# Comments on Holographic Entanglement Entropy and RG Flows

###### Abstract:

Using holographic entanglement entropy for strip geometry, we construct a candidate for a c-function in arbitrary dimensions. For holographic theories dual to Einstein gravity, this c-function is shown to decrease monotonically along RG flows. A sufficient condition required for this monotonic flow is that the stress tensor of the matter fields driving the holographic RG flow must satisfy the null energy condition over the holographic surface used to calculate the entanglement entropy. In the case where the bulk theory is described by Gauss-Bonnet gravity, the latter condition alone is not sufficient to establish the monotonic flow of the c-function. We also observe that for certain holographic RG flows, the entanglement entropy undergoes a ‘phase transition’ as the size of the system grows and as a result, evolution of the c-function may exhibit a discontinuous drop.

^{†}

^{†}preprint: arXiv:1202.2068 [hep-th]

## 1 Introduction

Zamolodchikov [1] showed that renormalization group (RG) flows of two-dimensional quantum field theories were governed by a remarkable underlying structure. One important feature was that there exists a positive definite function , which decreases monotonically along the RG flows. At the fixed points of the RG flow, this function is stationary and coincides with the central charge of the conformal field theory (CFT) describing the fixed point. A direct consequence for any RG flow connecting two such fixed points is then that

(1) |

More recently, Casini and Huerta [2] developed an elegant reformulation of Zamolodchikov’s c-theorem in terms of entanglement entropy in two dimensions. In their construction, the c-function was defined as

(2) |

where denotes the entanglement entropy for an interval of length . Then it follows that from the strong subadditivity property of entanglement entropy, as well as the Lorentz symmetry and unitarity of the underlying quantum field theory (QFT). Therefore, as the QFT is probed at longer distance scales, i.e., one increases , this c-function (2) decreases monotonically. Further, for a two-dimensional CFT, the entanglement entropy is given by [3, 4]

(3) |

where is the central charge, is a short-distance regulator and is a non-universal constant (independent of ). Hence at RG fixed points.

As a generalization of the two-dimensional c-theorem, Cardy
[5] conjectured that the central charge associated with A-type
trace anomaly – see eq. (11) – should decrease monotonically
along RG flows for QFT’s in any even number of dimensions. Of course,
in two dimensions, this proposal coincides precisely with
Zamolodchikov’s result (1) since . Cardy’s
conjecture was extensively studied in and a great deal of support
was found with nontrivial examples, including perturbative fixed points
[6] and supersymmetric gauge theories
[7, 8, 9].^{1}^{1}1Note that a flaw was recently
found [10] in a proposed counter-example [11] to Cardy’s
conjecture. Recently, a remarkable new proof of this c-theorem was
presented for any four-dimensional RG flow connecting two conformal
fixed points [12]. This result draws on earlier work involving
the spontaneous breaking of conformal symmetry [13] and bounds
on couplings in effective actions [14]. It remains to determine,
however, how much more of the structure of two-dimensional RG flows
carries over to higher dimensions.^{2}^{2}2A related question which
has seen active discussion in the recent literature is whether or not
there exist interesting QFT’s in higher dimensions which exhibit scale
invariance but not conformal invariance [15, 16]. Of course,
in two dimensions, it is proven that scale invariant QFT’s are also
conformally invariant [17].

As we will review below, support for Cardy’s generalized c-theorem was also established using the AdS/CFT correspondence [18, 19, 20]. One of the advantages of the investigating RG flows in a such holographic framework is that the results are readily extended to arbitrary dimensions . In particular then, the analysis of holographic RG flows identified a certain quantity satisfying an inequality analogous to eq. (1) for any dimension, that is, for both odd and even numbers of spacetime dimensions. Since the trace anomaly is only nonvanishing for even , a new interpretation was required for odd . Ref. [20] identified the relevant quantity as the coefficient of a universal contribution to the entanglement entropy for a particular geometry in both odd and even . These holographic results then motivated a generalized conjecture for a c-theorem for RG flows of odd- and even-dimensional QFT’s. For even , this new central charge was shown to precisely match the coefficient of the A-type trace anomaly [20] and so this conjecture coincides with Cardy’s proposal. For odd , it was shown that this effective charge could also be identified by evaluating the partition function on a -dimensional sphere [21] and so the conjecture is connected to the newly proposed F-theorem [22].

