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 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.

v3:Version improved, second half of the paper re-written, relate the fine structure to the entanglement contour, references added, title changed, calculation details added. v4:published version 21 pages, 9 figures