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Aspects of Holographic Entanglement Entropy

Shinsei Ryu, Tadashi Takayanagi

TL;DR

The paper proposes and tests a holographic formula for entanglement entropy, S_A = Area(γ_A)/(4G_N^{(d+2)}), relating quantum entanglement in a CFT to the area of a minimal surface in AdS. It validates the construction in AdS3/CFT2 exactly via the replica trick and geodesic lengths, and provides strong evidence across higher dimensions by matching leading area-law divergences and universal subleading terms with Weyl anomalies and central charges. The authors compute explicit results for massless free fields, gauge fields, and supersymmetric theories (including N=4 SYM and M-theory backgrounds), revealing universal logarithmic terms tied to central charges and detailing finite pieces that depend on geometry and coupling. The work illuminates how entanglement entropy encodes degrees of freedom in strongly coupled QFTs and offers a geometrical, computable handle on quantum correlations, with implications for critical phenomena, phase transitions, and potentially condensed-matter systems.

Abstract

This is an extended version of our short report hep-th/0603001, where a holographic interpretation of entanglement entropy in conformal field theories is proposed from AdS/CFT correspondence. In addition to a concise review of relevant recent progresses of entanglement entropy and details omitted in the earlier letter, this paper includes the following several new results : We give a more direct derivation of our claim which relates the entanglement entropy with the minimal area surfaces in the AdS_3/CFT_2 case as well as some further discussions on higher dimensional cases. Also the relation between the entanglement entropy and central charges in 4D conformal field theories is examined. We check that the logarithmic part of the 4D entanglement entropy computed in the CFT side agrees with the AdS_5 result at least under a specific condition. Finally we estimate the entanglement entropy of massive theories in generic dimensions by making use of our proposal.

Aspects of Holographic Entanglement Entropy

TL;DR

The paper proposes and tests a holographic formula for entanglement entropy, S_A = Area(γ_A)/(4G_N^{(d+2)}), relating quantum entanglement in a CFT to the area of a minimal surface in AdS. It validates the construction in AdS3/CFT2 exactly via the replica trick and geodesic lengths, and provides strong evidence across higher dimensions by matching leading area-law divergences and universal subleading terms with Weyl anomalies and central charges. The authors compute explicit results for massless free fields, gauge fields, and supersymmetric theories (including N=4 SYM and M-theory backgrounds), revealing universal logarithmic terms tied to central charges and detailing finite pieces that depend on geometry and coupling. The work illuminates how entanglement entropy encodes degrees of freedom in strongly coupled QFTs and offers a geometrical, computable handle on quantum correlations, with implications for critical phenomena, phase transitions, and potentially condensed-matter systems.

Abstract

This is an extended version of our short report hep-th/0603001, where a holographic interpretation of entanglement entropy in conformal field theories is proposed from AdS/CFT correspondence. In addition to a concise review of relevant recent progresses of entanglement entropy and details omitted in the earlier letter, this paper includes the following several new results : We give a more direct derivation of our claim which relates the entanglement entropy with the minimal area surfaces in the AdS_3/CFT_2 case as well as some further discussions on higher dimensional cases. Also the relation between the entanglement entropy and central charges in 4D conformal field theories is examined. We check that the logarithmic part of the 4D entanglement entropy computed in the CFT side agrees with the AdS_5 result at least under a specific condition. Finally we estimate the entanglement entropy of massive theories in generic dimensions by making use of our proposal.

Paper Structure

This paper contains 45 sections, 140 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Basics of Entanglement Entropy
  3. Definition of Entanglement Entropy
  4. Properties
  5. Entanglement Entropy in QFTs and Area Law
  6. Entanglement Entropy in $2\mathrm{D}$ CFT
  7. How to Compute Entanglement Entropy
  8. Derivation of Entanglement Entropy in an Infinitely Long System
  9. Derivation of Entanglement Entropy on a Circle
  10. Derivation of Entanglement Entropy at Finite Temperature
  11. Massive Theories
  12. Entanglement Entropy in Higher Dimensional CFTs
  13. Entanglement Entropy of $d+1$ D Massless Free Fields
  14. Rough Estimation
  15. Exact Estimation from Entropic $c$-function
  16. ...and 30 more sections

Figures (6)

  • Figure 1: (a) The path integral representation of the reduced density matrix $[\rho_A]_{\phi_+\phi_-}$. (b) The $n$-sheeted Riemann surface $\mathcal{R}_n$. (Here we take $n=3$ for simplicity.)
  • Figure 2: Two different shapes of the submanifold $A$ considered in this paper. (a) The straight belt $A_S$ and (b) the circular disk $A_D$. (Here, $d=3$ for simplicity.)
  • Figure 3: The entropic $c$-functions $C(x)$ for free massive real scalar boson and free Dirac fermion in 1+1 D reproduced from Casini05b.
  • Figure 4: (a) AdS$_3$ space and CFT$_2$ living on its boundary and (b) a geodesics $\gamma_A$ as a holographic screen.
  • Figure 5: (a) Minimal surfaces $\gamma_A$ in the BTZ black hole for various sizes of $A$. (b) $\gamma_A$ and $\gamma_B$ wrap the different parts of the horizon.
  • ...and 1 more figures