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Universal features of left-right entanglement entropy

Diptarka Das, Shouvik Datta

TL;DR

This paper establishes a universal framework for left-right entanglement entropy (LREE) in (1+1)D CFTs on a circle, showing that regularized boundary states yield a universal UV-divergent term proportional to the central charge and a finite piece encoded by quantum dimensions, which reproduces the spatial entanglement entropy of (2+1)D TQFTs. Using Ishibashi and Cardy boundary states, the authors derive the LREE via a replica-trace calculation that leverages the modular S-matrix; for diagonal theories, the finite part is explicitly given in terms of S-matrix data. They connect the Ishibashi-state result to topological entanglement entropy and illustrate the formalism with concrete calculations in Ising, tricritical Ising, and SU(2)_k WZW models. The work highlights a deep bulk-edge correspondence between 2D CFT boundary data and 3D TQFT entanglement, with implications for topological phases and holographic ideas.

Abstract

We show the presence of universal features in the entanglement entropy of regularized boundary states for (1+1)-d conformal field theories on a circle when the reduced density matrix is obtained by tracing over right/left-moving modes. We derive a general formula for the left-right entanglement entropy in terms of the central charge and the modular S matrix of the theory. When the state is chosen to be an Ishibashi state, this measure of entanglement is shown to reproduce the spatial entanglement entropy of a (2+1)-d topological quantum field theory. We explicitly evaluate the left-right entanglement entropies for the Ising model, the tricritical Ising model and the su(2)_k WZW model as examples.

Universal features of left-right entanglement entropy

TL;DR

This paper establishes a universal framework for left-right entanglement entropy (LREE) in (1+1)D CFTs on a circle, showing that regularized boundary states yield a universal UV-divergent term proportional to the central charge and a finite piece encoded by quantum dimensions, which reproduces the spatial entanglement entropy of (2+1)D TQFTs. Using Ishibashi and Cardy boundary states, the authors derive the LREE via a replica-trace calculation that leverages the modular S-matrix; for diagonal theories, the finite part is explicitly given in terms of S-matrix data. They connect the Ishibashi-state result to topological entanglement entropy and illustrate the formalism with concrete calculations in Ising, tricritical Ising, and SU(2)_k WZW models. The work highlights a deep bulk-edge correspondence between 2D CFT boundary data and 3D TQFT entanglement, with implications for topological phases and holographic ideas.

Abstract

We show the presence of universal features in the entanglement entropy of regularized boundary states for (1+1)-d conformal field theories on a circle when the reduced density matrix is obtained by tracing over right/left-moving modes. We derive a general formula for the left-right entanglement entropy in terms of the central charge and the modular S matrix of the theory. When the state is chosen to be an Ishibashi state, this measure of entanglement is shown to reproduce the spatial entanglement entropy of a (2+1)-d topological quantum field theory. We explicitly evaluate the left-right entanglement entropies for the Ising model, the tricritical Ising model and the su(2)_k WZW model as examples.

Paper Structure

This paper contains 8 sections, 18 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Left-right entanglement entropy formulation for general CFT s
  3. Quantum dimensions and topological entanglement entropy
  4. Examples
  5. Ising model
  6. Tricritical Ising model
  7. The $\widehat{su}(2)_k$ WZW model
  8. Conclusions

Figures (1)

  • Figure 1: A TQFT on a sphere which is spatially bipartitioned into the upper and lower hemispheres. A non-chiral CFT lives in the interedge region. Calculating the left-right entanglement entropy of this CFT is equivalent to calculating the real-space entanglement entropy in (2+1)-$d$.