A note on entanglement entropy for topological interfaces in RCFTs

In this paper we calculate the entanglement entropy for topological interfaces in rational conformal field theories for the case where the interface lies at the boundary of the entangling interval and for the case where it is located in the center of the entangling interval. We compare the results to each other and also to the recently calculated left/right entropy of a related BCFT. We also comment of the entanglement entropies for topological interfaces for a free compactified boson and Liouville theory.


Introduction
In two dimensional conformal field theories the construction of conformal interfaces between CF T 1 and CF T 2 is equivalent to the construction of conformal boundary conditions in the folded CFT which is the tensor product CF T 1 ⊗ CF T 2 [1,2]. There is a special class of interfaces which are called topological interfaces. In the folded CFT the condition for an interface to be topological is [2] L (1) n −L This condition implies that the interface is completely transmissive and these interfaces are called topological since their position and shape can be deformed without cost of energy.
In addition one can define a fusion product of topological interfaces by bringing two of them close together [3]. It has been argued in [4] that topological interfaces can furnish spectrum generating symmetries. Topological interfaces have been constructed for the a single free boson in [2,5] and for n free bosons compactified on an n dimensional torus in [6]. Topological interfaces in orbifold theories have been studied in [7].
The entanglement entropy of a spatial region A is defined by the von Neumann entropy of a reduced density matrix which is obtained by integrating out the degrees of freedom localized in the complement of A. For two dimensional CFTs [8,9] the entanglement entropy, where A is a single interval of length L, is given by Here is an UV cutoff and we dropped terms which vanish as → 0. The logarithmically divergent term only depends on the central charge of the CFT. Generically the constant term is non-universal and dependent on the definition of the cutoff since a simple rescaling of the cutoff is equivalent to adding a constant term to the entanglement entropy.
In the presence of a boundary or an interface the constant term becomes physical as one can study the difference in entanglement entropy for a system with an interface and without and the dependence on the UV cutoff cancels in this case. In the case of a conformal interface there are two different geometric setups which one can consider. First if the interface is located at the center of the entangling region A then [9] the entanglement entropy is of the form given in (1.2) and the constant term is the so called g-factor or boundary entropy [1]. A second geometry is given by choosing the interface to be located at the boundary of the entangling region A. The entanglement entropy in this case has only been calculated for some specific cases, notably for generic conformal interfaces for a single free boson in [10] and the Ising model in [11]. The entanglement entropy in this case takes the form Where f (I) is a complicated function depending on the parameters of the conformal interface, which we schematically denote by I. The entanglement entropies for Janus like interface theories [13,14,15] have been calculated for the symmetric geometry in [16,17] and for the case where the interface is at the boundary of the entangling surface in [18].
The goal of this note is to calculate the entanglement entropies for both geometric setups for topological interfaces in rational CFTs. The structure of the paper is as follows: In section 2 we review the construction of topological interfaces in rational CFTs which goes back to the work of Petkova and Zuber [12]. In section 3 we adapt the calculation of Sakai and Satoh [10] to calculate the entanglement entropy for the case where the topological interface is at the boundary of the entangling surface In section 4 we adapt the calculation of Sakai and Satoh to re-derive the entanglement entropy (1.2) for the symmetric interface and also give an argument that the location of the interface does not change the result as long as it is a finite distance away from the boundaries of the entangling interval. In section 5 we compare the entanglement entropies for various specific cases. We also include the recently computed left/right entropy [19,20] in the comparison. In section 6 we compare results of entanglement entropies for compact bosons which were previously obtained in the literature. In section 7 we use the construction of topological interfaces in Liouville theory given in [21,22] to attempt a calculation of the entanglement entropy for this system. We close with a discussion of future directions of research in section 8.
Note added: While this paper was finalized a paper [23] appeared, which has significant overlap with the material presented in section 3.

Topological interfaces in RCFT
In this section we consider the construction of topological interfaces in rational CFTs. A rational CFT (RCFT) [24] contains a finite number of primary states and hence a finite number of representations of the Virasoro algebra labeled by i with characters χ i (q). The partition function on the torus is given by where Z ij are positive integers which denote how many times a representation appears in the spectrum of the theory. We mostly limit ourselves the the case where Z ij = δ ij and the theory has a diagonal spectrum. 3 The canonical examples for rational CFTs 1 are the unitary minimal models which have central charge and the primaries are labeled by two integers r = 1, 2, · · · , m − 1 and s = 1, 2, · · · r and have conformal dimension The simplest minimal model is the Ising model which has m = 3 and hence we have c = 1 2 and three primaries with h = 0, h = 1 2 and h = 1 16 . In [12,25]  In [12] a classification of such operators was given analogous to the construction of Cardy states [26]. For the diagonal theories one finds Here P iī is a projector on the space spanned by the i − th primary and it's descendants This means that in the simple diagonal case there are as many topological interfaces as there are primaries, where for simplicity we assume that each primary only appears once in the theory, a degeneracy can be easily included in the construction. The matrix S is the modular S matrix which denotes how the characters transform under modular transformation q = e 2πiτ The conjugate interface operator is given by While the motivation to introduce the operators I was slightly different it was pointed out in [21] that the condition (2.4) is identical to the condition that a topological interface operator satisfies. Hence the classification of [12,25] provides us with a classification of topological interfaces in RCFTs.

