arXiv:1608.04900 · Physics Letters B

On the logarithmic divergent part of entanglement entropy, smooth versus singular regions

Harald Dorn A

Published Oct 19, 2016• 8 citations

Abstract

The entanglement entropy for smooth regions A has a logarithmic divergent contribution with a shape dependent coefficient and that for regions with conical singularities an additional log 2 term. Comparing the coefficient of this extra term, obtained by direct holographic calculation for an infinite cone, with the corresponding limiting case for the shape dependent coefficient for a regularised cone, a mismatch by a factor two has been observed in the literature. We discuss several aspects of this issue. In particular a regularisation of A , intrinsically delivered by the holographic picture, is proposed and applied to an example of a compact region with two conical singularities. Finally, the mismatch is removed in all studied regularisations of A , if equal scale ratios are chosen for the limiting procedure.

DOI: 10.1016/j.physletb.2016.10.029

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arXiv:1608.04900 · Physics Letters B

On the logarithmic divergent part of entanglement entropy, smooth versus singular regions

Harald Dorn A

Published Oct 19, 2016• 8 citations

arXiv HTML PDF

Abstract

DOI: 10.1016/j.physletb.2016.10.029

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z

and

θ

. We study a curved corner in anisotropic space with arbitrary

θ

and

z

. We choose the region to be shape invariant under the scaling of spacetime. For this case, we show that the contribution of the singularity to the entanglement entropy depends on

z

and

θ

values. We identify the structure of various divergences that may appear, especially those which give rise to a universal contribution in the form of logarithmic or double logarithmic terms. In the range

z>1

, for values

z=2k/(2k-1)

with some integer

k

and

θ=0

, Lifshitz geometry, we find a double logarithmic term. In the range

z<0

, for values

θ=1-2n|z-1|

with some integer

n

we find a logarithmic term.

{S}_n^{\mathrm{univ}}=-\frac{1}{8\pi }{f}_b(n){\int}_{\gamma }{k}^2{\log}^2\left(R/\delta \right)

S n univ = − 1 8 π f b n ∫ γ k 2 log 2 R / δ , where γ is the curve formed by the intersection of the entangling surface with a unit sphere centered at the vertex, and k the trace of its extrinsic curvature. While for circular and elliptic cones this term reproduces the general-CFT result, it vanishes for polyhedral corners. For those, we argue that the universal contribution, which is logarithmic, is not controlled by a local integral, but rather it depends on details of the CFT in a complicated way. We also study the angle dependence for the entanglement entropy of wedge singularities in 3+1 dimensions. This is done for general CFTs in the smooth limit, and using free and holographic CFTs at generic angles. In the latter case, we show that the wedge contribution is not proportional to the entanglement entropy of a corner region in the (2 + 1)-dimensional holographic CFT. Finally, we show that the mutual information of two regions that touch at a point is not necessarily divergent, as long as the contact is through a sufficiently sharp corner. Similarly, we provide examples of singular entangling regions which do not modify the structure of divergences of the entanglement entropy compared with smooth surfaces.

Δ=\frac{d+2}{2}

generates a logarithmic universal term to the entanglement entropy. As the smooth case, when the boundary CFT deformed by a relevant operator, we find that the entanglement entropy of singular surface also contains a new logarithmic term which is due to relevant perturbation of the conformal field theory, and depends on the scaling dimension of relevant operator. We also find for extended singular surfaces,

c_{n}\times R^{m}

, as well as logarithmic term, the new universal double logarithmic terms may appear depending on the scaling dimension of relevant operator and spacetime dimensions. These new terms are due to relevant perturbation of the boundary theory.