Rényi entropy and conformal defects.

There has been a growing interest in entanglement and Rényi entropies as probes of complex interacting quantum systems in a variety of areas ranging from condensed matter physics, e.g., wenx ; kitaev ; haldane to quantum gravity, e.g., qg1 ; qg2 ; qg3 ; qg4 . While commonly regarded as a useful theoretical diagnostic, the past year has seen remarkable experimental advances where the Rényi entropy, as well as quantum purity and mutual information, of a system of delocalized interacting particles can be measured in the laboratory expEE . This experimental breakthrough strengthens the motivation to develop further theoretical insight into these entanglement measures, particularly in the framework of quantum field theory (QFT). In this paper, we focus our attention on Rényi entropies renyi1 ; renyi2 in the context of conformal field theories (CFTs). Conformal symmetry obviously introduces additional constraints on the Rényi entropy beyond those in a general QFT, however, CFTs are still a very important class of QFTs since they describe physics at the quantum critical points and also shed light on the structure of gravity through the AdS/CFT duality.

In the case of holographic CFTs with a gravitational dual, the Ryu-Takayanagi prescription Ryu:2006bv ; Ryu:2006ef — and its generalizations Hubeny:2007xt ; Hung:2011xb ; deBoer:2011wk ; Dong:2013qoa ; Camps:2013zua — provides an elegant and practical tool to evaluate the entanglement entropy across an arbitrary entangling surface. While the recent derivation Lewkowycz:2013nqa of this prescription presents a generalization to holographic Rényi entropies in principle, explicit holographic calculations of the Rényi entropy have been largely restricted to a spherical entangling surface Casini:2011kv ; Hung:2011nu — however, see recent progress in dong33 . Similarly, efficient computational tools to study the Rényi entropies for more general CFTs are rare. Numerical techniques have been developed to evaluate the Rényi entropy in lattice models describing critical theories, e.g., roger1 ; roger2 ; roger3 but these are demanding and must be adapted for the specifics of a given model. Beyond these numerical studies, the existing literature considers primarily the Rényi entropy across a spherical entangling surface for a CFT living in flat space, e.g., Casini:2011kv ; Casini:2010kt ; Perlmutter:2013gua ; Hung:2014npa ; Lee:2014zaa ; Lewkowycz:2014jia .

In this paper, we build on conformal defect techniques to develop a field theoretic framework which allows for quantitative studies of the Rényi entropy. Conformal defects have a long story, both in two and higher dimensions — see e.g. Cardy:1984bb ; McAvity:1995zd ; billo:confdef . In section 2, we begin from the basic definitions to draw a parallel between conformal defects and the twist operators, which enter the calculation of the Rényi entropy. This perspective demonstrates that the Rényi entropy readily lends itself to the application of defect CFT techniques. In particular, we define the so-called displacement operator for the twist operators, which then implements small deformations of the entangling surface.

Hence the displacement operator can be used for perturbative calculations of the Rényi entropy when small modifications are made in the geometry of the entangling surface. Our focus on the displacement operator in the following arises because recently there has been a great deal of interest in the shape dependence of Rényi and entanglement entropies, e.g., mark0 ; mark1 ; Bueno:2015rda ; Bueno:2015xda ; Bueno:2015qya ; Bueno:2015lza ; safdi1 ; aitor7 . First, we show below for a planar or spherical entangling surface that the second order variation of the Rényi  entropy is fixed by the two-point function of the displacement operator. Given this framework, one of our key results is then to identify a simple relation between the coefficient defining the two-point function of the displacement operator and the conformal weight of the twist operator, which unifies a number of distinct conjectures with regards to the shape dependence of the Rényi entropy. In particular, in section 3, we apply this approach to evaluate the second order variation of the Rényi  entropy for a spherical region. In the limit that the Rényi index goes to one, using the new relation, we precisely recover Mezei’s conjecture for variations of the entanglement entropy mark0 — see also mark1 . Further the displacement operator can also be used to examine the variation of the Rényi entropy for small but ‘singular’ deformations of the entangling surface. In section 4, we consider the case of a planar entangling surface which undergoes a singular deformation to create a small conical singularity. With the previous relation, our result for the change in the Rényi entropy matches previous conjectures with regards to cusp and cone geometries in the limit that the entangling surface is almost smooth Bueno:2015rda ; Bueno:2015xda ; Bueno:2015qya ; Bueno:2015lza . In section 5, we focus on the Rényi entropy across an arbitrary entangling surface in four spacetime dimensions. We are able to relate two coefficients in the universal contribution to the Rényi entropy to the conformal weight of the twist operator and to the coefficient in the two-point function of the displacement operator, respectively. Our relation between the latter two quantities then yields the equality of these coefficients, as was conjectured for all four-dimensional CFTs in safdi1 . Hence, interestingly, the relation between the coefficient of the two-point function of the displacement and the conformal weight of the twist operator underlies a number of existing conjectures in the literature about the Rényi  entropy.

