Boundary effects in entanglement entropy
We present a number of explicit calculations of Renyi and entanglement entropies in situations where the entangling surface intersects the boundary in -dimensional Minkowski spacetime. When the boundary is a single plane we compute the contribution to the entropy due to this intersection, first in the case of the Neumann and Dirichlet boundary conditions, and then in the case of a generic Robin type boundary condition. The flow in the boundary coupling between the Neumann and Dirichlet phases is analyzed in arbitrary dimension and is shown to be monotonic, the peculiarity of case is noted. We argue that the translational symmetry along the entangling surface is broken due to the presence of the boundary which reveals that the entanglement is not homogeneous. In order to characterize this quantitatively, we introduce a density of entanglement entropy and compute it explicitly. This quantity clearly indicates that the entanglement is maximal near the boundary. We then consider the situation where the boundary is composed of two parallel planes at a finite separation and compute the entanglement entropy as well as its density in this case. The complete contribution to entanglement entropy due to the boundaries is shown not to depend on the distance between the planes and is simply twice the entropy in the case of single plane boundary. Additionally, we find how the area law, the part in the entropy proportional to the area of entire entangling surface, depends on the size of the separation between the two boundaries. The latter is shown to appear in the UV finite part of the entropy.
Laboratoire de Mathématiques et Physique Théorique CNRS-UMR 7350,
Fédération Denis Poisson, Université François-Rabelais Tours,
Parc de Grandmont, 37200 Tours, France
e-mail: firstname.lastname@example.org, Sergey.Solodukhin@lmpt.univ-tours.fr
Entanglement entropy is a useful tool which plays an important role in modern physics. First introduced  in order to explain the black hole entropy, it was later shown to be very efficient in measuring the quantum entanglement between sub-systems separated by a surface. In infinite spacetime this surface is necessarily compact so that it divides the spacetime into two complementary regions. The correlations present in the quantum system across the entangling surface produce the non-trivial entropy which is essentially determined by the geometry of the surface. The geometrical nature of entanglement entropy explains why it finds so many applications in various fields of physics, from black holes and holography to integrable models and quantum computers . For some recent progress in measuring entanglement entropy see .
For conformal field theories, the entanglement entropy plays a special and important role since the logarithmic terms in the entropy are related to the conformal anomalies, as suggested in . In infinite spacetime, the anomaly appears only in even dimensions. In parallel, for compact entangling surfaces, only in even dimensions there appear the logarithmic terms in the entropy.
Recently there has been some progress in understanding the conformal anomalies in the case where the spacetime is not infinite but has some boundaries, , , ,  (for earlier works see ). It is interesting that in the presence of boundaries the integrated anomaly is non-vanishing in odd spacetime dimensions, the relevant contribution being produced by the boundary terms only, . Thus, it becomes an interesting and urgent problem to understand the precise structure of the entropy for entangling surface which intersects the boundary of a spacetime. In the holographic context, this and related problems were studied in , , and on the field theory side in . The precise calculation for free fields of various spin in dimension has been done in  where it was shown that the logarithmic term in the entropy in this case is proportional to the number of intersections the entangling surface has with the boundaries. In higher dimensions it was suggested that, unlike the case of compact closed surfaces, the logarithmic terms in the entropy of a surface intersecting the boundary are present in any, odd and even, dimensions.
The boundary phenomenon in entanglement entropy is certainly more general and is not restricted only to conformal field theories. Yet, the explicit calculations for arbitrary boundaries and surfaces are technically complicated, if even possible. Therefore, we find it instructive to first analyze the problem in some simple cases, where the spacetime is flat and the boundary is composed by a collection of planes. In this paper we present a number of explicit calculations, for a free massive scalar field, of entanglement entropy in the case where the entangling surface is a plane which crosses orthogonally the boundary. The main focus is made on the role of the boundary conditions. The latter can be viewed as some form of boundary interactions. The general Robin type condition then interpolates between the Neumann condition in the weak coupling regime and the Dirichlet condition in the strong coupling regime. We study the respective behavior of entanglement entropy when the boundary coupling passes between these two regimes.
