Modular Hamiltonian in flat holography in the framework of generalized minimal massive gravity

Abstract

Recently a general prescription for determining the vacuum modular flow generator and the corresponding modular Hamiltonian in the BMS-invariant field theories (BMSFTs) have provided by Apolo et. al \cite{Apolo}. According to this paper Einstein gravity in asymptotically flat three-dimensional spacetimes is dual to a BMSFT. In the present paper we extend this study to the generalized minimal massive gravity model (GMMG). So we study some aspects of $Flat_3/BMSFT$ holography in the framework of this model. We derive the modular Hamiltonian for single intervals in thermal and non-thermal BMSFTs and show its holographic dual is the gravitational charge associated with isometries that preserve the asymptotic form of the metric. Also using the modular Hamiltonian we derive the first law of entanglement entropy. Our results for GMMG model reduces to the result for Einstein gravity in $2+1$ dimension, when $\mu , m^2 \to \infty $, and $\sigma=-1$. These limiting cases, are a place in parameter space of model, where GMMG model reduces to the Einstein gravity.