d should be a dimensionless combination of R and other possible mass scales of the system.
Calculating R d for a domain wall geometry describing flows among two conformal fixed points, we find an interesting surprise. The second line of (1.15) can be understood from a local curvature expansion associated with a spherical entangling surface. Such curvature invariants altogether vanish for a strip and thus one may expect that for a strip only the first line of (1.15) should be present. We find instead find that R d has the large R behavior It would be interesting to see, whether it is possible to identify a geometric origin for the terms in the second line.
The paper is organized as follows. In Sec. II we discuss the holographic geometries to be considered, and outline a general strategy to obtain the large R expansion of REE for a spherical region for generic holographic geometries. In Sec. III we consider holographic theories which are gapped or whose IR fixed point does not have a good gravity description.
In Sec. IV we elaborate more on the physical interpretation of such geometries and consider some explicit examples. In Sec. V we consider domain wall geometries with an IR conformal fixed point. We conclude in Sec. VI with some applications of the formalism to the black hole geometry.

II. SETUP OF THE CALCULATION AND GENERAL STRATEGY
In this section we describe the basic setup for our calculations and outline the general strategy.
A. The metric The RG flow of a Lorentz-invariant holographic system in the vacuum can be described by a metric of the form where L is the AdS radius and near the boundary The null energy condition requires f to be monotonically increasing. The IR behavior, as z → ∞ can then have the following two possibilities: In this case, the IR geometry is given by AdS with radius L IR < L, and thus the system flows to an IR conformal fixed point. Near the IR fixed point, i.e. z → ∞, f can be expanded as 2. The spacetime becomes singular at z = ∞: f (z) = az n + · · · , a > 0, n > 0 . (2.5) Due to the singularity at z = ∞, one might be concerned, whether one could trust the holographic entanglement entropy obtained in such a geometry. We will see, however, that the results obtained in this paper only depend on the existence of the scaling behavior (2.5) for a certain range of z and are insensitive to how the singularity at z = ∞ is resolved.
Since n > 0, the singularity lies at a finite proper distance away and the naive expectation is the corresponding IR phase should be gapped. As we will discuss later, it turns out this is only true for n > 2, an example of which is the GPPZ flow [21]. For n < 2, the story is more intricate and there exist gapless modes in the IR. Below we will refer to n < In our subsequent discussion we will assume that there exists a crossover scale z CO such that (2.4) or (2.5) is valid for While in this paper we will be focusing on vacuum solutions (i.e. with Lorentz symmetry), since the holographic computation of the entanglement entropy for a static system only depends on the spatial part of the metric [20], the techniques we develop in this paper for calculating the large R behavior of the REE also apply to a more general class of metrics of the form This is in fact the most general metric describing a translational and rotational invariant boundary system including all finite temperature and finite chemical potential solutions. g does not directly enters the computation of the REE. Its presence is felt in the more general behavior allowed for f ; the null energy condition no longer requires f to be monotonically increasing. For example, for a black hole solution f decreases from the boundary value 1 to zero at the horizon. The null energy condition also allows n < 0 in (2.5) for certain g. One such example is the hyperscaling violating solution [22][23][24][25][26] (at T = 0), where the metric functions have the scaling form We will discuss the black hole case in section VI.

B. Holographic Entanglement entropy: strip
We first discuss the holographic entanglement entropy of the strip region (1.18). It is obtained by minimizing the action: where G N is the bulk Newton constant and A is the area functional [20,27]. If the spacetime is singular, as in the case of (2.5), the minimal surface can become disconnected. In this case, the minimal surface consists of two disconnected straight planes x(z) = ±R. The minimal surface area is independent from R due to the translational symmetry of the problem. If the surface is connected, its area is given by . (2.10) The shape of the entangling surface is specified by the boundary conditions Since the action has no implicit dependence on x, we have an associated conserved quantity: This reduces the equation of motion to first order: where z t = z(x = 0) gives the tip of the minimal surface. z t is determined by requiring (2.14) Inverting this implicit equation gives the relation z t (R). Using (2.13) we can also write (2.10) where δ is a UV cutoff.
Expanding (2.13) near the boundary z = 0, we find the expansion (2.16) Varying (2.10) with respect to R and using (2.16), we find that Thus to find R d it is enough to invert (2.14) to obtain z t (R). (2.18) was obtained before in [18].

