A Graph-Theoretic Framework for Free-Parafermion Solvability

TL;DR

This paper introduces a graph-theoretic method to determine when quantum spin models can be exactly solved using free parafermions, extending previous free-fermion solutions and applying to specific qudit models.

Contribution

It provides a novel graph-based criterion for free-parafermion solvability and an efficient algorithm to identify integrable models.

Findings

Characterization of free-parafermion solvability via frustration graphs.

Development of an efficient algorithm for model analysis.

Application to three qudit spin models.

Abstract

We present a graph-theoretic characterisation of when a quantum spin model admits an exact solution via a mapping to free parafermions. Our characterisation is based on the concept of a frustration graph, which represents the commutation relations between Weyl operators of a Hamiltonian. We show that a quantum spin system has an exact free-parafermion solution if its frustration graph is an oriented indifference graph. Further, we show that if the frustration graph of a model can be dipath oriented via switching operations, then the model is integrable in the sense that there is a family of commuting independent set charges. Additionally, we establish an efficient algorithm for deciding whether this is possible. Our characterisation extends that given for free-fermion solvability. Finally, we apply our results to solve three qudit spin models.