The paper is organized as follows. In Section 2 we review the standard replica method that uses the heat kernel and the conical singularity technology. We demonstrate how this method works for a simple case of infinite plane in infinite (without boundaries) Minkowski spacetime. This technology is then applied in Section 3 to the case of a single plane boundary with the Neumann (Dirichlet) boundary condition. The case of a general Robin type condition is considered in Section 4. We observe some inequalities for the entropy for different boundary conditions in Section 5. The monotonicity of the entropy with respect to the boundary coupling is demonstrated in Section 6. Two parallel boundaries and the effects of the finite size are considered in Section 7. In Section 8 we introduce a notion of the entanglement entropy density and calculate this quantity in all examples considered in the previous sections. We conclude in Section 9.
2 Replica method, heat kernel and entanglement entropy
Before proceeding, we remind the technical method very useful for calculation of entanglement entropy. This method is known as the replica method. One first observes that . The next observation is that the density matrix obtained by tracing over modes inside the surface is , where and is the partition function of the field system in question, considered on Euclidean space with a conical singularity at the surface . Thus one has that
One chooses the local coordinate system where is the Euclidean time, such that the surface is defined by the conditions and are the coordinates on . In the subspace it is convenient to choose the polar coordinate system and where angular coordinate changes in the limits . The conical space in question is then defined by making the coordinate periodic with the period , where is very small.
In order to calculate the effective action we use the heat kernel method. Consider a quantum bosonic field described by a field operator so that . Then the effective action is defined as
where is an UV cut-off, and is expressed by means of the trace of the heat kernel satisfying the heat kernel equation
In the Lorentz invariant case, the heat kernel (where we skip the coordinates other than the angle ) on regular flat space depends on the difference . The heat kernel on space with a conical singularity is then constructed from this quantity by applying the Sommerfeld formula 
The contour consists of two vertical lines, going from to and from to , and intersecting the real axis between the poles of : , and , respectively. For the integrand in (4) is a -periodic function and the contributions of these two vertical lines cancel each other. Thus, for a small angle deficit the contribution of the integral in (4) is proportional to .
In -dimensional spacetime, for a massive scalar field described by the operator , , where is the Euclidean time, the heat kernel is known explicitly,
We take a -surface to be the infinite plane defined by equations so that are coordinates on . In the polar coordinate system and we have for two points and that . The trace is defined as . For the contour integral over one finds (see )
Thus one obtains for the trace of the heat kernel
where is the volume of spacetime and is the area of the surface . The entanglement entropy is then easily obtained,
We stress that this is the entropy for an infinite plane in infinite (without boundaries) Minkowski spacetime. For the UV divergent part of the entropy we have
In even dimension the term with becomes a logarithm.
The Rényi entropy is defined by the formula
Thus, in order to compute this entropy one needs to keep finite in (6) and (7). One finds that in our example of infinite plane in Minkowski spacetime the Renyi entropy is simply proportional to the entanglement entropy,
In all examples considered in this paper we have a similar relation between the two entropies. In what follows we thus keep our focus on computing the entanglement entropy.
3 Single plane boundary: Neumann and Dirichlet boundary conditions
Consider -dimensional flat spacetime with coordinates and a plane boundary at . We define the entangling surface by the equations: . It crosses the boundary orthogonally, the intersection is -surface with coordinates . We impose Neumann or Dirichlet boundary condition at ,
where the plus (minus) corresponds to Neumann (Dirichlet) condition. Then we are supposed to go through the same steps as before. Taking the trace, i.e. identifying and , and taking the contour integral over and the integration over , and , we find
where the first term is the same as in infinite (without boundaries) spacetime and is the area of . We then use that . Applying the replica trick and computing the integration over proper time we arrive at the following form of the entanglement entropy,
Here is the entropy in infinite spacetime, defined in (8), and is the part of the entropy which is entirely due to the intersection of the entangling surface and the boundary . For the UV divergent part of this entropy one finds
In particular, for dimensions we find
The case was already considered in . We see from (16) that there appears a logarithmic term in if spacetime dimension is odd. Thus, there always appears a logarithm in the complete entanglement entropy: either due to in even dimension or due to in odd dimension .