C. Holographic Entanglement entropy: sphere
Writing d x 2 = dρ 2 +ρ 2 dΩ 2 d−2 in polar coordinates, the entanglement entropy for a spherical region of radius R can be written as where ω d−2 is the area of a unit (d − 2)-dimensional sphere and A is obtained by minimizing the surface area where z t denotes the tip of the minimal surface. The boundary conditions are As discussed in [1], for (2.5) it is also possible for the minimal surface to have the cylinder topology, for which z t = ∞ and the IR boundary conditions become with ρ 0 a finite constant. The equation of motion can be written as In general, ρ(z) can be expanded near the boundary in small z as where all coefficients except c d (R) can be determined locally (or in terms of c d ). One can where · · · denotes non-universal terms which drop out when acted on with the differential operator in (1.6), and e d is a constant, which is nonvanishing only for d = 4, 8, · · · .
Using (2.26) one can express the REE (1.6) in terms of c d (R). For example, for d = 3 where C is determined by requiring that S 3 (R = 0) reduces to the value at the UV fixed point, and for d = 4, (2.28) One could also obtain S d by directly evaluating the action (2.20) and then taking the appropriate derivatives (1.6).

D. Strategy for obtaining the entanglement entropy for a sphere
In general it is not possible to solve (2.23) or (2.24) exactly. Here we outline a strategy to obtain the large R expansion of S(R) (or S d (R)) via a matching procedure: Note that the above expansion applies to the vacuum. For a black hole geometry one should include all integer powers of 1/R as we will discuss in more detail in section VI.
The expansion (2.29) should be considered as an ansatz, motived by (1.15) one wants to show, but should be ultimately confirmed by the mathematical consistency of the expansion itself (and the matching described below).
Depending on the IR behavior of a system, the large R expansion (2.29) can contain terms which are not odd powers of 1/R. We have denoted the exponent of the first such term in (2.29) as ν, whose value will be determined later. The expansion is valid       From (2.29) we see that c d (R) in (2.25) takes the following expansion where b n andb are some R-independent constants. It follows from (2.26) and (1.6) that a term proportional to 1/R n in (2.30) contributes to S d a term of order 1/R n−d+1 , whose We now examine more explicitly the UV expansion (2.29) for the sphere, which is the same for all geometries of the form (2.1). The IR expansion and matching will be discussed in later sections case by case.
The equation for ρ i (z) can be written as where s i denotes a source from lower order terms with, for example, The equation for ρ 1 can be readily integrated to give where b 1 is an integration constant and ρ hom is the homogenous solution to (2.32) In particular because its unique R-dependence there are no source terms forρ(z), thus it takes the form:ρ As z → 0, ρ 1 andρ has the leading behavior (for d ≥ 2) Note that the normalization of ρ hom in (2.36) was chosen such that the contribution to c d (R), read off from (2.37), gives the term appearing in (2.30).

III. GAPPED AND SCALING GEOMETRIES
In this section we consider the large R behavior of the REE for holographic systems, whose IR geometry is described by (2.5). As mentioned below (2.5) there is an important difference between n > 2 and n ≤ 2, to which we refer as gapped and scaling geometries respectively. For comparison we will treat them side by side. We will first consider the strip and then the sphere case.

A. Strip
In (2.14) to leading order in large z t , we can replace f (z) in the integrand by its large z behavior f (z) = az n , leading to For small z t we can replace f (z t v) in (2.14) by 1 and thus For n > 2, the function R(z t ) then goes to zero for both z t → 0 and z t → ∞, and thus must have a maximum in between at some z (max) t . Introducing we conclude that for R > R max there is no minimal surface with strip topology. Instead, the minimal surface is just two disconnected straight planes x(z) = ±R. The minimal surface area is independent from R due to the translational symmetry of the problem. We conclude that for n > 2 in the R → ∞ limit S becomes independent of R, hence R d (R > R max ) = 0.
For n = 2, R(z t ) → const at large z t , and again in this case there is no minimal surface of strip topology and R d (R > R max ) = 0.
For n < 2, inserting (3.1) into (2.18) we find that This result also applies to a hyperscaling violating geometry (2.8), and agrees with the scaling derived in [26].