The above developments motivated the present paper which examines the the connections between entanglement entropy and RG flows in a holographic framework. Earlier work in this direction can be found in [23, 24, 25]. Here, we make a simple generalization of the c-function in eq. (2) to higher dimensions and then use a holographic framework to examine its behaviour in RG flows. We are able to show that subject to specific conditions, the flow of the c-function is monotonic for boundary theories dual to Einstein gravity. In examining specific flow geometries, we also find that the entanglement entropy undergoes a ‘first order phase transition’ as the size of the entangling geometry passes through a critical value. That is, in our holographic calculation, there are competing saddle points and the dominant contribution shifts from one saddle point to another at the critical size.

An overview of the paper is as follows: In section 2, we review the standard derivation of holographic c-theorems with both Einstein gravity and Gauss-Bonnet gravity in the bulk. We stress that in either case, the monotonic flow of the c-function requires that the matter fields driving the holographic RG flow must satisfy the null energy condition. In section 3, we discuss the holographic entanglement entropy for the ‘strip’ or ‘slab’ geometry and construct a c-function which naturally generalizes eq. (2) to higher dimensions. In section 4, we show that for an arbitrary RG flow solution in Einstein gravity, this c-functions decreases monotonically if the bulk matter fields satisfy the null-energy condition. Section 5 considers explicit examples of holographic RG flows and demonstrates that in certain cases, the entanglement entropy undergoes a ‘phase transition.’ As a result, the c-function exhibits a discontinuous drop along these RG flows. In section 6, we examine holographic RG flows with Gauss-Bonnet gravity and there, we find that the null-energy condition is insufficient to constrain the flow of our c-function to be monotonic. We conclude with a brief discussion of our results and future directions in section 7. Appendix A presents certain technical details related to the discussion in section 4. In the appendix B, we discuss holographic RG flow solutions in Gauss-Bonnet gravity. Finally, appendix C describes the construction of a bulk theory for which the holographic flow geometries examined in section 5 would be solutions of the equations of motion.

While we were in the final stages of preparing this paper, we learned of a similar study of entanglement entropy and holographic RG flows appearing in [26].

## 2 Review of holographic c-theorems

Here we begin with a review of the holographic c-theorem as originally studied by [18, 19] for Einstein gravity. These references begin by constructing a holographic description of RG flows. The simplest case to consider is (+1)-dimensional Einstein gravity coupled to a scalar field:

(4) |

We assume that the potential has various critical points where the potential energy is negative, i.e.,

(5) |

Here is some convenient scale while the dimensionless parameters distinguish the different fixed points. At these points, the gravity vacuum is simply AdS with the curvature scale given by .

Now in the context of the AdS/CFT correspondence, the bulk scalar above is dual to some operator and the fixed points (5) of the scalar potential represent the critical points of the boundary theory. In particular then, with an appropriate choice of the bulk potential, will be a relevant operator for a certain fixed point and so an RG flow will be triggered by perturbing the corresponding critical theory by this operator in the UV. Of course, the holographic description of this RG flow is that the scalar field acquires an nontrivial radial profile which connects two of the critical points in eq. (5). The bulk geometry for this solution can be described with a metric of the following form [18, 19]

(6) |

Here, the radial evolution of the geometry is entirely encoded in the conformal factor . At a fixed point where the geometry is AdS, the conformal factor is simply where again is the AdS curvature scale. Implicitly, we will assume that asymptotic UV boundary is at while the IR part of the solution corresponds to . Hence for an RG flow between two fixed points as described above, the metric (6) approaches that of AdS in both of these limits.

Now following [18, 19], we define:

(7) |

where ‘prime’ denotes a derivative with respect to . Then for general solutions of the form (6), one finds

Above in the second equality, Einstein’s equations were used to
eliminate in favour of components of the stress
tensor.^{3}^{3}3Note that for the scalar field theory in
eq. (4), we have . The
final inequality assumes that the matter fields obey the null energy
condition [27]. Now given the usual connection between and
energy scale in the CFT, eq. (2) indicates that is
always increasing as we move from low energies to higher energy scales.
Further, if the flow function (7) is evaluated for an AdS
background, one finds a constant:

(9) |

Hence if we compare this constant for the UV and IR fixed points of the holographic RG flow, we find the holographic c-theorem:

(10) |

To make closer contact with the dual CFT, we recall the trace anomaly [28, 29],

(11) |

which defines the central charges for a CFT in an even number of spacetime dimensions. Each term on the right-hand side is a Weyl invariant constructed from the background geometry. In particular, is the Euler density in dimensions while the are naturally written in terms of the Weyl tensor (as well as its covariant derivatives), e.g., see [30]. Note that in eq. (11), we have ignored the possible appearance of a conformally invariant but also scheme-dependent total derivative.