Entanglement entropy at a topological interface
The entanglement entropy for an interval A is given by the von Neumann entropy of the reduced density matrix ρ A = trĀ|0 0|, whereĀ is the complement of A. The replica trick relates the entanglement entropy to the Renyi entropies as follows The Renyi entropies are calculated by a path integral over a K-sheeted Riemann surface with branch cuts running along A. The entanglement entropy can then be derived from the partition function Z(K) on the K-sheeted Riemann surface by In [10] it has been argued that the interface can be included in the K-sheeted partition function Z(K). The interface gets mapped via z = ln w to a covering space (see figure 1).
Introducing an UV cutoff and an IR cutoff L and imposing periodic boundary conditions for simplicity, the K-th replica partition function can be expressed as a trace where the "time" is related to the cutoffs by t = 2π 2 log(L/ ) (3.4) and is the Hamiltonian of the CFT. The K-th partition function with a topological interface (2.5) labeled by a primary a inserted is where we have introduced q =q = e −t . In the second line we have used (2.4) to commute X a through the Hamiltonian and in the third line we used the fact that the P i,i in (2.5) are projectors to the i-th representation and the trace produces the associated character χ i .
Since we are interested in taking the U V cutoff → 0 (or alternatively take L → ∞), we have to evaluate (3.6) in the limit q → 1. With the identification of a new modular parameter τ by with t given in (3.4), the limit can taken by performing a modular transformation on the characters lim q→1 In the limit t → 0 the leading contribution in (3.8) will come from the vacuum characters In that case the partition function (3.6) becomes where the dots indicate terms which vanish as the cutoff is taken to zero. Further calculating where we have repeatedly used the fact that S is real, symmetric, unitary and in particular the relation j |S aj | 2 = 1. Putting everything together we arrive at the following expression for the entanglement entropy at a topological interface

Symmetric and left/right entanglement entropy
For an interface which is located symmetrically on the entangling interval A the entanglement entropy has been calculated by [9] and is given by where g B is the boundary g-factor which when the interface is folded into a boundary state is determined by the overlap of the boundary state corresponding to the doubled interface with the vacuum state [1,27] For the topological interface (2.5) the boundary state becomes Consequently the g factor is given by and the symmetric entanglement entropy becomes Up to now we have considered the symmetric case where the interface is located at the center of the entangling interval A. There is however a simple argument showing that for topological interfaces the location of the interface does not change the result as long as it is a finite distance away from the boundary of the entangling interval. We illustrate the argument in figure 2. We start in the ζ plane with a finite interval A with boundary at ζ = 0 and ζ = l, where the interface is located along ζ = y + iξ, ξ ∈ R. We map the ζ plane into the w plane by the map This maps the finite interval to the positive real axis, the interface I gets mapped to an off centre circle. Finally we perform the replica map to the z coordinate via z = ln w and impose periodic boundary conditions as before at the cutoff z = ln and z = ln L. This produces again a torus. Unlike the case of the interface at the boundary here the interface is mapped into a vertical curve on the torus. For a topological interface it is clear that the shape can be changed and changing the location along the real part of z corresponds to changing the original location y of the interface. This shows that partition function on the K-th sheeted Riemann surface is independent of y as log as the interface is a finite distance away form the cutoffs. We can be more specific and evaluate the partition function where q =q = e − π 2 Kt where t is again given by (3.4), Hence in the limit of vanishing cutoff the sum over representations in the partition function gets projected on the vacuum character and one has log L + · · · (4.8) Applying the replica formula (3.2) one obtains S symm a = c 6 log L + ln S a0 S 00 (4.9) Hence we have re-derived (4.5). Additionally it is clear that for a topological interface moving the interface along the real axis in the z coordinates does not change (4.7) as the interface operator commutes with the generator of these translations, which is the Hamiltonian.
A third type of entanglement entropy which takes a similar form is the so called left/right entanglement entropy [19,20,28]. This is defined for a boundary CFT, where the entanglement entropy is calculated with a reduced density obtained by tracing over left-moving modes. Interestingly for a boundary CFT defined by a Cardy state [26] (for a single copy of the CFT not the doubled one we are considering in the previous sections) where | j are the Ishibashi states [29] enforcing conformal boundary conditions. We quote the result of the calculation of the left/right entanglement entropy which is also labeled by a primary a in a RCFT, obtained in [20] S l/r a = The physical relation of this result to the other two cases is not clear at the moment, as one considers a boundary of a single CFT instead of an interface between two CFTs and the relation of the UV cutoffs in both cases is not obvious.

