Kosaki-Longo index and classification of charges in 2D quantum spin models

TL;DR

This paper links the classification of charges in 2D quantum spin models to the Kosaki-Longo index, providing bounds on charge classes and applying the theory to the toric code to identify four distinct charges.

Contribution

It introduces a method to bound the number of charge classes using the Kosaki-Longo index and applies it to the toric code, explicitly calculating the index and classifying charges.

Findings

The number of charge classes is bounded by the Kosaki-Longo index.

In the toric code, there are exactly four distinct irreducible charges.

A criterion for the non-degeneracy of charge sectors is established.

Abstract

We consider charge superselection sectors of two-dimensional quantum spin models corresponding to cone localisable charges, and prove that the number of equivalence classes of such charges is bounded by the Kosaki-Longo index of an inclusion of certain observable algebras. To demonstrate the power of this result we apply the theory to the toric code on a 2D infinite lattice. For this model we can compute the index of this inclusion, and conclude that there are four distinct irreducible charges in this model, in accordance with the analysis of the toric code model on compact surfaces. We also give a sufficient criterion for the non-degeneracy of the charge sectors, in the sense that Verlinde's matrix S is invertible.