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arXiv:2603.13013

Jones index from Rényi entropies in the Ising conformal field theory

Valentín Benedetti, Isai Davila-Cuba, Erik Tonni C

Published Mar 13, 2026• 0 citations
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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.

DOI
10.48550/arxiv.2603.13013

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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.
A bstract 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.
We study the Rényi entropies of N disjoint intervals in the conformal field theories given by the free compactified boson and the Ising model. They are computed as the 2N point function of twist fields, by employing the partition function of the model on a particular class of Riemann surfaces. The results are written in terms of Riemann theta functions. The prediction for the free boson in the decompactification regime is checked against exact results for the harmonic chain. For the Ising model, matrix product states computations agree with the conformal field theory result once the finite size corrections have been taken into account.
We study the entanglement and Rényi 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 , 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.
We study the Rényi entanglement entropies of two disjoint intervals in XY chains. We exploit the exact solution of the model in terms of free Majorana fermions and we show how to construct the reduced density matrix in the spin variables by taking the Jordan–Wigner string between the two blocks properly into account. From this we can evaluate any Rényi entropy of finite integer order. We study in detail critical XX and Ising chains and we show that the asymptotic results for large blocks agree with recent conformal field theory predictions if corrections to the scaling are included in the analysis correctly. We also report results for the gapped phase and after a quantum quench.
We study the scaling of the Renyi entanglement entropy of two disjoint blocks of critical lattice models described by conformal field theories with central charge c=1. We provide the analytic conformal field theory result for the second order Renyi entropy for a free boson compactified on an orbifold describing the scaling limit of the Ashkin-Teller (AT) model on the self-dual line. We have checked this prediction in cluster Monte Carlo simulations of the classical two dimensional AT model. We have also performed extensive numerical simulations of the anisotropic Heisenberg quantum spin-chain with tree-tensor network techniques that allowed to obtain the reduced density matrices of disjoint blocks of the spin-chain and to check the correctness of the predictions for Renyi and entanglement entropies from conformal field theory. In order to match these predictions, we have extrapolated the numerical results by properly taking into account the corrections induced by the finite length of the blocks to the leading scaling behavior.
We continue the study of the entanglement entropy of two disjoint intervals in conformal field theories that we started in J. Stat. Mech. (2009) P11001. We compute Trρ_A^n for any integer n for the Ising universality class and the final result is expressed as a sum of Riemann-Siegel theta functions. These predictions are checked against existing numerical data. We provide a systematic method that gives the full asymptotic expansion of the scaling function for small four-point ratio (i.e. short intervals). These formulas are compared with the direct expansion of the full results for free compactified boson and Ising model. We finally provide the analytic continuation of the first term in this expansion in a completely analytic form.