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Some Results on Mutual Information of Disjoint Regions in Higher Dimensions

John Cardy

TL;DR

This work extends the replica-based analysis of mutual information to higher-dimensional conformal field theories, deriving a universal expansion for the mutual Rényi information of two disjoint regions at large separation. The leading term scales as $(R_AR_B/r^2)^{d-1}$ with coefficients determined by the operator content of the CFT; for a free massless scalar, these coefficients factorize into region-dependent quantities that, for $n=2$, are identified with geometrical capacitances. For spherical regions in $d=2,3$, explicit forms for the leading $I^{(n)}$ are obtained, including the von Neumann limit, and a universal logarithmic correction to the area law for a massive scalar is derived. The results illuminate the structure of entanglement in higher-dimensional QFTs and provide exact benchmarks for holographic and numerical studies.

Abstract

We consider the mutual Renyi information I^n(A,B)=S^n_A+S^n_B-S^n_{AUB} of disjoint compact spatial regions A and B in the ground state of a d+1-dimensional conformal field theory (CFT), in the limit when the separation r between A and B is much greater than their sizes R_{A,B}. We show that in general I^n(A,B)\sim C^n_AC^n_B(R_AR_B/r^2)^a, where a the smallest sum of the scaling dimensions of operators whose product has the quantum numbers of the vacuum, and the constants C^n_{A,B} depend only on the shape of the regions and universal data of the CFT. For a free massless scalar field, where 2x=d-1, we show that C^2_AR_A^{d-1} is proportional to the capacitance of a thin conducting slab in the shape of A in d+1-dimensional electrostatics, and give explicit formulae for this when A is the interior of a sphere S^{d-1} or an ellipsoid. For spherical regions in d=2 and 3 we obtain explicit results for C^n for all n and hence for the leading term in the mutual information by taking n->1. We also compute a universal logarithmic correction to the area law for the Renyi entropies of a single spherical region for a scalar field theory with a small mass.

Some Results on Mutual Information of Disjoint Regions in Higher Dimensions

TL;DR

This work extends the replica-based analysis of mutual information to higher-dimensional conformal field theories, deriving a universal expansion for the mutual Rényi information of two disjoint regions at large separation. The leading term scales as with coefficients determined by the operator content of the CFT; for a free massless scalar, these coefficients factorize into region-dependent quantities that, for , are identified with geometrical capacitances. For spherical regions in , explicit forms for the leading are obtained, including the von Neumann limit, and a universal logarithmic correction to the area law for a massive scalar is derived. The results illuminate the structure of entanglement in higher-dimensional QFTs and provide exact benchmarks for holographic and numerical studies.

Abstract

We consider the mutual Renyi information I^n(A,B)=S^n_A+S^n_B-S^n_{AUB} of disjoint compact spatial regions A and B in the ground state of a d+1-dimensional conformal field theory (CFT), in the limit when the separation r between A and B is much greater than their sizes R_{A,B}. We show that in general I^n(A,B)\sim C^n_AC^n_B(R_AR_B/r^2)^a, where a the smallest sum of the scaling dimensions of operators whose product has the quantum numbers of the vacuum, and the constants C^n_{A,B} depend only on the shape of the regions and universal data of the CFT. For a free massless scalar field, where 2x=d-1, we show that C^2_AR_A^{d-1} is proportional to the capacitance of a thin conducting slab in the shape of A in d+1-dimensional electrostatics, and give explicit formulae for this when A is the interior of a sphere S^{d-1} or an ellipsoid. For spherical regions in d=2 and 3 we obtain explicit results for C^n for all n and hence for the leading term in the mutual information by taking n->1. We also compute a universal logarithmic correction to the area law for the Renyi entropies of a single spherical region for a scalar field theory with a small mass.

Paper Structure

This paper contains 11 sections, 78 equations, 1 figure.

Table of Contents

  1. Introduction
  2. General form of the expansion for $R_{A,B}\ll r$.
  3. Free scalar field theory
  4. Free scalar field theory
  5. The case $n=2$
  6. Free field theory when $A$ and $B$ are spherical, general $n$.
  7. The case $d=3$
  8. The case $d=2$
  9. Universal correction for a single spherical region
  10. Conclusions
  11. Thomson's argument for general $d$

Figures (1)

  • Figure 1: Newton's demonstration that the field at any point in the interior of a spherically symmetric charge distribution vanishes. [Reproduced from Newton's Principia, www.gutenberg.org/files/28233].