arXiv:1905.01675

Curved Corner Contribution to the Entanglement Entropy in an Anisotropic Spacetime

Mostafa Ghasemi, Shahrokh Parvizi

Published May 5, 2019• 1 citation

Abstract

In this article, we explore the divergences and universal terms of the holographic entanglement entropy for singular regions in anisotropic and nonconformal theories that are holographically dual to geometries with a hyperscaling violation, parameterized by two parameters $z$ 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.

DOI: 10.48550/arxiv.1905.01675

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arXiv:1905.01675

Curved Corner Contribution to the Entanglement Entropy in an Anisotropic Spacetime

Mostafa Ghasemi, Shahrokh Parvizi

Published May 5, 2019• 1 citation

arXiv HTML PDF

Abstract

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.

DOI: 10.48550/arxiv.1905.01675

Related papers

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arXiv:1709.08169Journal of High Energy PhysicsFeb 1, 20188 citations

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

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.