Comparison of entanglement entropies
For the m-th unitary minimal models the modular S matrix is given by (see e.g. [30] ).
The entanglement entropies then take the following values The modular S-matrix is given by where s 1 and s 2 are given by The entanglement entropies then take the following values

Entanglement entropies for a compact boson
A general class of conformal interfaces for a free boson was constructed in [2] The interfaces are characterized by two radii R 1 , R 2 and two relatively prime integers k 1 , k 2 . They correspond to an interface between two compactified bosons where the compactification radius jumps from R 1 to R 2 across the interface. In the doubled BCFT description the interface corresponds to a euclidean D0 brane stretched on a rectangular torus with radii R 1 and R 2 where its one dimensional world volume wraps k 1 times around the R 1 circle and k 2 times around the R 2 circle. In general these interfaces are not topological but for special values of the radii where the following condition is satisfied We can compare this to the symmetric entanglement entropy, the g B factor for a general interface is given by [2,3] and hence the symmetric entanglement entropy for the topological interface which satisfies (6.1) becomes Where the constant part is half the value of S. Note that if we consider the case of an interface between identical CFTs, we have R 1 = R 2 and by (6.1) k 1 = k 2 for topological interfaces.

Remarks on entanglement entropies for Liouville theory
In [21,22] topological interfaces for the Liouville CFT (see [31,32] for reviews with references to the original literature) were constructed following the procedure which was used for RCFTs. There are two type of defects which are both of the form dα D(P )P α (7.1) Where we integrate P over the positive real line, i.e. P ∈ (0, ∞). And one has Q = b + 1/b, which determines the central charge as C = 1 + 6Q 2 . Here P is a projector on the continuum of primary states labelled by P and their descendants.
As shown in [21] one can distinguish the two D by associating them with the discrete degenerate primary states labelled by two integers The second interface is labelled by a continuous real parameter s and given by We can now calculate the K-sheeted partition function (3.3) with the interface (7.1) inserted.
Using the fact that the projectors satisfies P α P β = δ(α − β)P α (7.5) and the fact that the interface operator I satisfies (2.4) we arive at where χ P (q) is the character of the non-degenerate Liouville primary field labelled by P and given by Where q 2πiτ and we can use the following formula for the modular transformation of the character (7.7) With the identification (3.7) and t given by (3.4) as before, the modular transformed Ksheeted partition function becomes ∞ 0 dP e 4πiPP χP (i π Kt ) (7.9) In the limit t → 0 we can replace the full character χ P (q) by its leading term q P 2 − c 24 and perform the gaussian integrals over P andP which produce the same result. Hence we arrive at ∞ 0 dP D(P ) K e −4P 2 Kt + · · · (7.10) Where the dots denote terms which vanish as t goes to zero. We would now like to use this expression to calculate the entanglement entropy using the replica formula (3.2). Note that for the case where D is labelled by a continuous parameter s and given by (7.4) D(P ) K in the integral (7.10) vanishes for large P . It is therefore legitimate to drop the exponent e −4P 2 Kt in the integral and the non vanishing terms in entanglement entropy for this case are given by We notice two curious features of this result. First, the logarithmically divergent term is multiplied by 1 3 which is what one would expect for a c = 1 CFT, whereas the central charge of the Liouville theory is given by C L = 1 + 6Q 2 . Second, apart from finite terms as t → 0 we also obtain an additonal divergent term of the form ln(ln L ) from the second term (7.11). The significance and interpretation of this term is not clear at this point and a more careful treatment of the cutoff might be necessary. For the interfaces labelled by discrete integers m, n defined in (7.3), D mn diverges for large P and the full integral has to be evaluated first in order to obtain the entanglement entropy. We leave this problem for future work.

Discussion
In this paper we have discussed entanglement entropies in the presence of topological defects in two geometric settings, namely when the interface is located at the boundary of the entangling interval A and when it is in the center of the entangling interval. For topological defects in RCFTs the logarithmic part of the entanglement entropy is always universal (this is not the case for general conformal interfaces) and the constant term can be expressed in a compact form in terms of the modular matrix S. We have also compared these entropies to the recently obtained left/right entanglement entropy for a related BCFT, but the physical relation of this quantity to the two entanglement entropies we have discussed is not clear at this moment.
There are several directions in which our results can be generalized. We have limited ourselves to RCFTs with diagonal partition functions. The construction of [12] also includes non-diagonal theories and it would be interesting to understand the entanglement entropy for this case. We also only considered CFTs which are rational with respect to the Virasoso algebra, it would also be very interesting to repeat the analysis for RCFTs with respect to extended chiral algebras.
Since the large m limit of minimal models is conjectured to approach a non-rational c = 1 CFT which is different from a free boson [33] it would be interesting to study the continuation of the minimal model entanglement entropy.
It would also be interesting to understand the entanglement entropy for Liouville theory better. Apart from the calculations sketched in section 7 one might also consider semiclassical limits where b → 0 and analyze the role of topological defect in classical Liouville theory following [34,22].