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Holographic Entanglement Entropy for General Higher Derivative Gravity

Xi Dong

TL;DR

The paper derives a general holographic entanglement entropy functional for theories dual to higher-derivative gravity whose Lagrangian is built from contractions of the Riemann tensor. This functional combines Wald-like contributions with extrinsic-curvature (anomaly) corrections obtained via regularized squashed cones and a careful treatment of would-be logarithmic divergences, providing a covariant formulation. The authors verify the formula in several key theories: $f(R)$ gravity, general four-derivative gravity, and Lovelock gravity, recovering known results such as Wald entropy and Jacobson–Myers entropy, and they show the entangling surface can be located by minimizing the functional in several cases. The work offers a practical framework to compute holographic EE in broad classes of higher-derivative theories and suggests a minimization principle as an efficient method to determine the relevant bulk surface, with potential implications for entanglement structure and black hole entropy.

Abstract

We propose a general formula for calculating the entanglement entropy in theories dual to higher derivative gravity where the Lagrangian is a contraction of Riemann tensors. Our formula consists of Wald's formula for the black hole entropy, as well as corrections involving the extrinsic curvature. We derive these corrections by noting that they arise from naively higher order contributions to the action which are enhanced due to would-be logarithmic divergences. Our formula reproduces the Jacobson-Myers entropy in the context of Lovelock gravity, and agrees with existing results for general four-derivative gravity. We emphasize that the formula should be evaluated on a particular bulk surface whose location can in principle be determined by solving the equations of motion with conical boundary conditions. This may be difficult in practice, and an alternative method is desirable. A natural prescription is simply minimizing our formula, analogous to the Ryu-Takayanagi prescription for Einstein gravity. We show that this is correct in several examples including Lovelock and general four-derivative gravity.

Holographic Entanglement Entropy for General Higher Derivative Gravity

TL;DR

The paper derives a general holographic entanglement entropy functional for theories dual to higher-derivative gravity whose Lagrangian is built from contractions of the Riemann tensor. This functional combines Wald-like contributions with extrinsic-curvature (anomaly) corrections obtained via regularized squashed cones and a careful treatment of would-be logarithmic divergences, providing a covariant formulation. The authors verify the formula in several key theories: gravity, general four-derivative gravity, and Lovelock gravity, recovering known results such as Wald entropy and Jacobson–Myers entropy, and they show the entangling surface can be located by minimizing the functional in several cases. The work offers a practical framework to compute holographic EE in broad classes of higher-derivative theories and suggests a minimization principle as an efficient method to determine the relevant bulk surface, with potential implications for entanglement structure and black hole entropy.

Abstract

We propose a general formula for calculating the entanglement entropy in theories dual to higher derivative gravity where the Lagrangian is a contraction of Riemann tensors. Our formula consists of Wald's formula for the black hole entropy, as well as corrections involving the extrinsic curvature. We derive these corrections by noting that they arise from naively higher order contributions to the action which are enhanced due to would-be logarithmic divergences. Our formula reproduces the Jacobson-Myers entropy in the context of Lovelock gravity, and agrees with existing results for general four-derivative gravity. We emphasize that the formula should be evaluated on a particular bulk surface whose location can in principle be determined by solving the equations of motion with conical boundary conditions. This may be difficult in practice, and an alternative method is desirable. A natural prescription is simply minimizing our formula, analogous to the Ryu-Takayanagi prescription for Einstein gravity. We show that this is correct in several examples including Lovelock and general four-derivative gravity.

Paper Structure

This paper contains 24 sections, 89 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Review: derivation of the Ryu-Takayanagi prescription
  3. Two methods
  4. Variation of the action
  5. A general entropy formula
  6. Regularized squashed cones
  7. Derivation of the formula
  8. First term (Wald's formula)
  9. Second term ("anomaly")
  10. Covariant form
  11. Examples
  12. f(R) gravity
  13. General four-derivative gravity
  14. Lovelock gravity
  15. Minimization of the entropy formula
  16. ...and 9 more sections

Figures (2)

  • Figure 1: Left: the $n$-fold cover $M_n$ for a $1+1$ dimensional field theory. Right: the $Z_n$ orbifold which has a conical defect $C_n$ in the bulk dual.
  • Figure 2: A pictorial way of filling in the tip of the cone and writing $S_{\rm outside}=S_{\rm total}-S_{\rm inside}$.