Fine structure in holographic entanglement and entanglement contour

TL;DR

This paper investigates the detailed structure of holographic entanglement entropy in AdS3/CFT2, introducing a new contour function based on modular flows that decomposes entanglement contributions from subintervals.

Contribution

It introduces a novel slicing of the entanglement wedge using modular planes, establishing a correspondence between boundary subintervals and RT surface segments, leading to a new entanglement contour function.

Findings

The length of RT surface segments encodes subinterval entanglement contributions.

The proposed contour function can be expressed as a linear combination of single-interval entropies.

The approach is validated through several non-trivial tests.

Abstract

We explore the fine structure of the holographic entanglement entropy proposal (the Ryu-Takayanagi formula) in AdS$_3$/CFT$_{2}$. With the guidance from the boundary and bulk modular flows we find a natural slicing of the entanglement wedge with the modular planes, which are co-dimension one bulk surfaces tangent to the modular flow everywhere. This gives an one-to-one correspondence between the points on the boundary interval $\mathcal{A}$ and the points on the Ryu-Takayanagi (RT) surface $\mathcal{E}_{\mathcal{A}}$. In the same sense an arbitrary subinterval $\mathcal{A}_2$ of $\mathcal{A}$ will correspond to a subinterval $\mathcal{E}_2$ of $\mathcal{E}_{\mathcal{A}}$. This fine correspondence indicates that the length of $\mathcal{E}_2$ captures the contribution $s_{\mathcal{A}}(\mathcal{A}_2)$ from $\mathcal{A}_2$ to the entanglement entropy $S_{\mathcal{A}}$, hence gives the contour function for entanglement entropy. Furthermore we propose that $s_{\mathcal{A}}(\mathcal{A}_2)$ in general can be written as a simple linear combination of entanglement entropies of single intervals inside $\mathcal{A}$. This proposal passes several non-trivial tests.