B. Sphere
Since for d = 2, the sphere and strip coincide (the answer is then given by (3.5)), we will restrict our discussion below to d ≥ 3.

IR expansion
We first consider the behavior of the minimal surface in the IR geometry (2.5). Plugging f (z) = az n into (2.23) we notice that ifρ(z) satisfies the resulting equation with a = 1, then Solutions of two different topologies are possible. As discussed in [1], for n > 2, in the large R limit the minimal surface has the topology of a cylinder, while for n ≤ 2, the minimal surface has the topology of a disk. See Fig. 2(a) and Fig. 2(b).
For a solution of cylinder topology (i.e. for n > 2) the IR solution satisfies Introducing a solutionρ c (z) to (2.23) with a = 1, which satisfies the condition we can write a general ρ(z) in a scaling form From (2.23), ρ(z) has the large z expansion (see also Appendix C of [1]) For a solution of disk topology (i.e. for n ≤ 2), there should exist a z t < ∞, where Now introducing a solutionρ d (z) to (2.23) with a = 1, which satisfies the boundary condition ρ d (1) = 0, we can write ρ(z) in a scaling form (3.14) Note that by taking z t sufficiently large, u can be small even for z z CO , where (2.5) applies. Expandingρ d in small u one finds that where η was introduced in (3.2) and , · · · .
This is all the information we need about the IR solution. Note that the above expansion applies to the range of z, which satisfies The small u expansion (3.15) is singular for n = 2, as can be seen from (3.16). Hence the n = 2 case should be treated separately, see Appendix A.

Matching
We first examine the UV solutions (2.34) and (2.36) for a sufficiently large z so that (2.5) applies. At leading order in large z, we then find that can be extended to arbitrary z without breaking down, which can be verified by showing that higher order terms are all finite for any z. This is also intuitively clear from Fig. 2(a) where for large R the minimal surface has a large radius at any z. Note that, sinceb = 0, the non-integer ν term in (2.29) is not present.
For n < 2, where the minimal surface has the topology of a disk, the UV expansion is destined to break down at certain point before the tip of the minimal surface is reached. In the region (3.19) both the IR and UV expansions apply, and by comparing (3.20) and (3.21) with (3.17), we find that they match precisely provided that From (3.18) we conclude that z t scales with R as Again, the story for n = 2 is discussed in Appendix A with equation (3.24) replaced by (3.25)

Asymptotic expansion of the REE
We will now obtain the leading order behavior of the REE in the large R limit.