However, at this point, we must add that recent holographic calculations dong34 imply that the proposed relation does not hold for general values of the Rényi index in four-dimensional holographic CFTs dual to Einstein gravity. Hence it becomes an interesting question to ask for precisely which CFTs does such a constraint hold. For example, our calculations in Appendix C confirm that it does in fact hold for free massless scalars in four dimensions.

In section 6, we pose the question whether it may be possible to define the twist operator through the operator product expansion generically available in the presence of defects, and we point out an intriguing universal feature of the fusion of the stress-tensor with this specific extended operator. We conclude with a discussion of our results and possible future directions in section 7. We also have a number appendices where we present various technical calculations whose results are used in the main text. In Appendix A, we derive a set of Ward identities in the presence of a twist operator. In Appendix B, we list some useful formulas which describe variation of various geometrical objects around the flat space. We devote Appendix C to the example of a four-dimensional free scalar field, where the displacement operator can be given a precise identity in terms of the elementary field.

Finally, let us add that while this paper was in the final stages of preparation, ref. new appeared. Although their discussion only considers the entanglement entropy, some of the results overlap with aspects of the present paper.

A central object for our discussion will be the twist operator which naturally arises in evaluating Rényi entropies in quantum field theory Hung:2014npa ; Calabrese:2004eu ; Cardy:2007mb .¹¹1In two-dimensional CFTs, twist fields associated to branch points were first introduced in Knizhnik:1987xp ; Dixon:1986qv . Therefore, let us start by recalling the definition of our main player. We begin with a generic QFT in flat $d$-dimensional spacetime. On a given time slice, the QFT is in a global state described by the density matrix $\rho$ — in fact, shortly we will restrict our attention to the vacuum state. We consider the density matrix $\rho_{A}$ obtained when the state is restricted to a particular region $A$, i.e., obtained by tracing over the degrees of freedom in the complementary region $\bar{A}$ of the time slice:

The one-parameter family of Rényi entropies associated to the reduced density matrix $\rho_{A}$ is defined as follows renyi1 ; renyi2 :

The entanglement entropy, i.e., the von Neumann entropy, is recovered with the limit:

Here we have implicitly considered the Rényi index $n$ in (2.2) to be a real number. However, specifically for integer $n$ (with $n>1$), a path integral construction, which is widely known as the replica trick, allows us to evaluate the Rényi entropies for a QFT. An analytic continuation is then required to make contact with the entanglement entropy but we will have nothing to add about the conditions under which the continuation is reliable.

The replica trick begins by evaluating the reduced density matrix $\rho_{A}$ in terms of a (Euclidean) path integral on $\mathbb{R}^{d}$ but with independent boundary conditions fixed over the region $A$ as it is approached from above and below in Euclidean time, e.g., with $t_{\textrm{\tiny E}}\to 0^{\pm}$. The expression $\textup{Tr}(\rho_{A}^{n})$ is then evaluated by extending the above to a path integral on a $n$-sheeted geometry Calabrese:2004eu ; Cardy:2007mb , where the consecutive sheets are sewn together on cuts running over $A$. Denoting the corresponding partition function as $Z_{n}$, we can write the Rényi entropy (2.2) as

where the denominator $Z_{1}^{n}$ is introduced here to ensure the correct normalization, i.e., $\textup{Tr}[\rho_{A}]=1$. The partition function $Z_{n}$ has an important symmetry. That is, even if in the above construction we chose to glue the copies together along the codimension-one submanifold $A$ on the $t_{\textrm{\tiny E}}=0$ slice, the precise location of the cut between different sheets is meaningless — see for instance section 3.1 of Caraglio:2008pk . Hence the only source of breaking of translational invariance on each sheet is at the location of the entangling surface, i.e., the boundary of $A$. Since the modification is local in this sense, it can be reinterpreted as the insertion of a twist operator $\tau_{n}$. In defining $\tau_{n}$, the above construction is replaced by a path integral over $n$ copies of the underlying QFT on a single copy of the flat space geometry. The twist operator is then defined as a codimension-two surface operator in this $n$-fold replicated QFT, which extends over the entangling surface and whose expectation value yields

Hence eq. (2.5) implies that $\tau_{n}$ opens a branch cut over the region $A$ which then connects consecutive copies of the QFT in the $n$-fold replicated theory. Note that here and in the following, we omit the $A$ dependence of $\tau_{n}$ to alleviate the notation.