A holographic description of the trace anomaly was developed [31] and can be applied to the AdS stationary points in the present case (for even ). These calculations show that , the value of the flow function at the fixed points, precisely matches the A-type central charge in eq. (11), i.e., for even [20]. Hence with the assumption that the matter fields obey the null energy condition, the holographic CFT’s dual to Einstein gravity satisfy Cardy’s conjecture of a c-theorem for quantum field theories in higher dimensions [5]. Of course, one must add the caveat that for these holographic CFT’s, i.e., those dual to Einstein gravity, all of the central charges in eq. (11) are equal to one another [31]. Hence the holographic models (4) considered above can not distinguish between the behaviour of and in RG flows.

It has long been known that to construct a holographic model where the various central charges are distinct from one another, the gravity action must include higher curvature interactions [32]. In part, this motivated the recent holographic studies of Gauss-Bonnet (GB) gravity [33] — for example, see [34]. In section 6, we will extend our discussion of holographic RG flows to GB gravity with the following action

(12) |

where

(13) |

As before, we again assume the scalar potential has various stationary points as in eq. (5), where the energy density is negative. Note that for convenience, we are using the same canonical scale which appears for the critical points in eq. (5) in the coefficient of the curvature-squared interaction in eq. (12). Hence the strength of this GB term is controlled by the dimensionless coupling constant, . We write the curvature scale of the AdS vacuum as where the constant satisfies [20]

(14) |

In general, eq. (14) has two solutions but we only consider the smallest positive root

(15) |

with which, in the limit , we recover and , as discussed above for Einstein gravity. One would find that graviton fluctuations about the AdS solution corresponding to the second root are ghosts [35, 36] and hence the boundary CFT would not be unitary. The theory (12) is further constrained by demanding that the dual boundary theory respects micro-causality or alternatively, that it does not produce negative energy fluxes [34, 37].

For our present purposes, the most important feature of GB gravity (12) is that the dual boundary theory will have two distinct central charges. To facilitate our discussion for arbitrary , we would like to define two central charges that appear in any CFT for any – including odd – and hence for our pursposes, the trace anomaly is not a useful definition of the central charges. Following [37, 20], we consider:

(16) | |||||

(17) |

The first charge is that controlling the leading singularity of
the two-point function of the stress tensor.^{4}^{4}4Here, as in
[38], we have normalized so that in the limit ,
. This choice is slightly different from that originally
presented in [37], i.e.,

(18) |

Now assuming the existence of bulk solutions describing holographic RG
flows for the GB theory (12),^{5}^{5}5Appendix
B includes a discussion of one approach to constructing
such solutions. we can establish a holographic c-theorem following the
analysis of [20]. We begin by constructing two flow functions
[20]:

(19) | |||||

(20) |

These expressions were chosen as the simplest extensions of eq. (7) which yield the two central charges above at the fixed points, i.e., and — recall that for the AdS vacua. Now let us examine the radial evolution of in a holographic RG flow:

Here, the equations of motion for GB gravity (see eq. (147)) have
been used to trade the expression in the first line for the components
of the stress tensor appearing in the second line. As before with
Einstein gravity, we assume the null energy condition applies for the
matter fields for the final inequality to hold. In eq. (12),
the matter contribution is still a conventional scalar field action and
so just as before . With this
assumption, it then follows^{6}^{6}6We note that some additional
arguments are needed to ensure that there are no problems with
for odd [20]. that evolves
monotonically along the holographic RG flows and we can conclude that
the central charge is always larger at the UV fixed point than
at the IR fixed point. Hence we recover precisely the same holographic
c-theorem found previously with Einstein gravity, namely,

