The Markov gap in the presence of islands

TL;DR

This paper computes the Markov gap in various gravity models with islands, revealing its origin from boundary CFT OPE coefficients and proposing a boundary-based lower bound related to entanglement structure.

Contribution

It explicitly calculates the Markov gap in defect extremal surface models, JT gravity, and 2D extremal black holes, and introduces a boundary counting method for its lower bound.

Findings

Markov gap computed in multiple gravity models with islands.

Lower bound of the Markov gap related to boundary regions and entanglement.

Explicit connection between Markov gap and boundary CFT OPE coefficients.

Abstract

The Markov gap \cite{Hayden:2021gno}, namely the difference between reflected entropy and mutual information, is explicitly computed in the defect extremal surface model, JT gravity, and the generic 2d extremal black holes, in vacuum states. The phases that contain various island contributions are considered, and their existence is carefully checked. Moreover, we show explicitly how the Markov gap originates from the OPE coefficient of the boundary CFT. And, as a generalization of \cite{Hayden:2021gno}, the lower bound of the Markov gap is given by $\frac{c}{3}\log 2$ times the number of EWCS boundaries on minimal surfaces. We propose a boundary way of counting the lower bound for the Markov gap, which states that the lower bound is given by $\frac{c}{3}\log 2$ times the number of gaps between two boundary regions in vacuum states. We discuss the limitation and possible generalization of the boundary counting, and its relation to tripartite entanglement.