Shape dependence of entanglement entropy in conformal field theories

  • Thomas Faulkner
  • Robert G. Leigh
  • Onkar Parrikar
Open Access
Regular Article - Theoretical Physics


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 CT 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.


AdS-CFT Correspondence Field Theories in Higher Dimensions 


Open Access

This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.


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Copyright information

© The Author(s) 2016

Authors and Affiliations

  • Thomas Faulkner
    • 1
  • Robert G. Leigh
    • 1
  • Onkar Parrikar
    • 1
  1. 1.Department of PhysicsUniversity of IllinoisUrbanaU.S.A.

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