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Entanglement entropy in higher derivative holography

Arpan Bhattacharyya, Apratim Kaviraj, Aninda Sinha

TL;DR

This work probes holographic entanglement entropy for higher-derivative gravity, focusing on Gauss-Bonnet theory as the tractable four-derivative case. By extending the Lewkowycz–Maldacena generalized gravitational entropy approach, it shows that the resulting entangling-surface equation in GB matches the proposed JM functional and aligns with Brown–York stress-tensor considerations in the regime where cubic extrinsic-curvature terms are subleading. It also demonstrates that the JM area functional acts as a natural counterterm to remove power-law divergences in the GB Euclidean action, reproducing the universal piece of the entanglement entropy proportional to the anomaly coefficient $a$. For general four-derivative theories, however, the analysis reveals that a pure geometric area functional does not generically exist, with Weyl-squared contributions obstructing such a formulation, illuminating limits on extending Ryu–Takayanagi-type prescriptions beyond GB gravity.

Abstract

We consider holographic entanglement entropy in higher derivative gravity theories. Recently Lewkowycz and Maldacena arXiv:1304.4926 have provided a method to derive the equations for the entangling surface from first principles. We use this method to compute the entangling surface in four derivative gravity. Certain interesting differences compared to the two derivative case are pointed out. For Gauss-Bonnet gravity, we show that in the regime where this method is applicable, the resulting equations coincide with proposals in the literature as well as with what follows from considerations of the stress tensor on the entangling surface. Finally we demonstrate that the area functional in Gauss-Bonnet holography arises as a counterterm needed to make the Euclidean action free of power law divergences.

Entanglement entropy in higher derivative holography

TL;DR

This work probes holographic entanglement entropy for higher-derivative gravity, focusing on Gauss-Bonnet theory as the tractable four-derivative case. By extending the Lewkowycz–Maldacena generalized gravitational entropy approach, it shows that the resulting entangling-surface equation in GB matches the proposed JM functional and aligns with Brown–York stress-tensor considerations in the regime where cubic extrinsic-curvature terms are subleading. It also demonstrates that the JM area functional acts as a natural counterterm to remove power-law divergences in the GB Euclidean action, reproducing the universal piece of the entanglement entropy proportional to the anomaly coefficient . For general four-derivative theories, however, the analysis reveals that a pure geometric area functional does not generically exist, with Weyl-squared contributions obstructing such a formulation, illuminating limits on extending Ryu–Takayanagi-type prescriptions beyond GB gravity.

Abstract

We consider holographic entanglement entropy in higher derivative gravity theories. Recently Lewkowycz and Maldacena arXiv:1304.4926 have provided a method to derive the equations for the entangling surface from first principles. We use this method to compute the entangling surface in four derivative gravity. Certain interesting differences compared to the two derivative case are pointed out. For Gauss-Bonnet gravity, we show that in the regime where this method is applicable, the resulting equations coincide with proposals in the literature as well as with what follows from considerations of the stress tensor on the entangling surface. Finally we demonstrate that the area functional in Gauss-Bonnet holography arises as a counterterm needed to make the Euclidean action free of power law divergences.

Paper Structure

This paper contains 15 sections, 83 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Entanglement entropy proposals in GB gravity
  3. Modified area functional $S_{JM}$
  4. Stress tensor considerations
  5. Generalized gravitational entropy method
  6. $S_{JM}$ as a counterterm
  7. Discussion
  8. Stress Tensor split
  9. EE for cylinder using Euclidean method
  10. Details of the Calculation of Section 3
  11. Metric
  12. Christoffel symbols
  13. Riemann tensors
  14. Ricci tensors
  15. Ricci Scalar

Figures (1)

  • Figure 1: $T_{tt}=0$ determines the shape which also implies the ground state of the configuration. This has a similarity with the MERA procedure in which all the states in a particular level are in the ground state.