The spectrum of spin model angle operators

TL;DR

This paper investigates the spectrum of the angle operator associated with complex Hadamard matrices and subfactors, revealing a connection between the spectrum and principal graph spectrum, and classifying certain Hadamard matrices as producing infinite depth subfactors.

Contribution

It identifies the angle operator within the symmetric enveloping algebra, computes its trace, and establishes a spectrum equivalence with the principal graph spectrum for amenable subfactors, linking Hadamard matrices to subfactor depth.

Findings

The angle operator spectrum matches the principal graph spectrum up to a constant for amenable subfactors.

Paley type II Hadamard matrices produce infinite depth subfactors.

Petrescu's 7x7 complex Hadamard matrices also yield infinite depth subfactors.

Abstract

Complex Hadamard matrices are biunitaries for spin model commuting squares. The corresponding subfactor standard invariant can be identified with the $1$-eigenspace of the angle operator defined by Jones. We identify the angle operator as an element of the symmetric enveloping algebra and compute its trace. We then show the angle operator spectrum coincides with the principal graph spectrum up to a constant iff the subfactor is amenable. We use this to show Paley type $II$ Hadamard matrices and Petrescu's $7 \times 7$ family of complex Hadamard matrices yield infinite depth subfactors.