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.
Highlights
The study of entanglement entropy, which describes the correlation structure of a quantum system, has played a central role in the study of modern theoretical physics in the last decade
For a static subregion A in the boundary CFT and a minimal surface EA in the dual AdS bulk that anchored on the boundary ∂A of A, the RT formula states that the entanglement entropy of A is measured by the area of EA in Planck units, SEE
This work shows that a fine correspondence between quantum entanglement and space-time geometry can be extracted from the RT formula by the bulk and boundary modular flows
Summary
The study of entanglement entropy, which describes the correlation structure of a quantum system, has played a central role in the study of modern theoretical physics in the last decade. Transformations using the symmetries of the quantum field theory generalize the Rindler method to holographic models beyond AdS/CFT. Prescription [10] (see [11] for its covariant generalization) which extend the replica trick [12] into the bulk, and calculate the entanglement entropy using the on-shell partition function on the replicated bulk geometry. SA 1⁄4 −n∂nðlog Zn − n log Z1Þjn1⁄41; ð2Þ where Zn is the partition function of the quantum field theory on Bn. Assuming holography and the unbroken replica symmetry in the bulk, the LM prescription manages to construct the bulk dual of Bn, which is a replicated bulk geometry Mn with its boundary being Bn. Zn can be calculated by path integral on Mn on the gravity side. We relate the fine structure to the entanglement contour, which characterizes the spatial structure of entanglement entropy
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