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Symmetry-Resolved Entanglement Entropy in Higher Dimensions

Yuanzhu Huang, Yang Zhou

TL;DR

This work develops a universal method to compute symmetry-resolved entanglement entropy (SREE) for spherical regions in higher-dimensional CFTs by mapping the entanglement problem to hyperbolic-space thermodynamics via the CHM construction. It applies the approach to both free field theories and holographic CFTs, deriving a common large-volume expansion for SREE: the leading entropy term S, a subleading logarithmic correction in the hyperbolic-volume parameter V, and constant-order equipartition up to that order, with charge q-dependence entering only at higher orders. The authors provide explicit results in 4D and 2D free scalars, and holographic results in arbitrary dimensions, demonstrating that equipartition holds up to constant order across frameworks; they also develop a rigorous asymptotic (Laplace/steepest-descent) analysis to establish the universal structure. These findings reveal a robust, dimension-independent pattern for symmetry-resolved entanglement in large subsystems and lay groundwork for extensions to non-Abelian symmetries and other expansion schemes, with potential implications for quantum information in quantum field theories and holography.

Abstract

We present a method to compute the symmetry-resolved entanglement entropy of spherical regions in higher-dimensional conformal field theories. By employing Casini-Huerta-Myers mapping, we transform the entanglement problem into thermodynamic calculations in hyperbolic space. This method is demonstrated through computations in both free field theories and holographic field theories. For large hyperbolic space volume, our results reveal a universal expansion structure of symmetry-resolved entanglement entropy, with the equipartition property holding up to the constant order. Using asymptotic analysis techniques, we prove this expansion structure and the equipartition property in arbitrary dimensions.

Symmetry-Resolved Entanglement Entropy in Higher Dimensions

TL;DR

This work develops a universal method to compute symmetry-resolved entanglement entropy (SREE) for spherical regions in higher-dimensional CFTs by mapping the entanglement problem to hyperbolic-space thermodynamics via the CHM construction. It applies the approach to both free field theories and holographic CFTs, deriving a common large-volume expansion for SREE: the leading entropy term S, a subleading logarithmic correction in the hyperbolic-volume parameter V, and constant-order equipartition up to that order, with charge q-dependence entering only at higher orders. The authors provide explicit results in 4D and 2D free scalars, and holographic results in arbitrary dimensions, demonstrating that equipartition holds up to constant order across frameworks; they also develop a rigorous asymptotic (Laplace/steepest-descent) analysis to establish the universal structure. These findings reveal a robust, dimension-independent pattern for symmetry-resolved entanglement in large subsystems and lay groundwork for extensions to non-Abelian symmetries and other expansion schemes, with potential implications for quantum information in quantum field theories and holography.

Abstract

We present a method to compute the symmetry-resolved entanglement entropy of spherical regions in higher-dimensional conformal field theories. By employing Casini-Huerta-Myers mapping, we transform the entanglement problem into thermodynamic calculations in hyperbolic space. This method is demonstrated through computations in both free field theories and holographic field theories. For large hyperbolic space volume, our results reveal a universal expansion structure of symmetry-resolved entanglement entropy, with the equipartition property holding up to the constant order. Using asymptotic analysis techniques, we prove this expansion structure and the equipartition property in arbitrary dimensions.

Paper Structure

This paper contains 20 sections, 122 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Methods
  3. Free Field Computation
  4. 4D scalar field
  5. 2D scalar field
  6. In sectors with charge $q \sim V$
  7. Holographic Computation
  8. Universal Structure of SREE
  9. Expansion structure of SREE
  10. Expansion structure via asymptotic analysis
  11. Case 1: $F'(\mu)=0$ at $\mu=0$
  12. Case 2: $F'(\mu)\ne0$ at $\mu=0$
  13. Discussion
  14. Review of Symmetry-Resolved Entanglement
  15. Entropy and the coarse-graining property
  16. ...and 5 more sections

Figures (3)

  • Figure 1: Plot of $F_n(\mu)$ for complex free scalar field in 4D for $n=1$. The qualitative shape and the minimum point ($\mu=0$) remain unchanged for different values of $n$.
  • Figure 2: Plot of $F_n(\mu)$ for complex free scalar field in 2D for $n=1$. The qualitative shape and the minimum point ($\mu=0$) remain unchanged for different values of $n$.
  • Figure 3: Illustration for microstates (big black dots) and sub-microstates (small dots in small circle). The "microstate" $i$ appears with probability $p_i$ and has its own entropy $S_i$. The actual total entropy comes from uncertainty of probability distribution $\{p_i\}$ and uncertainty within state $i$.