4 Single plane boundary: Robin boundary condition
We now generalize the above analysis and consider a more general boundary condition of the Robin type,
where is the boundary coupling constant. Value corresponds to the Neumann boundary condition while the limit corresponds to the Dirichlet boundary condition. The corresponding solution to the heat kernel equation (3) takes the form (see ),
where is the coordinate orthogonal to the boundary and we skip all other coordinates. The trace of this heat kernel considered on spacetime with a conical singularity reads
where is the error function. Respectively we find for the entanglement entropy,
For positive boundary coupling and in the limit of large the function which appears under the integral in (21) behaves as
Therefore, the integral in (21) converges in the upper limit in dimension , even in the massless case () if the coupling is positive. On the other hand, for negative one has
and the integral in (21) converges in the upper limit only if the mass is sufficiently large, .
On the other hand, for small we find
Therefore, we note that in dimension the integral in (21) is divergent when the lower limit is taken to zero and thus the regularization with is needed. This is of course the usual UV divergence. However, in dimension the integral in (21) has a regular limit if is taken to zero. Thus, for any finite the integral in (21) is UV finite. The integration can be performed explicitly in dimension and one finds
It is interesting that this is the exact result. We see that in this case the boundary coupling appears only in the UV finite term in the entropy. We notice that the entropy (25) is divergent if . This is a IR divergence: the integral in (21) diverges in the upper limit if is negative and .
In higher dimensions the integration can be done in a form of an expansion in powers of ,
More generally, we find the expansion in arbitrary dimension ,
To leading order in we find in any dimension ,
In dimension the power law is replaced by a logarithm as in (26).
The integral (21) is divergent in the upper limit if . Therefore the entropy shows a divergence when goes to zero. This is a IR divergence. In dimension this divergence is logarithmic. In higher dimension the divergence is milder. The entropy takes a finite value if . However, the derivatives of sufficiently high order diverge there
so that the entropy is not an analytic function of at the point . This may signal for some type of a phase transition. We, however, do not elaborate on this idea here.
The other useful forms of (21) are
that compares the Robin entropy with the entropy in the case of the Dirichlet boundary condition, and
that compares it with the entropy in the case of infinite (without boundaries) spacetime. This equation generalizes (3) for arbitrary boundary coupling .
5 Some inequalities
Here we formulate some inequalities relating the entropies for various boundary conditions. The first obvious inequality follows from equation (3). Indeed, it simply indicates that the entropy for a field with the Neumann boundary condition is strictly larger than that for the Dirichlet boundary condition,
Including the entropy computed for a plane of the same area in infinite spacetime, we have
The other inequalities come from the comparison with the entropy for the Robin boundary condition. Comparing the entropy for the Neumann and the Robin boundary conditions we use equation (21). The function , introduced in (22), that appears in the integral in (21) is positive for positive values of and negative for negative values,
On the other hand, the comparison with the entropy for the Dirichlet boundary condition uses equation (31). The function that appears in the integral in this case is negative for any (positive or negative) values of ,
while for negative values of ,
Thus, increasing the negative values of one makes the entanglement entropy larger than it is for the Neumann boundary condition. However, one cannot make as negative as one wants since, as we have shown, the integral in (21) is not convergent for large if .
On the other hand, for positive , increasing the value of to infinity one arrives at the entanglement entropy for the Dirichlet boundary condition. Indeed, using that
one finds from equation (31) that
This relation indicates that in the limit the Robin entropy approaches the Dirichlet entropy,
We stress that this limit is valid only if so that should be smaller than the UV cut-off. The case of is special. In this case, the integral in (31) goes from to so that one necessarily includes the integration over small values of . Therefore, the approximation (39) cannot be justified for all values of . In fact, the integration over can be performed explicitly. The result (25) of this integration shows that in this case the limit is divergent and the Robin entropy does not approach the Dirichlet entropy. We stress once again that this is a peculiarity of three dimensions. Taking this observation it seems that the claim made in  that in CFT the RG flow which starts in the Neumann phase should end in the Dirichlet phase should probably be taken with some caution.