In proceeding, we restrict our attention to the case where the QFT of interest is a conformal field theory and the state is simply the flat space vacuum state. Now let us take a closer look at the residual symmetry group in the presence of the twist operator. In doing so, we restrict ourselves to a very symmetric situation where we choose $A$ to be half of the space. That is, we choose $\tau_{n}$ to lie on a flat ($d\!-\!2$)-dimensional plane, which we denote as $\Sigma$. For concreteness, we parametrize $\mathbb{R}^{d}$ with coordinates $(x^{1},\dots,x^{d})$, and we locate the twist operator at $x^{1}=0=x^{2}$. In the following, we will denote directions orthogonal to $\Sigma$ with Latin indices from the beginning of the alphabet ($a,\,b,\dots$) and parallel directions with Latin indices from the middle of the alphabet ($i,\ j,\dots$), while $\mu=(i,a)$. Let us explicitly notice that since a spherical entangling surface can be obtained from the planar one by means of a conformal transformation, the following applies equally well to the spherical case.

Now, the stabilizer of a $(d\!-\!2)$-dimensional plane within the $d$-dimensional conformal algebra is the subalgebra $so(d-1,1)\times u(1)$. The first factor comprises the conformal transformations in $(d\!-\!2)$ dimensions, while the second consists of rotations in the transverse space. Let us choose the cut to lie along a half-plane in $\mathbb{R}^{d}$, e.g., $x^{1}<0$ (and $x^{2}=0$), then a moment’s thought is sufficient to realize that the gluing condition is preserved only if the same conformal transformation is applied to all the copies at the same time.²²2In the density matrix language, this is rephrased in the statement that the transformation $U\rho_{A}U^{-1}$ is a symmetry of $Z_{n}$ only if applied to the $n$ factors of $\rho_{A}$ appearing in the trace (2.2). The rotations in the transverse plane, on the other hand, move the cut, which can be brought back to the original position through the symmetry of the partition function which we referred to above. This leads to a remark on the structure of the symmetry group. A rotation of an angle $2\pi$ has the net effect of shifting by one the labeling of the replicas: in a correlation function, an operator inserted in the $i$-th copy ends up in the $(i+1)$-th one. Therefore, the $u(1)$ algebra exponentiates in the $n$-fold cover of the group $O(2)$. Up to this subtlety, we see that the symmetry group preserved by the twist operator is the same as the one preserved by a flat conformally invariant extended operator, i.e., a conformal defect.

The symmetry algebra places constraints on observables, which in many cases have been worked out in the context of defect CFTs. In particular, this is the case for correlation functions of local operators billo:confdef . The twist operator is a conformal defect placed in the tensor product $(\textup{CFT})^{n}$ rather than in the original conformal field theory. Therefore, it is especially interesting to consider the consequences of interactions among replicas, which distinguish this setup from a mere local modification of a CFT on $\mathbb{R}^{d}$: these are probed by correlation functions of operators belonging to different copies of the theory. Such correlators do not escape the defect CFT framework and in particular can be handled with the classical tool available in any conformal field theory: the existence of an operator product expansion (OPE), which converges inside correlation functions. In the presence of a defect, bulk excitations can be brought close to the extended operator, and be expressed as a sum over local operators on the defect. This corresponds to a new OPE channel, usually referred to as the defect OPE. If we denote defect operators with a hat, the defect OPE of a bulk scalar of scaling dimension $\Delta$ takes the following form:

The meaning of the label $0$ given to the defect operator will become clear in a moment. Let us stress that operators in our formulae will always be thought of as inserted on a single copy of the CFT and when present, sums over replicas will be written explicitly. Now consider a correlator of bulk primaries which belong to different factors of the $n$-fold replicated CFT. We can substitute to each of them the respective defect OPE, and since the latter converges inside correlation functions, the resulting sum over two-point functions of defect operators must reproduce the original correlator. In particular, we see that the expression on the right hand side of eq. (2.6) must retain the information about the copy in which the primary on the left hand side was inserted. This is possible thanks to the global structure of the symmetry group — that is, the fact that rotations around the defect are combined non-trivially with the $\mathbb{Z}_{n}$ replica symmetry. The rotational symmetry around an extended operator is a global symmetry from the point of view of the theory on the defect. As a consequence, defect operators carry a $u(1)$ quantum number $s$. In our case, this transverse spin is rational: $s=k/n$, $k$ being an integer. We see that the defect OPE of a bulk scalar contains in general terms of the form

where $\phi\in[0,2\pi n)$ is the angle in a plane orthogonal to the defect and provides the information about the replica on which the bulk primary has been inserted. In appendix C, we shall see explicit examples of OPEs of the form (2.7) in the free scalar theory, and how they allow us to decompose correlation functions of bulk primaries placed in arbitrary positions.

The breaking of translational invariance in the directions transverse to the entangling surface gives rise to an operator of transverse spin $s=1$, which is always present on defects in local theories. Indeed, the Noether current which generates translations fails to be conserved only at the position of the defect, so that a new contact term should be present in the Ward identities of the stress-tensor. This defines the displacement operator $D^{a}$:

where the index m runs over the replicas³³3We stress again that in general, our calculations will refer to bulk operators in a single copy of the replicated CFT. Hence $T_{\textrm{\tiny(m)}}^{\mu\nu}$ here denotes the stress tensor in the m’th copy of the CFT and the total stress tensor for the full theory would be given by $T_{\textrm{\tiny tot}}^{\mu\nu}=\sum_{{\rm m}=1}^{n}T_{\textrm{\tiny(m)}}^{\mu\nu}$. However, in order to reduce clutter in expressions below, we will drop the subscript (m) but $T^{\mu\nu}$ still denotes the single-copy stress tensor. The total stress tensor will always be denoted as $T_{\textrm{\tiny tot}}^{\mu\nu}$. and $\delta_{\Sigma}$ denotes the delta function in the transverse space with support on the $\Sigma$. The sum over replicas appears because, as mentioned, symmetry transformations should be applied to all the sheets in the same way, resulting in a sum over insertions of the stress-tensor. Eq. (2.8) is written in a somewhat loose notation, which highlights the properties of the displacement. The right hand side should be intended as an additional contribution arising when the left hand side is inserted in a correlation function. We refer to appendix A for a derivation, and we content ourselves here with a few remarks: It is important that the quantum numbers of $D^{a}$ are fixed by the Ward identity, i.e., its scaling dimension is $\Delta=d-1$ and it carries one unit of spin under rotations around the defect. Notice that its normalization is also fixed by (2.8). Therefore, its Zamolodchikov norm $C_{D}(n)$ is a property of the defect under consideration:

Here and in the following the subscript $n$ applied to expectation values implies the presence of the twist operator:

Let us finally mention that generic defects might have a more complicated structure of contact terms showing up in the divergence of the stress-tensor: more operators might be present, associated with derivatives of $\delta$-functions appearing in eq. (2.8). They can be written down systematically billo:confdef but we will not need this information here.

A consequence of the Ward identity (2.8) is that a small deformation $\delta x^{a}(x^{i})$ of the defect, is obtained by integrating the displacement operator in the action. The first order variation under such a deformation can be written

where $X$ is an arbitrary product of local operators. As already pointed out, a flat twist operator preserves a subgroup of the conformal transformations which includes dilatations. As an immediate consequence, scale invariance prevents defect operators from acquiring an expectation value in this particular case. Hence, the first order variation of the partition function (2.5) vanishes for a flat (or spherical) entangling surface, or more precisely, it is non-universal. The second order variation is then related directly to $C_{D}$. Indeed, denoting the variation as $\epsilon\,\delta x^{a}$, we find

The double integration will contain divergences which must be regulated. However, power-law divergences can be unambiguously tuned away, and finite or logarithmically divergent parts are universal well defined quantities, proportional to $C_{D}$.

