Fine structure in holographic entanglement and entanglement contour

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.