Non-Abelian fusion rules from Abelian systems with SPT phases and graph topological order

TL;DR

This paper proves that non-Abelian fusion rules are essential for SPT phase transitions in Abelian models, highlighting their role in symmetry breaking and condensation mechanisms within topological phases.

Contribution

It provides a proof that non-Abelian fusion rules are necessary for SPT phase transitions in Abelian systems with topological order.

Findings

Non-Abelian fusion rules are linked to SPT phase transitions.

These rules are necessary for phase transitions via condensation or symmetry breaking.

Supports the emergence of non-Abelian features in Abelian models.

Abstract

Since Ref. [1] shows the emergence of non-Abelian fusion rules in some examples of a class of Abelian models, but does not prove whether these rules also exist in other cases, the purpose of this paper is to present such proof emphasizing the importance of the existence of these rules. By the way, as the ground state of these models can be degenerate as a function of their algebra and, hence, they can support some symmetry-protected topological (SPT) phases, we prove that these non-Abelian fusion rules are always necessary for these SPT phase transitions to occur via a condensation mechanism or/and some global symmetry breaking.