Entanglement entropy across a deformed sphere

I study the entanglement entropy (EE) across a deformed sphere in conformal field theories (CFTs). I show that the sphere (locally) minimizes the universal term in EE among all shapes. In the work of Allais and Mezei [Phys. Rev. D 91, 046002 (2015)] it was derived that the sphere is a local extremum, by showing that the contribution linear in the deformation parameter is absent. In this paper I demonstrate that the quadratic contribution is positive and is controlled by the coefficient of the stress tensor two-point function, $C_T$. Such a minimization result contextualizes the fruitful relation between the EE of a sphere and the number of degrees of freedom in field theory. I work with CFTs with gravitational duals, where all higher curvature couplings are turned on. These couplings parametrize conformal structures in stress tensor $n$-point functions; hence I show the result for infinitely many CFT examples.

Figure 1

A $d=3$ example of an entangling surface $\mathrm{\Sigma }$, which is a deformed circle (1.4). EE is the entropy of the reduced density matrix of region $V$, $\mathrm{\Sigma }=\partial V$. The change of the universal term in EE, $s_{\text{univ}}$, due to a deformation of amplitude $\epsilon$ is $\mathcal{O}({\epsilon }^2)$) and is given by (1.1) in a ${\mathrm{CFT}}_3$.