Fine structure in holographic entanglement and entanglement contour

Abstract

We explore the fine structure of the holographic entanglement entropy proposal (the Ryu-Takayanagi formula) in ${\mathrm{AdS}}_{3}/{\mathrm{CFT}}_{2}$. 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 codimension 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 nontrivial tests.