n > 2
Let us first consider n > 2. From the discussion below (2.30), we expect the leading order term for odd d to be proportional to 1/R, which comes from the 1/R d term in the expansion of c d . For even d, the leading term can in principle be 1/R 0 , which comes from the 1/R d−1 term in the expansion of c d . Note, however, since this a gapped system, we expect the order 1/R 0 term to vanish. So, for even d, the leading term should come from the 1/R d+1 term.
Since even for d = 3 we would need to know c 3 (R) to 1/R 3 order, and we only worked out ρ 1 (which only determines c 3 (R) to 1/R), our results seem insufficient to determine the 1/R contribution to S 3 . However, the 1/R contribution to S 3 can be obtained by directly evaluating the on-shell action [11], as the 1/R piece is the next to leading term in the large R expansion of S. For d = 4, we can use ρ 1 to verify that the 1/R 0 term (in the REE) vanishes as expected for a gapped system. With due diligence, it is straightforward to work out higher order terms, but will not be attempted here.
For d = 3, plugging (2.29) into (2.20) we have the expansion where in the second line we have used (2.32). Integrating by parts the second term in the integrand we find that where the boundary terms vanishe due to (2.37) and (3.12). We thus find that (3.28) It is desirable to make work with dimensionless coefficients that only depend on ratios of scales. We can useμ ≡ a 1/n (3.29) as an energy scale and define the dimensionless coefficient where all integration variables are dimensionless, and s 1 only depends on ratios of scales, e.g. (μ z CO ). Finally, we obtain This result agrees with those in [11]. It is interesting to note that the coefficient of 1/R term depends on the full spacetime metric, i.e. in terms of the boundary theory, the full RG trajectory.
For d = 4, the expansion of A has the form with δ a UV cutoff. Neither of the first two terms indicated in (3.32) will contribute to S 4 after differentiations in (1.6). As expected, a 0 ∼ 1/δ 2 is UV divergent. a 2 contains a logarithmic UV divergence log δµ, where µ is mass scale controlling the leading relevant perturbation from the UV fixed point. At large z, from (3.20) and (3.22) ρ 1 ∼ z 2−n , hence the integrand for a 2 goes as ∼ z −1−3n/2 , and the integral is convergent at the IR end. An IR divergent a 2 would signal a possible log R term. Thus we conclude that the leading order contribution for d = 4 is of order 1/R 2 , consistent with our expectation that the system is gapped.

n ≤ 2
For n < 2,b in (2.30) is nonzero and its contribution to S d can be directly written down from (3.23) where η and β were defined in (3.2) and (3.6) respectively, and (3.35) For n = 2 the first term in (3.34) should be replaced by (see (A9) and Appendix A) (3.36) Below for convenience we will refer to the first term in (3.34) (or (3.39)) as "non-analytic", while terms of inverse odd powers in odd dimensions (and even inverse powers in even dimensions) as "analytic." Note that the non-analytic term is the leading contribution in the large R limit when (3.37) in which case one can check that the coefficient e n is positive. In Fig. 3 we plotted e n for d = 2, 3 and 4. Note that for odd d, e n diverges as n → n c , while for even d it stays finite. 7 Despite appearances the numerical factors multiplyingh in (3.35) do not diverge at n = n c , hence the features described in Fig. 3 are caused byh.  Let us consider the n → n c limit of (3.34) for odd d. Because e n diverges as n → n c , in order for (3.34) to have a smooth limit, we expect the coefficient of the 1/R term in (3.34) 7 For d = 2 apart from the numerical results, we can analyze the analytic answer given in (3.5).
to diverge too, in a way that the divergences cancel resulting in a logarithmic term The coefficient of the logarithmic term is given by the residue of (3.35) in the limit n → n c .
In contrast, for even d, e n is finite at n = n c . Thus, the leading term will simply be of order 1/R 2 with no logarithmic enhancement (there can still be logarithmic terms at higher orders).
For d = 3 one can calculate the coefficient of 1/R term in (3.34) similar to n > 2 case discussed. See Appendix B for a derivation. One finds where s 1 is given by (3.28) for n > 2 3 , and for n < 2 3 by (3.40) In this case, we can work out explicitly how the divergence in the limit n → n c cancels between the coefficients of the analytic and non-analytic pieces. Note that the divergence in s 1 comes from the second term in the integrand in (3.40) 8 The numerical results presented in Fig. 3 are consistent with the behavior to 1% precision. Plugging into (3.39) then gives We can perform the same calculation with n = 2/3 fixed from the beginning, and we get the same result, see (B14).
We now briefly summarize the results by comparing between the strip and the sphere, and between n < 2, n > 2 and n = 2 geometries.
The presence of analytic terms for the sphere can be expected from the general structure of local contributions to the entanglement entropy [1,4], which implies the existence of terms of the form 1/R + 1/R 3 + · · · for odd dimensions and 1/R 2 + 1/R 4 + · · · for even dimensions. For n < 2 geometries, non-analytic terms are present for both the strip and the sphere, and have the same scaling. We note that the non-analytic terms (including the coefficients) are solely determined by the IR geometry. From the boundary perspective they can be interpreted as being determined by the IR physics. The presence of these non-analytic terms (despite the fact that they could be subleading compared to analytic terms) imply that the IR phase described by (2.5) is not fully gapped, and some IR gapless degrees of freedom are likely responsible for the non-analytic scaling behavior. For this reason we refer to such geometries as scaling geometries. Note that due to the singularity at z = ∞, we should view the region (2.5) as describing an intermediate scaling regime. It likely does not describe the genuine IR phase, which depends on how the singularity is resolved. Thus our discussion above should be interpreted as giving the behavior of S(R) for an intermediate regime.
We will see some explicit examples in the next section.
In contrast for n > 2, there is no non-analytic term and we expect the dual system to be fully gapped in the IR.
For n = 2 the strip and sphere entanglement entropies show different behaviors as emphasized recently by [28]. For R → ∞ the minimal surface for a strip is disconnected, and hence there is no non-analytic term in the expansion of R d . However, for a spherical entangling 9 In (3.28), (3.40) the upper limits of the integrals are ∞, as we are considering R → ∞ limit.
surface the topology of the minimal surface is a disc, and S d contains an exponentially small term (3.36). In next section, by examining the spectral function of a scalar operator, we argue that an n = 2 geometry describes a gapped phase, but with a continuous spectrum above the gap.