(22) |

One can also consider the behaviour of along RG flows

(23) |

but there is no clear way to establish that has a
definite sign. Hence this holographic model (12) seems to
single out as the central charge which satisfies a c-theorem.
This result has also been extended to holographic models with more
complex gravitational theories in the bulk:^{7}^{7}7Similar results
were also found to apply in the context of cosmological solutions
[40]. quasi-topological gravity [20], general
Lovelock theories [41, 42], higher curvature theories with
cubic interactions constructed with the Weyl tensor [20] and
gravity [41]. The result is also established for
holographic models where the RG flow is induced by a double-trace
deformation of the boundary CFT [43]. Given the relation
in even dimensions, these holographic results support Cardy’s
proposal [5] that the central charge (rather than any
other central charge) evolves monotonically along RG flows. However, it
is even more interesting that these results suggest that a similar
behaviour also occurs for the central charge in odd dimensions.
Further while the original field theory definition of involved a
calculation of entanglement entropy [20], it was shown that the
same charge can also be identified by evaluating the partition function
on [21]. Hence the exciting new field theoretic results
of [22] provide further evidence for the same c-theorem in odd
dimensions.

In any event, a key requirement for the holographic c-theorem to hold for Einstein gravity (10) or for GB gravity (22) is that the matter fields obey the null energy condition. Of course, this holds when these gravitational theories are coupled to a simple scalar field, as in eqs. (4) and (12), this constraint is trivially satisfied. However, phrasing the constraint in terms of the null energy condition allows for more general scenarios for the matter fields driving the holographic RG flow. We should add that the same constraint also ensures the holographic c-theorem holds for all of the extensions of the bulk gravity theory mentioned above. We might mention that violations of the null energy condition quite generally lead to instabilities [44] and so it is a natural constraint to define a reasonable holographic model. In the following, we will also see that the same constraint can be related to the monotonic flow of a holographic c-function defined in terms of entanglement entropy.

## 3 Holographic entanglement entropy and a c-function

Before beginning our holographic analysis, we must first identify a candidate c-function using entanglement entropy for . Recall that [2] identifies such a c-function for two-dimensional quantum field theories as

(24) |

where is the entanglement entropy for an interval of length on an infinite line. As described above, using the result (3) for the entanglement entropy of CFT’s, one finds at any fixed points of the RG flows, i.e., eq. (24) yields the central charge of the underlying CFT at the fixed points. We would like to emulate this construction in higher dimensions. However, one should recall that in general the entanglement entropy for field theories in higher dimensions will contain many (non-universal) power law divergences depending on the geometry of the entangling surface, e.g., see eq. (123). Hence we expect a simple derivative with respect to some scale characteristic of the entangling surface will typically yield a result which depends on the cut-off. While there may be various strategies to avoid this outcome – see further discussion in section 7 – here we take the following simple approach: First we note that, at the fixed points, the power law divergences are geometric in origin and all but the leading area-law terms vanish if the geometries of the background and the entangling surface are both flat. Hence we consider a ‘strip’ or ‘slab’ geometry, where the entangling surface consists of two parallel flat (–2)-dimensional planes separated by a distance in a flat background spacetime. The entanglement entropy (of a CFT) then takes the simple form [45, 46]

(25) |

where and are dimensionless numerical factors and is a(n infrared) regulator distance along the entangling surface – we assume that . That is, is the area for each of the planes comprising the entangling surface and so the first contribution in eq. (25) is simply the usual area law term. The coefficient of the second finite term is proportional to a central charge in the underlying -dimensional CFT, which we denote . Hence we can isolate this central charge by writing

(26) |

Hence we are naturally lead to consider the quantity

(27) |

as a candidate for a c-function along the RG flows, so that at the fixed points of the flow. We will identify the precise value of the coefficient with our holographic calculations below — see eq. (34). Comparing eqs. (24) and (27), we can view the latter expression as the simplest generalization of the two-dimensional c-function (24) to higher dimensions. At the outset, we wish to say that we will find below that will only be able to prove that this candidate c-function actually decreases monotonically along RG flows for holographic models with Einstein gravity in the bulk. However, another goal in the following analysis is to connect the behaviour of this c-function defined using holographic entanglement entropy with the standard discussions of holographic c-theorems [18, 19, 20]. We should also mention that eq. (27) was previously suggested as a c-function in [46].

### 3.1 Holographic entanglement entropy on an interval

In this section, we derive some of useful results to evaluate eq. (27) for holographic RG flows in following sections. The holographic models in sections 4 and 5 will be described by Einstein gravity in the bulk, while we will consider GB gravity [33] in section 6.

