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 AdS3/CFT2. 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 A and the points on the Ryu-Takayanagi (RT) surface EA. In the same sense an arbitrary subinterval A2 of A will correspond to a subinterval E2 of EA. This fine correspondence indicates that the length of E2 captures the contribution sA(A2) from A2 to the entanglement entropy SA, hence gives the contour function for entanglement entropy. Furthermore we propose that sA(A2) in general can be written as a simple linear combination of entanglement entropies of single intervals inside A. This proposal passes several nontrivial tests.