IV. MORE ON SCALING GEOMETRIES
In this section, we discuss further the properties of a scaling geometry with n ≤ 2 by examining the behavior of a probe scalar field. We show that the system has gapless excitations in the IR. We emphasize that here the term IR is used in a relative sense, i.e.

A. Correlation functions
Consider a probe scalar field in a spacetime (2.1) with (2.5). A similar analysis was done in [29] for two specific flows in d = 4 dimensions with n = 3 and n = 2 respectively, 10 and more recently in [26] in the context of hyperscaling violating geometries.
The field equation for a minimally coupled scalar in momentum space can be written as where k µ is the energy-momentum along the boundary spacetime directions and k 2 = η µν k µ k ν .
First, consider the gapped case, corresponding to n > 2. For z → ∞ the two allowed 10 There the scalar fields of interest mixed with the metric, here we assume no mixing.
behaviors for the scalar field are: where we have set a = 1 for simplicity of notation. The null energy condition requires that n/2 < d [26], hence only φ + is regular. Near z → 0, the normalizable solution φ norm (z) can be written as a linear superposition of φ ± , i.e. φ norm (z) are some functions of k 2 . Requiring both regularity at z → ∞ and normalizability at the boundary then leads to A − (k) = 0, which implies that the system has a discrete spectrum.
This is in agreement with the findings of [29] in specific examples, and is consistent with our discussion at the end of last section that such a geometry should be describe a gapped theory.
For n = 2, in the scaling region (4.1) can be solved analytically where I is the modified Bessel function of the first kind. For k 2 < −∆ 2 , ν is imaginary and φ ± behave as plane waves near z → ∞. Then following the standard story [30], choosing an infalling solution leads to a complex retarded Green function and a nonzero spectral function. We thus conclude that in this case, there is nonzero gap ∆ = d−1 2 and the system has a continuous spectrum above the gap. The presence of a continuum above a gap is presumably responsible for the exponential behavior (3.36) in the entanglement entropy. Now we consider n < 2. For k 2 < 0 and z → ∞, the solutions to (4.1) have the "plane wave" form Thus in this case one finds a continuous spectrum all the way to k 2 → 0 − . The corresponding spectral function can be extracted from [26] ρ( This continuous spectrum should be the origin of the "non-analytic" behavior in (3.34) for a sphere and (3.5) for a strip. It is also interesting to note that the exponents β in (3.5), (3.34) and γ in (4.6) satisfy a simple relation It would be interesting to understand further the origin of such a relation.

