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Holographic Entanglement Entropy in Cutoff AdS

Chanyong Park

TL;DR

This work analyzes holographic entanglement entropy (EE) for deformed CFTs dual to cutoff AdS, focusing on a boundary boost and the $T \bar{T}$ deformation. It shows that a boost can be reinterpreted as a shift of the UV cutoff in two dimensions, leading to a geodesic EE that increments by a term proportional to $\log(1 - v^2)$ and a $c$-function that decreases along the RG flow, in line with Zamolodchikov's theorem. In higher dimensions, the boost no longer corresponds to a simple UV cutoff shift, instead it deforms the entangling surface from a disk to an ellipse, altering the leading and subleading EE terms. The $T \bar{T}$ deformation, analyzed on $S^2$ and via cutoff AdS$_3$ holography, yields a closely related EE and RG flow, with a monotonic $c$-function; together these results unify the understanding of finite-cutoff holography and irrelevant deformations in the entanglement structure of quantum field theories.

Abstract

We investigate the holographic entanglement entropy of deformed conformal field theories which are dual to a cutoff AdS space. The holographic entanglement entropy evaluated on a three-dimensional Poincare AdS space with a finite cutoff can be reinterpreted as that of the dual field theory deformed by either a boost or $T \bar{T}$ deformation. For the boost case, we show that, although it trivially acts on the underlying theory, it nontrivially affects the entanglement entropy due to the length contraction. For a three-dimensional AdS, we show that the effect of the boost transformation can be reinterpreted as the rescaling of the energy scale, similar to the $T \bar{T}$ deformation. Under the boost and $T \bar{T}$ deformation, the $c$-function of the entanglement entropy exactly shows the features expected by the Zamoldchikov's $c$-theorem. The deformed theory is always stationary at a UV fixed point and monotonically flows to another CFT in the IR fixed point. We also show that the holographic entanglement entropy in a Poincare cutoff AdS space can reproduce the exact same result of the $T \bar{T}$ deformed theory on a two-dimensional sphere.

Holographic Entanglement Entropy in Cutoff AdS

TL;DR

This work analyzes holographic entanglement entropy (EE) for deformed CFTs dual to cutoff AdS, focusing on a boundary boost and the deformation. It shows that a boost can be reinterpreted as a shift of the UV cutoff in two dimensions, leading to a geodesic EE that increments by a term proportional to and a -function that decreases along the RG flow, in line with Zamolodchikov's theorem. In higher dimensions, the boost no longer corresponds to a simple UV cutoff shift, instead it deforms the entangling surface from a disk to an ellipse, altering the leading and subleading EE terms. The deformation, analyzed on and via cutoff AdS holography, yields a closely related EE and RG flow, with a monotonic -function; together these results unify the understanding of finite-cutoff holography and irrelevant deformations in the entanglement structure of quantum field theories.

Abstract

We investigate the holographic entanglement entropy of deformed conformal field theories which are dual to a cutoff AdS space. The holographic entanglement entropy evaluated on a three-dimensional Poincare AdS space with a finite cutoff can be reinterpreted as that of the dual field theory deformed by either a boost or deformation. For the boost case, we show that, although it trivially acts on the underlying theory, it nontrivially affects the entanglement entropy due to the length contraction. For a three-dimensional AdS, we show that the effect of the boost transformation can be reinterpreted as the rescaling of the energy scale, similar to the deformation. Under the boost and deformation, the -function of the entanglement entropy exactly shows the features expected by the Zamoldchikov's -theorem. The deformed theory is always stationary at a UV fixed point and monotonically flows to another CFT in the IR fixed point. We also show that the holographic entanglement entropy in a Poincare cutoff AdS space can reproduce the exact same result of the deformed theory on a two-dimensional sphere.

Paper Structure

This paper contains 5 sections, 59 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Entanglement entropy of a boosted CFT
  3. Boost in a higher dimensional AdS space
  4. Entanglement entropy deformed by a $T \bar{T}$ operator
  5. Discussion

Figures (1)

  • Figure 1: The energy scale dependence of the $c$-function when a two-dimensional CFT is deformed by either a $T \bar{T}$ deformation (blue-dashed) or a boost transformation (black-solid).