Quantum graphs and spin models

TL;DR

This paper develops a framework for quantum graphs and spin models, providing explicit examples and applications including quantum groups and highly regular quantum graphs with no classical counterparts.

Contribution

It introduces quantum graphs and spin models, constructs explicit examples, and connects them to quantum groups and combinatorial structures, advancing the understanding of quantum symmetries.

Findings

Constructed quantum versions of classical strongly regular graphs.

Developed methods to deform graphs preserving quantum automorphisms.

Built a quantum group monoidally equivalent to SO_q(5) with property (T).

Abstract

We quantize the regularity properties of classical graphs that determine spin models for singly-generated Yang-Baxter planar algebras, including the Kauffman polynomial, and construct explicit examples. A source of examples comes from deforming graphs using higher-idempotent splittings of quantum isomorphisms for which we prove that the relevant algebraic, combinatorial, and topological properties of the original graphs are preserved along with the quantum automorphism group. We also obtain exotic examples of highly regular quantum graphs using the quantum Fourier transform and a method of iterated convolution. Our examples include quantum versions of the strongly regular $9$-Paley, $16$-Clebsch and the Higman-Sims graphs, yielding new models for their regularity parameters. As applications, we construct a compact quantum group that is monoidally equivalent to $SO_q(5)$ at the square of the golden ratio, whose dual is infinite with property (T), and exhibit a highly-regular quantum graph with no classical analogue. Finally, we introduce quantum spin models, construct explicit examples and make contact with quantum Hadamard matrices.