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Entanglement entropy of two disjoint intervals from fusion algebra of twist fields

M. A. Rajabpour, F. Gliozzi

TL;DR

The paper addresses entanglement in two disjoint intervals within 1+1D minimal-model CFTs by exploiting branch-point twist-field correlators. It develops a practical framework based on the conformal-block expansion and Zamolodchikov's elliptic recursion, combined with twist-field fusion rules, to produce an analytic approximation for the Rényi entropies via the function $\mathcal{F}_n(x)$ of cross-ratio $x$, and then analytically continues to $n\to1$ to obtain the von Neumann entropy. The approach yields explicit approximate formulas that agree well with exact results and numerical data in the Ising model, and reveals a direct relation between $\mathcal{F}_2(x)$ and the torus partition function $Z(\tau)$. The framework offers a controlled route to improve accuracy by including higher fusion channels and demonstrates a deep connection between twist-field correlators and the underlying CFT geometry. Overall, this work advances analytic access to multi-interval entanglement in minimal models and clarifies the role of modular/elliptic structures in entanglement computations.

Abstract

We study the entanglement and Renyi entropies of two disjoint intervals in minimal models of conformal field theory. We use the conformal block expansion and fusion rules of twist fields to define a systematic expansion in the elliptic parameter of the trace of the n-th power of the reduced density matrix. Keeping only the first few terms we obtain an approximate expression that is easily analytically continued to n->1, leading to an approximate formula for the entanglement entropy. These predictions are checked against some known exact results as well as against existing numerical data.

Entanglement entropy of two disjoint intervals from fusion algebra of twist fields

TL;DR

The paper addresses entanglement in two disjoint intervals within 1+1D minimal-model CFTs by exploiting branch-point twist-field correlators. It develops a practical framework based on the conformal-block expansion and Zamolodchikov's elliptic recursion, combined with twist-field fusion rules, to produce an analytic approximation for the Rényi entropies via the function of cross-ratio , and then analytically continues to to obtain the von Neumann entropy. The approach yields explicit approximate formulas that agree well with exact results and numerical data in the Ising model, and reveals a direct relation between and the torus partition function . The framework offers a controlled route to improve accuracy by including higher fusion channels and demonstrates a deep connection between twist-field correlators and the underlying CFT geometry. Overall, this work advances analytic access to multi-interval entanglement in minimal models and clarifies the role of modular/elliptic structures in entanglement computations.

Abstract

We study the entanglement and Renyi entropies of two disjoint intervals in minimal models of conformal field theory. We use the conformal block expansion and fusion rules of twist fields to define a systematic expansion in the elliptic parameter of the trace of the n-th power of the reduced density matrix. Keeping only the first few terms we obtain an approximate expression that is easily analytically continued to n->1, leading to an approximate formula for the entanglement entropy. These predictions are checked against some known exact results as well as against existing numerical data.

Paper Structure

This paper contains 9 sections, 49 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Four point function of twist fields and conformal block technique
  3. Zamolodchikov's recursion relation
  4. Fusion rules and structure constants of twist operators
  5. Rényi entropy
  6. Von Neumann entropy
  7. The special case of two replicas
  8. Conclusion
  9. The functions $s_2(n,\alpha)$ and $s_4(n,\alpha)$

Figures (3)

  • Figure 1: $\mathcal{F}_{2}(x)$ for the Ising model. The blue line is the exact result obtained in ACT and the red one comes from the equation (\ref{['key formula']}).
  • Figure 2: $\mathcal{F}_{3}(x)$ for the Ising model. The blue line is the exact result obtained in CCT and the red one comes from the equation (\ref{['key formula']}).
  • Figure 3: The function $\mathcal{F}_{VN}(x)$ for the Ising model as a function of the cross-ratio $x$. The solid line is the plot of Eq. (\ref{['Von Neumann Ising']}), while the data are taken form Ref. ACT. The two integers in the legend are respectively the size $L$ of the system and the size of the interval $\ell$ in units of lattice spacing.