Hence eq. (2.12) shows very explicitly that the displacement operator is the key element of the defect CFT living on the twist operator, which governs the shape dependence of the Rényi entropy, which has been extensively studied in the recent literature e.g., mark0 –aitor7 . A key result of this paper is unify a variety of conjectures related to this shape dependence in terms of a constraint on $C_{D}$, the coefficient defining the two-point function (2.9) of the displacement operator. In particular, these conjectures imply that the value of $C_{D}$ is entirely determined by the one-point function of the stress tensor in presence of the defect, also called the conformal dimension of the twist operator. The latter, dubbed $h_{n}$, is defined by the leading singularity of the one-point function $\braket{T_{\mu\nu}}_{n}\equiv\braket{T_{\mu\nu}\,\tau_{n}}/\braket{\tau_{n}}$. For a planar conformal defect in Euclidean flat geometry, this leading singularity is easily identified as it is completely fixed by symmetry

Here $n_{a}$ is a unit normalized vector normal to the entangling surface and $r=|x^{a}|$ the transverse distance. The factor $n$ in the denominator appears so that $h_{n}$ is the coefficient in the one-point function for the total stress tensor (summed over all of the replicas), e.g., as defined in Hung:2014npa . In the following, we demonstrate that if, in a $d$-dimensional CFT, the values of $C_{D}(n)$ and $h_{n}$ are constrained to obey the following equality

then the Rényi entropy satisfies a number of interesting properties, outlined below, with regards to shape dependence.

One immediate consequence of this relation is $C_{D}(1)=0$, which must hold since the defect disappears for $n=1$. Further, if we analytically continue (2.14) to real $n$, we can consider the first order variation around $n=1$:

where we used the relation

first observed in Hung:2011nu for holographic theories and then proven in Hung:2014npa for general CFTs. Implicity, the recent results of new imply that eq. (2.15) holds for generic CFTs. Hence in the vicinity of $n=1$, the proposed relation (2.14) is a constraint that holds for general CFTs.

Moving away from $n=1$, the constraint in eq. (2.14) produces an number of interesting properties for the shape dependence of the Rényi entropy, which have appeared previously in the literature as conjectures:

• •

  In section 3, we calculate the second order correction to the Rényi entropy induced by small perturbations of a perfect sphere. In the limit $n\to 1$, the formula (2.15) reproduces the variation of the entanglement entropy across a deformed sphere conjectured in mark0 for arbitrary dimensions, which was recently proven in new .

• •

  Eq. (2.15) also allows one to compute the universal contribution to the Rényi entropy for an entangling surface with a (hyper)conical singularity of opening angle $\Omega$. The leading coefficient in an expansion around the smooth entangling surface has been conjectured to be related the conformal weight $h_{n}$ Bueno:2015lza — see also Bueno:2015rda ; Bueno:2015xda ; Bueno:2015qya . In section 4, we prove the equivalence of that conjecture and formula (2.15).

• •

  With $d=4$, eq. (2.14) implies the equivalence of the coefficients $f_{b}(n)$ and $f_{c}(n)$ in the universal part of the four-dimensional Rényi entropy for general $n$, as discussed in Lewkowycz:2014jia ; safdi1 . This is demonstrated in section 5 by relating $f_{b}$ to $C_{D}$, and $f_{c}$ to $h_{n}$. However, we re-iterate that dong34 recently showed that the proposed equivalence $f_{b}(n)=f_{c}(n)$ does not hold for four-dimensional holographic CFTs dual to Einstein gravity.

The latter result demonstrates that eq. (2.14) is not a universal relation that holds in all CFTs (for general values of $n$). However, it is then interesting to ask for precisely which CFTs does such a constraint hold. It seems that free field theories are a good candidate for such a theory. Certainly, the results of Bueno:2015qya ; Dowker1 ; Dowker2 imply that eq. (2.14) holds for free scalars and fermions in three dimensions. Further, our calculations in Appendix C confirm that it also holds for free massles scalars in four dimensions. We hope to return to this question in future work future .

In this section, we study shape dependence of the Rényi entropy for a generic CFT in flat space. In particular, we calculate the second order correction to the Rényi entropy induced by small perturbations of a perfect sphere. In the limit $n\to 1$, our findings agree with the holographic results previously found in mark0 .

Starting from (2.12), we note that upon slightly deforming a spherical entangling surface with $\epsilon\,\delta x^{a}$, the leading correction to $S_{n}$ appears at second order and is given by

Here, the two integrals run over the original spherical entangling surface of radius $R$. We will restrict the deformation to the $t_{\textrm{\tiny E}}=0$ time slice and denote $\delta\vec{x}=f(x)\,\hat{r}$ where $\hat{r}$ is a unit vector in the radial direction. The relevant correlator (2.9) then beomes

with $\gamma$ being the angle between $x,x^{\prime}\in S^{d-2}$.

