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On entanglement entropy functionals in higher derivative gravity theories

Arpan Bhattacharyya, Menika Sharma

TL;DR

This work tests the proposed holographic entanglement entropy (HEE) functional for higher-derivative gravity (the FPS/ Dong–Camps expression) by deriving the minimal-surface equations from extremizing the functional and by applying the Lewkowycz–Maldacena (LM) method, in both general $R^2$ gravity and quasi-topological (R^3) gravity. It finds that while the FPS functional reproduces the correct universal terms for entanglement entropy in these theories, the LM method does not generally yield the same surface equations, exhibiting limit ambiguities and extra divergences; GB theory is a notable exception where leading terms can coincide under certain limits. In quasi-topological gravity, the universal terms again agree with prior results, but the LM-derived surface equation deviates from the FPS-derived one, indicating limitations of the LM approach outside Lovelock theories. The results underscore the need for additional tests or modified methods to validate entropy functionals in generic higher-derivative gravities and motivate further exploration of the relationship between entropy functionals, conical singularity regularization, and brane interpretations.

Abstract

In arXiv:1310.5713 and arXiv:1310.6659 a formula was proposed as the entanglement entropy functional for a general higher-derivative theory of gravity, whose lagrangian consists of terms containing contractions of the Riemann tensor. In this paper, we carry out some tests of this proposal. First, we find the surface equation of motion for general four-derivative gravity theory by minimizing the holographic entanglement entropy functional resulting from this proposed formula. Then we calculate the surface equation for the same theory using the generalized gravitational entropy method of arXiv:1304.4926. We find that the two do not match in their entirety. We also construct the holographic entropy functional for quasi-topological gravity, which is a six-derivative gravity theory. We find that this functional gives the correct universal terms. However, as in the four-derivative case, the generalized gravitational entropy method applied to this theory does not give exactly the surface equation of motion coming from minimizing the entropy functional.

On entanglement entropy functionals in higher derivative gravity theories

TL;DR

This work tests the proposed holographic entanglement entropy (HEE) functional for higher-derivative gravity (the FPS/ Dong–Camps expression) by deriving the minimal-surface equations from extremizing the functional and by applying the Lewkowycz–Maldacena (LM) method, in both general gravity and quasi-topological (R^3) gravity. It finds that while the FPS functional reproduces the correct universal terms for entanglement entropy in these theories, the LM method does not generally yield the same surface equations, exhibiting limit ambiguities and extra divergences; GB theory is a notable exception where leading terms can coincide under certain limits. In quasi-topological gravity, the universal terms again agree with prior results, but the LM-derived surface equation deviates from the FPS-derived one, indicating limitations of the LM approach outside Lovelock theories. The results underscore the need for additional tests or modified methods to validate entropy functionals in generic higher-derivative gravities and motivate further exploration of the relationship between entropy functionals, conical singularity regularization, and brane interpretations.

Abstract

In arXiv:1310.5713 and arXiv:1310.6659 a formula was proposed as the entanglement entropy functional for a general higher-derivative theory of gravity, whose lagrangian consists of terms containing contractions of the Riemann tensor. In this paper, we carry out some tests of this proposal. First, we find the surface equation of motion for general four-derivative gravity theory by minimizing the holographic entanglement entropy functional resulting from this proposed formula. Then we calculate the surface equation for the same theory using the generalized gravitational entropy method of arXiv:1304.4926. We find that the two do not match in their entirety. We also construct the holographic entropy functional for quasi-topological gravity, which is a six-derivative gravity theory. We find that this functional gives the correct universal terms. However, as in the four-derivative case, the generalized gravitational entropy method applied to this theory does not give exactly the surface equation of motion coming from minimizing the entropy functional.

Paper Structure

This paper contains 15 sections, 104 equations.

Table of Contents

  1. Introduction
  2. Proposed entropy functional for general theories of gravity
  3. Test of the entropy functional for R$^2$ theory
  4. Minimal surface condition from the entropy functional
  5. Minimal surface condition from the Lewkowycz-Maldacena method
  6. Gauss-Bonnet theory revisited
  7. The general case
  8. The stress-energy tensor from the brane interpretation
  9. Quasi-topological gravity
  10. The entropy functional
  11. Universal terms
  12. Minimal surface condition
  13. Discussion
  14. Conical Metric
  15. Curvature Tensors