B. Explicit examples: near horizon Dp-brane geometries
We now consider the near-horizon Dp-brane geometries [31], which exhibit the scaling geometry (2.5) in some intermediate regime. EE in these geometries was analyzed previously in [26]. While these geometries are not asymptotically AdS, our earlier result for the nonanalytic term in (3.34) is nevertheless valid, since it only relies on the geometry of the scaling region. We will focus on this leading non-analytic contribution in 1/R.
The near horizon extremal black p-brane metric in the string frame can be written as where g and l s are the string coupling and string length respectively. As we will only be interested in the qualitative dependence on R and couplings, here and below we omit all numerical factors. We will restrict our discussion to p ≤ 5, for which a field theory dual exists. After dimensional reduction and going to the Einstein frame, the metric can be written as (see also [26]) which is of the same form as (2.1) and (2.5) with Now plugging (4.11) into the "non-analytic" term in (3.34) for sphere (or similarly (3.5) for strip) we find that where λ ef f (R) is the effective dimensionless t' Hooft coupling at scale R, In terms of λ ef f equation (4.13) can also be written as (4.16) For p = 1, 2, λ ef f increases with R but appears in S with a negative power. For p = 4, the opposite happens. In all cases S decreases with R. The p = 5 case, for which n = 2, has to be treated differently and one finds from (3.36) . (4.17)

V. DOMAIN WALL GEOMETRY
We now consider the large R behavior of the REE for holographic systems, whose IR geometry is described by (2.3), i.e. the system flows to a conformal IR fixed point. We will again consider the strip story first.

A. Strip
Again we start with (2.14) which can be written as . (5. 2) The leading behavior in large z t limit of the integral in (5.1) depends on the value of and find .

(5.6)
Note thatb d is positive for any d > 1. Forα ≥ d 2 , the term on RHS of (5.4) leads to a divergence in (5.1) near v = 0 and should be treated differently. 11 In particular, the divergence indicates that the leading contribution should come from the integration region v 1. We will thus approximate the factor 1/ √ 1 − v 2(d−1) in the integrand of (5.1) by 1, leading to Inverting (5.5) and (5.7) we find from (2.18) .

(5.9)
We discuss the physical implication of this result in Sec. V C.

B. Sphere
With (2.3) as z → ∞ the system flows to a CFT in the IR, and, as discussed in [1], to leading order in the large R expansion the REE S d approaches a constant, that of the IR CFT. Here we confirm that the subleading terms have the structure given in (1.15).

IR expansion
Since the IR geometries approaches AdS, in the large R limit the IR part of the minimal surface should approach that in pure AdS. In particular, in the limit R → ∞, we expect most part of the minimal surface to lie in the IR AdS region, hence the IR solution z(ρ) can be written as z 0 (ρ) is the minimal surface with boundary radius R in a pure AdS with f = f ∞ . z 1 and · · · in (5.10) denote subleading corrections which are suppressed compared with z 0 by some inverse powers of R. Below we will determine the leading correction z 1 (ρ) by matching with the UV solution.
Plugging (5.10) into (2.23), and expanding to linear order in z 1 , we find that where the source term s(ρ) is given by The homogenous equation, obtained by setting s(ρ) to zero in (5.11), has the following linearly independent solutions .