6 Monotonicity of flow between Neumann and Dirichlet phases
Above we have shown that, in dimension , varying the boundary coupling from zero to plus infinity, the Robin entropy changes from the Neumann entropy to the Dirichlet entropy. An interesting question is whether this evolution of the entropy is monotonic? The answer to this question is affirmative as we now show. Indeed, the derivative with respect to of the Robin entropy (21)
is negative as follows form the fact that
is positive for positive values of . Thus, the entropy is monotonically decreasing provided one changes the boundary coupling from zero to . It goes from the Neumann entropy for to the Dirichlet entropy for .
This demonstration is also valid in dimension . In fact, the monotonicity in this case can be seen directly from the exact formula (25). However, in the limit it does not approach the Dirichlet entropy. This is consistent with the discussion we made above.
7 Two parallel plane boundaries
We now want to analyze whether the boundary part in the entanglement entropy is affected by the finite size of the system. We start with a simple case of two parallel plane boundaries, at and . At each boundary one may impose either Neumann or Dirichlet boundary condition so that we have three cases to consider
7.1 Neumann-Neumann (Dirichlet-Dirichlet) boundary conditions
The explicit form for the corresponding heat kernel is
where the plus (minus) corresponds to Neumann (Dirichlet) condition and we keep only the dependence on coordinate orthogonal to the boundaries. As before, we define the entangling surface by equations: , . Repeating the conical space construction for this heat kernel we arrive at the following trace
where is the intersection of the entangling surface with both boundaries, so that it has two disconnected components, at each of the boundary. Respectively we find for the entanglement entropy
The integration over can be performed explicitly,
The sum over images then will give us
Remarkably, this result does not depend on the size . We conclude that the part in the entropy that is due to the intersection of the entangling surface with the boundary is not sensitive to the finite size . The whole effect of the presence of the second boundary is that this part in the entropy simply doubles,
so that the entropy is proportional to the complete area of the disjoint components of the intersection . In dimension , is the number of intersections of the line with the two boundaries.
The size , however, will appear in the area law, the part proportional to the area of the entire surface . In fact, this is the UV finite part of the entropy that will depend on . Indeed, in the sum over images the term with will produce the UV divergence already analyzed above and the terms with will give us a UV finite contribution. In order to identify this contribution we may interchange the order of the integration over and summation over . The integration (for ) then gives us
Thus we find
Some particular cases are worth mentioning.
1. In the massless case () in dimension one has
so that it decays by a power law. For the zeta-function in (56) diverges. This is yet another manifestation of the IR divergence in dimensions that we have already discussed.
2. In dimension the integral (53) produces elementary function,
so that the sum over in (55) can be easily evaluated and we find that
We see that it decays exponentially for large and approaches a logarithm for small ,
Similarly, one can analyze the massless limit in (58). In this limit there exists the IR divergence we have already discussed. Therefore, a IR regulator should be kept. We find that in this limit in the UV finite part in the entropy there appears a new logarithmic term,
where we used that the area and that in dimension . We see that this term is in fact not determined by the area of surface . It is due to a combination of two factors: the intersection of entangling surface with the boundary and the finite size of the system. The logarithmic term (60) resembles the entanglement entropy in two dimensions. It would be interesting to understand better the origin of this logarithmic term. Since in the massless case the theory becomes conformal, the logarithmic term (60) may be related to conformal symmetry.
3. In dimension , the sum over in (55) for gives
where is the polylogarithmic function. The asymptotics are given below for any .
4. In dimension we have
We see that for the boundary conditions of the same type (NN or DD) the UV finite part in the area law (55) is a positive quantity.
7.2 Mixed boundary conditions
Now we impose the mixed boundary conditions (46). The explicit form for the corresponding heat kernel is
Making this heat kernel -periodic and computing the trace we find
and for the entanglement entropy