As in the Lorentz invariant case to compute the entanglement entropy associated with a surface we choose spatial coordinates , where is the coordinate orthogonal to the surface and are the coordinates on the surface . Then, after going to Euclidean time , we switch to the polar coordinates, , . In the Lorentz invariant case the conical space, which is needed for calculation of the entanglement entropy, is obtained by making the angular coordinate periodic with period by applying the Sommerfeld formula (22) to the heat kernel. If Lorentz invariance is broken, as it is for the operator (45), there are certain difficulties in applying the method of the conical singularity when one computes the entanglement entropy. The difficulties come from the fact that the wave operator , if written in terms of the polar coordinates and , becomes an explicit function of the angular coordinate . As a result of this, the operator is not invariant under shifts of to arbitrary . Only shifts with , where is an integer are allowed. Thus, in this case one cannot apply the Sommerfeld formula since it explicitly uses the symmetry of the differential operator under shifts of angle . On the other hand, a conical space with angle deficit is exactly what we need to compute for the reduced density matrix. In [184], by using some scaling arguments it was shown that the trace of the heat kernel on a conical space with periodicity, is

where is the bulk contribution. By the arguments presented in Section 2.7 there is a unique analytic extension of this formula to non-integer . A simple comparison with the surface term in the heat kernel of the Lorentz invariant operator, which was obtained in Section 2.12, shows that the surface terms of the two kernels are identical. Thus, we conclude that the entanglement entropy is given by the same formula where is defined in Eq. (40), as in the Lorentz invariant case (41). A similar property of the entanglement entropy was observed for a non-relativistic theory described by the Schrödinger operator [205] (see also [59] for a holographic derivation). For polynomial operators, , some scaling arguments can be used [205] to get the form of the entropy that follows from Eq. (47).In the rest of the review we shall mostly focus on the study of Lorentz invariant theories, with field operator quadratic in derivatives, of the Laplace type, .

Living Rev. Relativity 14, (2011), 8
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