(5.15)
Note that there is an expression for φ 2 in terms of hypergeometric functions for all dimensions, but we find it more instructive to display explicit expressions in various dimensions.
The final results will be written down in general d.
In order for z(ρ) to be regular at ρ = 0, z 1 should be regular there, and can be written as where c is an integration constant. Note that the first integral above is convergent in the upper integration limit only forα < 1. Forα ≥ 1 some additional manipulations are required. For example for 1 <α < 2, we should replace the first integral by where c 1 is the numerical constant appearing in the limit φ 2 (r) W (r) s(r) → c 1 µ 2α Rα(R−r)α + · · · as r → R, and is given by Forα > 2 further subtractions may be needed. We will not write these separately, as they are irrelevant for our discussion below.
where the · · · on the right hand side of the second inequality includes all other scales of the system. The UV expansion we discussed earlier in Sec. II E applies to the region δ 1.
Thus the IR and UV expansions can be matched for ρ satisfying (5.20).
Let us now consider the behavior of (5.17) in the overlapping region (5.20). The first integral gives where for allα The second integral in (5.17) gives where Putting the two expansions together we get: where One could consider the next order in the IR expansion, i.e. including a z 2 in (5.10).
The equation for z 2 only differs from (5.11) by having a different source term, and the corresponding terms in (5.25) coming from the source will be proportional to (μR) −4α .
Similarly, the corresponding terms at the nth order are proportional (μR) −2nα .
Now including z 0 in the region (5.20), we have the expansion which can be considered as a double expansion in z/R and 1/(μz) and This above expression agrees with that obtained in [1] for two closely separated fixed points, which we review and extend in Appendix E. As discussed in the Introduction this can be anticipated on the grounds that the coefficient of the non-analytic term should depend only on the physics at the IR fixed point.
As discussed earlier our UV expansion (2.29) was designed to produce the second line of (1.15), and the fact that the UV expansion is consistent with the IR expansion confirms the second line of (1.15).
In d = 3 using ρ 1 and z 1 obtained in last subsection we can obtain the coefficient of 1/R term by directly evaluating the action as we have done for the gapped and scaling geometries. The calculation is given in Appendix D. The final answer is: where s 1 is given by (D11): The expressions for smaller values ofα are similar but require more subtractions. s 1 (and the integration variables, z and v) is dimensionless, hence only depends on ratios of RG scales.
As a consistency check, we apply these formulae to closely separated fixed points in Appendix E. We recover (E6) that is obtained using different methods. Another consistency check is that the f ∞ → ∞ limit of (5.39) recovers s 1 for the scaling geometries (3.40). This had to be the case, as a scaling geometry can be viewed as a limit of domain walls with increasing f ∞ .

C. Discussion
We conclude this section making a comparison between the result for the strip (1.21), (5.9) and that for the sphere (1.15).
First, let us look at the strip result (5.9). Whenα < d 2 , R d can be written in terms of an effective dimensionless irrelevant coupling g ef f (R) = (μR) −α as with a coefficient # only depending on the data at the IR fixed point. As for the sphere case (1.16), such a term can be expected from conformal perturbations around a fixed point. Forα > d 2 , we see that the leading approach to the IR value saturates at R −d no matter what the dimension of the leading irrelevant operator is. In particular, the coefficient b d (5.8) involves an integral over all spacetime, suggesting this term receives contributions from degrees of freedom at all length scales (not merely IR degrees of freedom). This term may be considered as the counterpart for a strip of the second line in (1.15). But note that for a sphere the second line of (1.15) can be associated with a curvature expansion of a spherical entangling surface, while for a strip all such curvature terms are absent.

VI. BLACK HOLES
In this section we consider the large R expansion of the entanglement entropy for strip and sphere for a holographic system at a finite temperature/chemical potential, which is described by a black hole on the gravity side. Compared with examples of earlier sections, there are some new elements in the UV and IR expansions. The setup is exactly the same as discussed in Sec. II B and Sec. II C except that now the function f (z) has a zero at some In our discussion below, we will assume f 1 is nonzero. For an extremal black hole, f 1 vanishes, which requires a separate treatment and will be given elsewhere. For notational simplicity, we will set z h = 1 below, which can be easily reinstated on dimensional grounds.
We also introduce 2) which will appear in many places below.

A. Strip
We again look at the strip first. As R → ∞ we expect the tip of the minimal surface z t to approach the horizon z h = 1. This can be seen immediately from equation (2.14): with z t = 1, due to f (1) = 0, the integrand develops a double pole and the integral becomes divergent. To obtain the large R behavior, we thus take and expand the integral in . From (2.14) we find that where γ was introduced in (6.2) and .
Then we can express as a function of R: Reinstating z h , from (2.18) The entanglement entropy itself can be written as At finite temperature, we do not expect non-integer power law terms in 1/R in (6.9), except exponentially small terms. Here will focus on the lowest two terms in (6.9).
The equations for ρ 0 and ρ 1 are The expansion (6.9) should break down for small ρ when ρ 0 or higher order terms become comparable to R. As in the strip case we again expect that the tip of the surface z(ρ = 0) ≡ z t approaches the horizon z = 1, when R is large. We thus expect the UV expansion to break down near the horizon. This indicates that we should choose a = 1 .
which from (2.26) immediately gives where V sphere is the volume of the sphere and we have reinstated z h . In Sec. VI D we generalize this result to an arbitrary shape.
In the above region equation (6.21) can be expanded in large R as One can show that z 2 has a similar structure, i.e.