The seminal work of Ryu and Takayanagi [45, 46] provided a holographic construction to calculate entanglement entropy. In the -dimensional boundary field theory, the entanglement entropy between a spatial region and its complement is given by the following expression in the (+1)-dimensional bulk spacetime:

(28) |

where indicates that is a bulk surface that is homologous
to the boundary region [47, 48]. In particular, the
boundary of matches the ‘entangling surface’ in the
boundary geometry. The symbol ‘min’ indicates that one should extremize
the area functional over all such surfaces and evaluate it for the
surface yielding the minimum area.^{8}^{8}8We are using ‘area’ to
denote the (–1)-dimensional volume of . If eq. (28) is
calculated in a Minkowski signature background, any extremal surfaces
are saddle points of the area functional and one should choose the
extremum with the minimum area. However, if one first Wick rotates to
Euclidean signature, the extremization procedure actually corresponds
to finding the global minimum of the area functional.
Eq. (28) assumes that the bulk physics is described by
(classical) Einstein gravity and we have adopted the convention:
Hence the functional which is extremized on
the right-hand side of eq. (28) matches the standard
expression for the horizon entropy of a black hole. While this proposal
passes a variety of consistency tests, e.g., see [46, 47, 49],
there is no general derivation of this holographic formula
(28). However, a derivation was recently provided for the
special case of a spherical entangling surface in [21].

In [23, 49], the above expression (28) was extended to
holographic theories dual to GB gravity (12) in the bulk.
The new prescription still extremizes over bulk surfaces which
connect to the entangling surface at the asymptotic boundary, however,
the entropy functional to be extremized becomes^{9}^{9}9This
expression was motivated by the construction of black hole entropy for
Lovelock gravity appearing in [50]. We note that when
evaluated on a general surface this functional will not match the Wald
entropy [51]. However, the two agree when evaluated on a
stationary black hole horizon.

(29) | |||||

Here, () is the induced metric on (the boundary of) the bulk surface , is the Ricci scalar of this induced geometry and is the extrinsic curvature of the boundary at the asymptotic cut-off surface. Note that we only apply this expression for since it is only for these dimensions that the GB interaction (13) contributes to the gravitational equations of motion. Of course, if we set in the above expression, it reduces to and we recover eq. (28). Note that the extrinsic curvature term in eq. (29) plays the role of a ‘Gibbons-Hawking’ surface term to ensure that the variational principle is consistent.

Now let us begin to consider evaluating the holographic entanglement entropy for a general RG flow. As in the previous section, we assume the bulk metric takes the form given in eq. (6). Then the boundary geometry is simply flat space and we define the entangling surfaces as follows: First recall that the entangling surface divides a Cauchy surface (e.g., the constant time slice, ) into two regions. As described above, we wish to consider an interval of length and so we introduce two flat (hyper)planes at and , as shown in figure 1. We also introduce a regulator length to limit the size of the two planes along the directions, e.g., we can imagine the boundary is periodic in these directions with length . Hence the area of either plane is , as described at eq. (25). In calculating the holographic entanglement entropy, we consider bulk surfaces that end on the entangling surface as , as shown in figure 1. With the ‘slab’ geometry described here, the radial profile of these surfaces will only be a function of the coordinate . We will write the profile as where indicates the width of the interval which sets the boundary condition, i.e., in the present case, as . Of course, the holographic calculations are only well-defined if we introduce an asymptotic cut-off surface as some . The position of this surface is related to a short distance cut-off in the boundary theory, i.e., . The radial profile will define another useful UV scale with , i.e., the profile intersects the cut-off surface at . Another useful scale in the bulk surface is the minimal radius which it reaches in the bulk, which appears as .

Given the background metric (6) and our ansatz for the profile of the bulk surface, we find that eq. (29) reduces to the following simple expression

(30) |

where and . Now one may treat the
above expression as an action which is varied to find a second-order
differential equation to determine the profile . However,
since the integrand above has no explicit dependence on the coordinate
, the following is a conserved quantity along the radial
profile^{10}^{10}10If we denote the integrand in eq. (30) as
, then .

(31) |

This leaves us with a first-order equation for the profile, which should be easier to solve. In principle then, our goal is to solve for in a given holographic RG flow geometry, i.e., for a specific conformal factor , and then substitute the solution back into eq. (30) to calculate the entanglement entropy.

