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Jones index from Rényi entropies in the Ising conformal field theory

Valentin Benedetti, Isai Davila-Cuba, Erik Tonni

Abstract

We study the relation between the Jones index and the Rényi entropies of two disjoint intervals on the line and of the ground state for a generic value of the Rényi index in the two conformal field theory models given by the Ising model and a free Majorana fermion, where Haag duality is satisfied. The analytic expressions of the crossing asymmetry for all the submodels displaying a violation of the Haag duality that are closed under the fusion rules are obtained. In the limiting regime where the two intervals become adjacent, the leading term of the expansion of the crossing asymmetry provides the Jones global index, for any finite value of the Rényi index.

Jones index from Rényi entropies in the Ising conformal field theory

Abstract

We study the relation between the Jones index and the Rényi entropies of two disjoint intervals on the line and of the ground state for a generic value of the Rényi index in the two conformal field theory models given by the Ising model and a free Majorana fermion, where Haag duality is satisfied. The analytic expressions of the crossing asymmetry for all the submodels displaying a violation of the Haag duality that are closed under the fusion rules are obtained. In the limiting regime where the two intervals become adjacent, the leading term of the expansion of the crossing asymmetry provides the Jones global index, for any finite value of the Rényi index.
Paper Structure (16 sections, 173 equations, 7 figures)

This paper contains 16 sections, 173 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Jones index in the Ising CFT$_2$
  3. Completeness and modular invariance
  4. The Jones index as a measure of completeness
  5. The subalgebras of the Ising CFT$_2$
  6. Jones index from the crossing asymmetry in CFT$_2$
  7. Ising CFT$_2$: Jones index from the $n=2$ Rényi entropy
  8. Ising CFT$_2$: Jones index from the $n\geqslant 2$ Rényi entropies
  9. Crossing asymmetry
  10. Properties of the crossing asymmetry and Jones index
  11. Free fermion: Jones index from the $n\geqslant 2$ Rényi entropies
  12. Conclusions
  13. Details on the higher genus characters for the Ising CFT$_2$
  14. Internal loops
  15. Genus two: Characters and partition functions
  16. ...and 1 more sections

Figures (7)

  • Figure 1: Subalgebras of the Ising CFT$_2$, enclosed by the blue boxes, and the corresponding values of the global Jones index (\ref{['mu-global-index-def']}).
  • Figure 2: The torus corresponding to the Riemann surface $\mathscr{R}_2$, with two equivalent choices for the canonical homology basis, denoted by $\{ (a_1, b_1) \}$ and $\{ (\tilde{a}_1, \tilde{b}_1) \}$.
  • Figure 3: The Riemann surface $\mathscr{R}_4$ (see (\ref{['eq:replicaSurface']})) with a specific choice of the canonical homology basis that provides the period matrix (\ref{['periodm']}).
  • Figure 4: The Riemann surface $\mathscr{R}_4$ (see (\ref{['eq:replicaSurface']})) with a specific canonical homology basis that provides the period matrix (\ref{['periodma']}).
  • Figure 5: The crossing asymmetries ${A}_{\mathbb{Z}_2\!,\,n} ({\xi})$ and ${A}_{\textrm{su}(2)_2\!,\,n} ({\xi})$ as functions of the cross ratio ${\xi}$ (in the top and bottom panel respectively), for different values of $n$, evaluated through their analytic expressions given in (\ref{['asyft-z2']}) and (\ref{['asyft-su2']}) respectively. The horizontal dashed gray lines highlight the value of the corresponding global Jones index (see (\ref{['limtsmu']})).
  • ...and 2 more figures