Entanglement Entropy in Flat Holography

TL;DR

This work extends flat-space holography to compute entanglement and Rényi entropies in BMS3-invariant theories by generalizing the Rindler method. It establishes a bulk interpretation in which a bulk modular flow fixes a spacelike geodesic γ connected to the boundary by null rays γ±, yielding EE as Length(γ)/4G and Rényi entropy related by a universal 1/2 factor to EE. The authors provide explicit calculations in 3D Einstein gravity and Topologically Massive Gravity, including flat limits of AdS3, and verify results against field theory Cardy-like expressions. The approach offers a geometric, bulk-consistent picture of holographic entanglement in non-AdS, flat spacetimes and clarifies the role of CS terms in TMG via CS corrections to holographic entanglement entropy.

Abstract

BMS symmetry, which is the asymptotic symmetry at null infinity of flat spacetime, is an important input for flat holography. In this paper, we give a holographic calculation of entanglement entropy and Rényi entropy in three dimensional Einstein gravity and Topologically Massive Gravity. The geometric picture for the entanglement entropy is the length of a spacelike geodesic which is connected to the interval at null infinity by two null geodesics. The spacelike geodesic is the fixed points of replica symmetry, and the null geodesics are along the modular flow. Our strategy is to first reformulate the Rindler method for calculating entanglement entropy in a general setup, and apply it for BMS invariant field theories, and finally extend the calculation to the bulk.

Figures (1)

• Figure 1: The red line $\mathcal{A}$ on the future null infinity $\mathcal{I}^+$ is the boundary interval. The blue line $\gamma$ is a spacelike geodesic, on which the bulk modular generator vanishes, $k_t^{\text{bulk}}=0$. The two green lines $\gamma_+$, $\gamma_-$ are null geodesics. The tangent vector on $\gamma_+$ and $\gamma_-$ are $k_t^{\text{bulk}}$. The union $\gamma_{\mathcal{A}}=\gamma\cup\gamma_+\cup\gamma_-$ is invariant under the modular flow $k_t^{\text{bulk}}$. The entanglement entropy is given by $S_{\text{HEE}}= {\text{Length} (\gamma) \over 4G}={\text{Length} (\gamma_{\mathcal{A}}) \over 4G}$.