Before going on to consider the entanglement entropy and c-function for
RG flow geometries, let us first examine the results when the bulk
geometry is simply AdS space, i.e., at a fixed point of the flow where
the boundary theory is conformal. Recall that for the AdS vacuum
. Let us begin by setting and considering the
results for Einstein gravity in the bulk.^{11}^{11}11Note that in this
case, the AdS curvature is given by simply . The case of
three-dimensional AdS or a boundary CFT is special since the
entanglement entropy yields a logarithmic UV divergence

(32) |

If we recall that the central charge of the boundary CFT is given by , we see that this expression precisely reproduces the expected result (3) for the entanglement entropy of a two-dimensional CFT. Next turning to Einstein gravity with , the entanglement entropy for the interval is given by [45]

(33) |

Here we see the general structure given in eq. (25) with two terms, a power law divergence proportional to and a finite contribution proportional to . Next for , both of the central charges in eqs. (16) and (17) are identical and we use this fact to define in eq. (25) for Einstein gravity: . As described previously then, we can extract this central charge from the above entanglement entropy using eq. (26), which yields

(34) |

Hence we have identified the precise value of (for ) which appears as the coefficient in eq. (27) of the c-function.

Finally let us apply the above formulae to calculate holographic entanglement entropy with the strip geometry for the boundary CFT dual to the AdS vacuum in GB gravity (12). To simplify the final results, it is convenient to first treat as the independent variable, in which case to fix the profile of the bulk surface, we must determine . Next we choose a new radial coordinate and define . Note that as , and further one can show at , . Now with these choices, eq. (31) becomes

(35) |

In general, this equation yields three roots for and the relevant solution is the real root which can be continuously connected to the solution: . Now it is straightforward to see that the entanglement entropy (30) can be written as

(36) | |||||

where . Then applying
eq. (34), we can express the central charge in the finite
contribution as^{12}^{12}12Note that analogous results were given for the
case in [23]. However, we note that the calculations
presented there did not include the ‘Gibbons-Hawking’ surface term in
eq. (29) and hence their expressions do not match those
presented here. However, we have verified numerically that the
effective central charge in [23] agrees with eq. (37)
when . We also observe that the leading divergent term in
eq. (36) is proportional to while without the
‘Gibbons-Hawking’ term, this term is proportional to .

(37) |

Regrettably, we do not have a closed analytic expression for in terms of the two central charges and . Hence we have numerically evaluated the above expression and plotted as a function of in figure 2a for several values of . Note that in this figure, at for all of the values of since this corresponds to or Einstein gravity in the bulk. From these curves, we can infer that is a complicated nonlinear function of both and . We can also illustrate this fact as follows: In the vicinity of or , we can make a linearized analysis of eq. (37) to find

(38) |

where

(39) |

Here, we have defined as the linear combination of the two central charges in eqs. (16) and (17) which yields an expansion which precisely matches that for . Next, we consider the ratio of and over the full (physical) range of . Since can vanish in this range, it is convenient plot the ratio as a function of , as shown in figure 2b. This figure illustrates even more dramatically our previous observation that is a complicated nonlinear function of both and . At this point, let us add that since , the central charge identified in [20] as satisfying a c-theorem, we might not expect that our new effective central charge will always flow monotonically in holographic RG flows for general .

(a) | (b) |

## 4 Holographic flow of c-function with Einstein gravity

In this section, we examine the behavior of the c-function (27) in a general holographic RG flow dual to Einstein gravity. We will first discuss the flow of the c-function in and then generalize it to arbitrary dimensions.

For , the bulk theory is three-dimensional Einstein gravity coupled to, e.g., a scalar field with a nontrivial potential, as described in section 2. Our holographic expression (30) for the entanglement entropy of a strip can be written as

(40) |

Note that the integration above runs over half of the range, i.e., . Further the conserved charge (31) simplifies to

(41) |

To calculate , we note that and , i.e., the profile of the extremal surface implicitly depends on the strip width . If we vary with respect to , keeping the UV cut-off fixed, we will get two contributions: one coming from change in the limits of the integration and second from change in the solution . We write them as

(42) |

where we have used the equation of motion for

(43) |

to cancel the bulk contribution. Since the UV cut-off is fixed while performing the variation, we get some extra constraints between and at the asymptotic boundary. Taking variation of relation with respect to , we get

(44) |

Substituting this relation, as well as eq. (41), into eq. (42) gives us following expression for :

(45) |

In the above relation, the partial derivatives of are evaluated near the asymptotic boundary. To further simplify this expression, we use the Fefferman-Graham expansion near the boundary [30]. In terms of the radial coordinate , this expansion takes the form [24]

