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Entanglement entropy across a deformed sphere

Márk Mezei

TL;DR

The paper analyzes how the ground-state entanglement entropy (EE) in a conformal field theory depends on the shape of the entangling surface, showing that a sphere minimizes the universal EE locally under small deformations. Using holography with higher-curvature gravity and the generalized Wald–Dong entropy functional, the author computes EE to second order in the deformation amplitude and finds a explicit quadratic shape correction $s_d^{(2)}( ext{Σ})$ proportional to the stress-tensor two-point function coefficient $C_T$, with a positive-definite shape functional. The linear-in-$ ext{deformation}$ term vanishes, establishing the sphere as a local minimum for the universal EE across nearby shapes. These results align with known field-theory data in $d=4$ and $d=6$ and integrate into perturbative analyses of EE, thereby linking EE to the number of degrees of freedom via $C_T$ and supporting sphere-based monotonicity considerations in CFTs.

Abstract

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 arXiv:1407.7249 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.

Entanglement entropy across a deformed sphere

TL;DR

The paper analyzes how the ground-state entanglement entropy (EE) in a conformal field theory depends on the shape of the entangling surface, showing that a sphere minimizes the universal EE locally under small deformations. Using holography with higher-curvature gravity and the generalized Wald–Dong entropy functional, the author computes EE to second order in the deformation amplitude and finds a explicit quadratic shape correction proportional to the stress-tensor two-point function coefficient , with a positive-definite shape functional. The linear-in- term vanishes, establishing the sphere as a local minimum for the universal EE across nearby shapes. These results align with known field-theory data in and and integrate into perturbative analyses of EE, thereby linking EE to the number of degrees of freedom via and supporting sphere-based monotonicity considerations in CFTs.

Abstract

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 arXiv:1407.7249 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, . 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 -point functions, hence I show the result for infinitely many CFT examples.

Paper Structure

This paper contains 12 sections, 44 equations, 1 figure.

Table of Contents

  1. Introduction and summary of results
  2. Introduction
  3. Summary of results
  4. Relation to field theory results
  5. Field theory results for EE
  6. Relation to the perturbative approach to EE
  7. EE of a deformed sphere in higher curvature gravity
  8. EE in higher derivative gravity
  9. Stress tensor correlators in higher derivative gravity
  10. Ryu--Takayanagi surface in Einstein gravity
  11. Simplification of the EE functional
  12. Evaluation of the EE of a deformed sphere

Figures (1)

  • Figure 1: A $d=3$ example of an entangling surface $\Sigma$, which is a deformed circle \ref{['GeomRegion']}. EE is the entropy of the reduced density matrix of region $V$, $\Sigma = \partial V$. The change of universal term in EE, $s_\text{univ}$ due to a deformation of amplitude $\epsilon$ is ${{\mathcal{O}}}(\epsilon^2$) and is given by \ref{['ShapeDep']} in a CFT$_3$.