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Holographic entanglement entropy for disconnected regions

Veronika E. Hubeny, Mukund Rangamani

TL;DR

The paper uses the holographic entanglement entropy framework to derive the EE for disconnected regions in 1+1D QFTs by analyzing bulk minimal surfaces and applying strong sub-additivity, reproducing known CFT results without relying on conformal symmetry. It extends the 1+1D findings to multiple disjoint intervals and shows a consistent linear combination of minimal-surface entropies s_ij reproduces the expected expressions. It then explores 2+1D scenarios, proposing a conjecture for generic non-simple regions that reduces to known limits and passes UV-divergence checks, while acknowledging the lack of a full proof. Overall, the work highlights the power of geometric holographic methods in computing entanglement entropy for complex region topologies and outlines clear directions for generalization and rigorous validation.

Abstract

We present a simple derivation of the entanglement entropy for a region made up of a union of disjoint intervals in 1+1 dimensional quantum field theories using holographic techniques. This generalizes the results for 1+1 dimensional conformal field theories derived previously by exploiting the uniformization map. We further comment on the generalization of our result to higher dimensional field theories.

Holographic entanglement entropy for disconnected regions

TL;DR

The paper uses the holographic entanglement entropy framework to derive the EE for disconnected regions in 1+1D QFTs by analyzing bulk minimal surfaces and applying strong sub-additivity, reproducing known CFT results without relying on conformal symmetry. It extends the 1+1D findings to multiple disjoint intervals and shows a consistent linear combination of minimal-surface entropies s_ij reproduces the expected expressions. It then explores 2+1D scenarios, proposing a conjecture for generic non-simple regions that reduces to known limits and passes UV-divergence checks, while acknowledging the lack of a full proof. Overall, the work highlights the power of geometric holographic methods in computing entanglement entropy for complex region topologies and outlines clear directions for generalization and rigorous validation.

Abstract

We present a simple derivation of the entanglement entropy for a region made up of a union of disjoint intervals in 1+1 dimensional quantum field theories using holographic techniques. This generalizes the results for 1+1 dimensional conformal field theories derived previously by exploiting the uniformization map. We further comment on the generalization of our result to higher dimensional field theories.

Paper Structure

This paper contains 9 sections, 21 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Disconnected regions in $1+1$ dimensional field theories
  3. Two disconnected regions
  4. Minimal surface and sub-additivity inequalities:
  5. Constraints on $x$:
  6. Deriving $x(s_{ij})$:
  7. Multiply disconnected regions
  8. Disconnected regions in $2+1$ dimensional QFTs
  9. Conjecture for generic non-simple regions

Figures (2)

  • Figure 1: Minimal surfaces $\gamma_{ij}$ in the bulk, whose length gives the entanglement entropy $s_{ij}$, corresponding to the boundary intervals $R_{ij}$ bounded by points $p_i$ and $p_j$. The disconnected boundary region $X$ is indicated in red.
  • Figure 2: Three basic types of finite 2-dimensional non-simple regions $X$ (shaded). Note that case (b) is special, and can be deformed into either of the general cases (a) and (c).