Let us now represent the two-point correlator (3.2) in the basis of spherical harmonics on $S^{N}$ ($N\equiv d-2$)

where $\ell_{N}\geq\ell_{N-1}\geq\ldots\geq|\ell_{1}|$ are integers and

For simplicity, we assume that one of the points is sitting at the north pole, in which case only spherical harmonics with $\ell_{N-1}=\ell_{N-2}=\ldots=\ell_{1}=0$ contribute

Hence, by assumption $\gamma=\theta_{N}$ in (3.2), and the following identity holds

where we introduced a new variable $z=\cos\gamma$.

The above integral diverges at $z=1$. This is not surprising given that the coefficients $A_{\ell_{N}}$ correspond to a spherical harmonic representation of a singular function (3.2). To regulate these coefficients let us modify the power of $(1-\cos\gamma)$ in (3.2) by introducing a new parameter $\alpha$ such that $A_{\ell_{N}}$ takes the form

where $y=(1-z)/2$ and we used (3.7) to express the associated Legendre polynomial in terms of the hypergeometric function. Now the integral can be readily evaluated assuming that $\alpha$ is large enough⁴⁴4As usual, small values of $\alpha$ are treated by analytic continuation.

For odd $N$ (odd $d$) the limit $\alpha\to 0$ is finite. However, it diverges for even $N$ (even $d$). Therefore we analyze these cases separately.

In this section, we consider the relation of the proposed constraint (2.14) to various conjectures about the universal contribution to the Rényi  entropy coming from singular deformations of entangling surfaces. In particular, Bueno:2015rda ; Bueno:2015xda proposed a conjecture for the universal corner contribution to the entanglement entropy in three-dimensional CFTs, and this conjecture was then extended to Rényi entropy in Bueno:2015qya . Finally, the discussion was extended to higher dimensions in Bueno:2015lza . In order to introduce the claim of these conjectures, we consider a deformation of a flat entangling surface which consists in creating a conical singularity. The three- and four-dimensional cases are shown in figure 1 of ref. Bueno:2015lza . The universal contribution to the Rényi (and consequently the entanglement) entropy is affected by this modification. In particular, if the twist operator is smooth, the universal contribution would be logarithmically divergent in even dimensions and constant (i.e., regulator independent) in odd dimensions. When a conical singularity is present an additional logarithm emerges and the universal contribution to the Rényi entropy takes the form

Here $\Omega$ is the opening angle of the cone, varying in the interval $[0,\frac{\pi}{2}]$ and approaching $\frac{\pi}{2}$ in the limit of smooth surface.⁵⁵5The angle $\Omega$ actually varies over the full range $[0,\pi]$, but, since the Rényi entropy evaluated for a pure state is equal for the region $A$ or for its complement $\bar{A}$, the function $a_{n}^{(d)}$ is symmetric for reflections with respect to $\Omega=\frac{\pi}{2}$, i.e. $a_{n}^{(d)}(\Omega)=a_{n}^{(d)}(\pi-\Omega)$ and we can consistently focus on the interval $[0,\frac{\pi}{2}]$ The function $a_{n}^{(d)}$ is the universal contribution to the Rényi entropy and depends on the angle $\Omega$ only. Further $\ell$ and $\delta$ are the IR and UV regulators, respectively. The former can be thought of as a (macroscopic) length scale characterizing the geometry of the entangling region $A$ (i.e., the region enclosed by the twist operator), whereas the latter can be taken to be a short-distance cut-off originating from the infinite number of short-distance correlations in proximity of the twist-operator. The cusp conjecture, in the most general formulation of Bueno:2015lza , states that, for an arbitrary conformal field theory, the leading contribution to $a_{n}^{(d)}$ for $\Omega\to\frac{\pi}{2}$ is controlled by the constant $h_{n}$ introduced in (2.13). Explicitly,

Restricting to the case $n=1$ and using (2.16), one finds the following relation between the small angle contribution to the entanglement entropy and the central charge $C_{T}$ of a CFT

In the following, we will apply the theoretical framework introduced in section 2 to this particular deformation and find a connection between $\sigma_{n}$ and $C_{D}$. This allows us to prove the equivalence of the cusp conjecture and eq. (2.14).