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Precision tests of bulk entanglement: $AdS_3$ vectors

Rayirth Bhat, Justin R. David, Semanti Dutta

Abstract

We consider single particle excitations of the massive Chern-Simons field of mass $M$ in $AdS_3$ and evaluate their contribution at the first sub-leading order in $G_N$ to the entanglement entropy across the Ryu-Takayanagi surface. Quantizing the Chern- Simons field in $AdS_3$, we evaluate the corrections to the holographic entanglement entropy using the Faulkner-Lewkowycz-Maldacena formula. The massive Chern-Simons field also obeys the equations of motion of a massive vector in $AdS_3$. The lowest energy single particle excitation of this field is dual to the primary operator of conformal dimensions $M + 1$ with spin one in the dual CFT, all other single particle excitations are dual to its global descendants. We compare the result for the entanglement entropy from the Faulkner-Lewkowycz-Maldacena formula to the single interval entanglement entropy in large charge holographic CFT obtained using the replica trick for the primary and its tower of holomorphic descendants. The two results agree precisely in the leading and sub-leading terms of the short interval expansion. On taking the massless limit the result coincides with the contribution of a $U(1)$ current to the single interval entanglement entropy.

Precision tests of bulk entanglement: $AdS_3$ vectors

Abstract

We consider single particle excitations of the massive Chern-Simons field of mass in and evaluate their contribution at the first sub-leading order in to the entanglement entropy across the Ryu-Takayanagi surface. Quantizing the Chern- Simons field in , we evaluate the corrections to the holographic entanglement entropy using the Faulkner-Lewkowycz-Maldacena formula. The massive Chern-Simons field also obeys the equations of motion of a massive vector in . The lowest energy single particle excitation of this field is dual to the primary operator of conformal dimensions with spin one in the dual CFT, all other single particle excitations are dual to its global descendants. We compare the result for the entanglement entropy from the Faulkner-Lewkowycz-Maldacena formula to the single interval entanglement entropy in large charge holographic CFT obtained using the replica trick for the primary and its tower of holomorphic descendants. The two results agree precisely in the leading and sub-leading terms of the short interval expansion. On taking the massless limit the result coincides with the contribution of a current to the single interval entanglement entropy.

Paper Structure

This paper contains 24 sections, 311 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Entanglement entropy of spin-1 excited states in CFT
  3. Massive Chern-Simons theory in $AdS_3$
  4. Dirac quantization of massive Chern-Simons theory
  5. Construction of single particle vector excitations
  6. Corrections to minimal area
  7. Back reacted geometry of single particle states
  8. Shift in minimal area
  9. Area shift for the state $| \psi_{1,0} \rangle$
  10. Area shift for the states $| \psi_{m,0} \rangle$
  11. Bulk entanglement entropy
  12. Bulk entanglement for $|\psi_{1, 0}\rangle$
  13. Bulk entanglement for $|\psi_{m, 0}\rangle$
  14. Conclusions
  15. Global $AdS_3$ and Rindler BTZ
  16. ...and 9 more sections

Figures (1)

  • Figure 1: The $t=0$ slice of $AdS_3$. We consider the entanglement between system $A$ and $\bar{A}$ in the boundary. The minimal surface $\gamma_A$ is the geodesic in the bulk connecting the end points of $A$. $\gamma_A$ splits the bulk into right(left) wedge denoted by $\Sigma_A(\Sigma_{\bar{A}})$. $\gamma_A$ consists of: Branch I, where $\varphi^\prime(r) < 0$ and Branch II with $\varphi^\prime(r) > 0$. When excitations breaks the isometry in $\varphi$, we need to evaluate the minimal area for the above branches separately.