arXiv:1407.2891 · Journal of High Energy Physics

Entanglement entropy, planar surfaces, and spectral functions

Vladimir Rosenhaus, Michael Smolkin

Published Sep 1, 2014• 47 citations

Abstract

We consider the universal part of entanglement entropy across a plane in flat space for a QFT, giving a non-perturbative expression in terms of a spectral function. We study the change in entanglement entropy under a deformation by a relevant operator, providing a pertrubative expansion where the terms are correlation functions in the undeformed theory. The entanglement entropy for free massive fermions and scalars easily follows. Finally, we study entanglement entropy across a plane in a background geometry that is a deformation of flat space, finding new universal terms arising from mixing of geometry and couplings of the QFT.

DOI: 10.1007/jhep09(2014)119

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arXiv:1407.2891 · Journal of High Energy Physics

Entanglement entropy, planar surfaces, and spectral functions

Vladimir Rosenhaus, Michael Smolkin

Published Sep 1, 2014• 47 citations

arXiv HTML PDF

Abstract

DOI: 10.1007/jhep09(2014)119

Related papers

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We study universal features in the shape dependence of entanglement entropy in the vacuum state of a conformal field theory (CFT) on ${\mathrm{\mathbb{R}}}^{1,d-1}$ . We consider the entanglement entropy across a deformed planar or spherical entangling surface in terms of a perturbative expansion in the infinitesimal shape deformation. In particular, we focus on the second order term in this expansion, known as the entanglement density. This quantity is known to be non-positive by the strong-subadditivity property. We show from a purely field theory calculation that the non-local part of the entanglement density in any CFT is universal, and proportional to the coefficient C T appearing in the two-point function of stress tensors in that CFT. As applications of our result, we prove the conjectured universality of the corner term coefficient $\frac{\sigma }{C_T}=\frac{\pi^2}{24}$ in d = 3 CFTs, and the holographic Mezei formula for entanglement entropy across deformed spheres.We study universal features in the shape dependence of entanglement entropy in the vacuum state of a conformal field theory (CFT) on $$ {\mathrm{\mathbb{R}}}^{1,d-1} $$ . We consider the entanglement entropy across a deformed planar or spherical entangling surface in terms of a perturbative expansion in the infinitesimal shape deformation. In particular, we focus on the second order term in this expansion, known as the entanglement density. This quantity is known to be non-positive by the strong-subadditivity property. We show from a purely field theory calculation that the non-local part of the entanglement density in any CFT is universal, and proportional to the coefficient C T appearing in the two-point function of stress tensors in that CFT. As applications of our result, we prove the conjectured universality of the corner term coefficient $$ \frac{\sigma }{C_T}=\frac{\pi^2}{24} $$ in d = 3 CFTs, and the holographic Mezei formula for entanglement entropy across deformed spheres.

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We study universal features in the shape dependence of entanglement entropy in the vacuum state of a conformal field theory (CFT) on

{\mathrm{\mathbb{R}}}^{1,d-1}

. We consider the entanglement entropy across a deformed planar or spherical entangling surface in terms of a perturbative expansion in the infinitesimal shape deformation. In particular, we focus on the second order term in this expansion, known as the entanglement density. This quantity is known to be non-positive by the strong-subadditivity property. We show from a purely field theory calculation that the non-local part of the entanglement density in any CFT is universal, and proportional to the coefficient C T appearing in the two-point function of stress tensors in that CFT. As applications of our result, we prove the conjectured universality of the corner term coefficient

\frac{\sigma }{C_T}=\frac{\pi^2}{24}

in d = 3 CFTs, and the holographic Mezei formula for entanglement entropy across deformed spheres.

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.