(46) |

where

(47) |

In this expansion, is the AdS radius in the UV region (i.e., as ) and where is the conformal weight of the operator dual to the bulk scalar field. Near the boundary, the coordinate is very large and hence it is sufficient to work with only the leading order term in the expansion (47). Although has a complicated profile deep inside the bulk, near the boundary it will have the simple form

(48) |

For this , eq. (41) can be re-expressed as the following equation of motion: . The latter is easily integrated to yield the following solution

(49) |

where the integration constant was chosen so that as . Next we differentiate the above solution with respect to and to find and , treating that as a function of – see appendix A for further details. Taking the limit in ratio of and appearing in eq. (45), we find that

(50) |

This relation not only simplifies eq. (45) but also ensures that the first derivative of is indeed finite for all RG flow solutions. Using this relation in eq. (45), we arrive at following elegant form of the c-function (24) for arbitrary RG flow backgrounds:

(51) |

The next step is to show that this c-function increases monotonically along holographic RG flows. Implicitly, the extremal bulk surfaces on which we are evaluating the Ryu-Takayanagi formula (28) extend to infinite at and pass through a minimum at . The latter radius gives us an indication of which degrees of freedom the entanglement entropy is probing, i.e., for smaller values of , we expect the entropy and responds more to the IR structure of the RG flow. Hence in the following, we will study behavior of as a function of the turning point radius and we wish to establish the ‘c-theorem’ as – at least for background geometries that satisfy appropriate constraints.

Comparing to the field theory construction of [2], we note that there the c-theorem was formulated as . Naively, this result matches with the holographic inequality which we wish to establish since we expect that as the width of the strip increases, the minimal area surface will explore deeper regions in the bulk geometry. The two inequalities would be rigorously connected if we could prove a second inequality for consistent holographic models. However, as we will see in the next section, in fact this inequality does not hold for all extremal surfaces. However, we will still find in all cases of interest. The violations of the previous inequality are associated with unstable saddle-points which do not contribute to the physical entanglement entropy. Hence, in section 5, we will find that the behaviour of the entanglement entropy in general holographic RG flows provides a richer story than might have been naively anticipated.

Returning to the flow of the c-function, we note that at the minimum of the bulk surface, we will have and . Hence considering eq. (41) at this turning point, we find

(52) |

Here it is natural to treat this constant of the motion as a function of , rather than . We will also work with width of the strip as function of . Then combining eqs. (51) and (52) yields

(53) |

Now to express in terms of , we begin with the relation

(54) |

Here in the final expression we have used eq. (41). Now above, we will apply integration by parts using

(55) |

to find that

(56) |

where we have used eq. (48) to evaluate at . Further we can differentiate this expression with respect to to get

(57) |

Now substituting eqs. (56) and (57) into eq. (53), we find

In the second line, we have used eq. (41) to convert the integration over to one over . In the last line, we have used Einstein’s equations to replace by the components of the stress tensor. As for the discussion of holographic c-theorems in section 2, the final inequality assumes that the bulk matter fields driving the holographic RG flow satisfy the null energy condition. The latter ensures that the integrand is negative. The overall inequality also requires and . The first condition is obvious from eq. (52) while the second can be established as follows: Given the null energy condition, it follows that which means that is everywhere a decreasing function of radial coordinate . Implicitly, we are assuming the bulk geometry approaches AdS space asymptotically, i.e., the dual field theory approaches a conformal fixed point in the UV. Hence with , we see the minimal value of is , where is the asymptotic AdS scale. Since this minimal value is positive, it must be that is positive everywhere along the holographic RG flow. Hence is positive and our two-dimensional c-function increases monotonically along the RG flow if the bulk matter satisfies the null energy condition.

We now turn to proving the monotonic flow of the c-function (27) for higher dimensions. The required analysis is a straightforward extension of the above calculations with . In particular, one finds that eq. (51) generalizes to

(59) |

with boundary dimensions. The conserved quantity (31) is now given by

(60) |

We have relegated the detailed derivation of eq. (59) to appendix A. However, we can see from this result that all the complexities of determining the c-function boil down to evaluating the conserved charge (60) for the minimal area surface. We might note that we can evaluate this expression at the minimal radius (where ) to find

(61) |

which generalizes eq. (52) to general .