C. Large R behavior of the entanglement entropy
By carrying out the procedure outlined above one could in principle obtain the large R expansion for the entanglement entropy to any desired order. As an illustration we now calculate the constant term (i.e. R-independent term) in S for d = 3.
We divide the area functional (2.20) into a UV and IR piece and calculate to O(R 0 ): where z * is an arbitrary point in the matching region and ρ(z * ) = ρ * and δ is a UV cutoff.
Plugging in the UV expansion (6.9) and (6.11) into A U V we get: This has an expression for small u * = 1 − z * : Note that ρ 1 (u * ) contains log u * and constant terms, but we chose not to expand it for later convenience. We isolated all u * and δ dependence, hence a U V is a finite term independent of u * . It includes finite area law terms. A IR is given by Plugging in the results of the IR expansion we find Adding together (6.37) and (6.40), we find that the u * dependence cancels which provides a nontrivial consistency check, and the final result is A = # R 2 2 + (area law terms) + a (6.41) b 0 is the constant term in the expansion (6.27) of ρ 0 , and it is given by (6.5).

D. Leading order result for an arbitrary shape
For arbitrary shape we cannot go into as much detail as for the sphere case. Here we demonstrate that at leading order in the large size limit the entanglement entropy goes to thethermal entropy in an explicit calculation. To the best of our knowledge this is the first demonstration using the holographic approach, although the result is widely expected.
We choose spherical coordinates on each z slice of the spacetime: where g i are just the conventional metric components: g 1 = 1 , g 2 = sin 2 θ 1 , g 3 = sin 2 θ 1 sin 2 θ 2 , . . . . (6.45) We will use the notation and denote the set of θ i 's as Ω.
The entanglement entropy is given by the minimal surface area: (Ω) + · · · + c d (R, Ω)z d + · · · + ∞ n=2,m=2 a nm (R, Ω)z n+mα . (6.50) r(Ω) and the functions appearing in higher orders can be determined by solving algebraic equations only involving r(Ω) and its derivatives. One can use the asymptotic data, c d (R, Ω) to obtain dA/dR, by using the Hamilton-Jacobi formalism [1]. We take z to be time, and introduce the canonical momentum and Hamiltonian As a result S d (R) can be solely expressed in terms of c d (R), and the same formulae apply as in section II C.
In the large R limit we consider the expansion ρ(z, θ i ) = R r(Ω) − ρ 0 (z, Ω) + · · · . 1 + (∂ i r(Ω)) 2 r 2 (Ω (6.56) (3.14) then implies that ρ(z) has the small z/z t expansion valid in the region (3.19): Let us turn our attention to the UV expansion (2.29). We have to modify it so thatρ is multiplied by a general function F (R), not R −ν . To obtain the large z behavior of ρ 1 (z) we go through the same steps as in (3.20) to get: We note that taking the n → 2 limit of (3.20) can also give us this result. Plugging in n = 2 into (3.21), and combining all this together in (2.29) gives: Matching this expansion to the IR solution (A2) determines where we plugged in the value of α 1 (3.16) and the UV expansion of ρ 1 (3.20). For n > 2/3 the two terms beautifully combine to give: For n = 2/3 there are no terms coming from (B3) that could contribute to the log R/R term of (B10). Hence we obtain: For n < 2/3 we have to apply subtractions, then a 1 is given by Note that in the main text we use a dimensionless version of a 1 denoted by s 1 . Because we set a = 1 in this appendix, plugging inμ = 1 in the expression of s 1 gives the result for a 1 obtained here.
where we used partial integration and assumed fast enough (α > 1 2 ) decay at infinity. If the decay is slower, we need additional subtractions. For the even dimensional case we encounter