A stabilizer code model with non-invertible symmetries: Strange fractons, confinement, and non-commutative and non-Abelian fusion rules

TL;DR

This paper introduces a novel stabilizer code model with non-invertible symmetries, revealing complex fractonic excitations, confinement phenomena, and non-commutative fusion rules, expanding understanding of topological quantum phases.

Contribution

It presents a new stabilizer code with non-invertible symmetries, demonstrating novel fractonic excitations and a non-commutative fusion category, advancing topological quantum matter theory.

Findings

Identification of confined fractonic excitations with emergent internal degrees of freedom.

Discovery of non-commutative, non-Abelian fusion rules in the model.

Existence of a deconfined fracton liquid phase and a magnetic string condensed phase.

Abstract

We introduce a stabilizer code model with a qutrit at every edge on a square lattice and with non-invertible plaquette operators. The degeneracy of the ground state is topological as in the toric code, and it also has the usual deconfined excitations consisting of pairs of electric and magnetic charges. However, there are novel types of confined fractonic excitations composed of a cluster of adjacent faces with vanishing flux. They manifest confinement, and even larger configurations of these fractons are fully immobile although they acquire emergent internal degrees of freedom. Deconfined excitations change their nature in presence of these fractonic defects. As for instance, fractonic defects can absorb magnetic charges making magnetic monopoles exist while electric charges acquire restricted mobility. Furthermore, some generalized symmetries can annihilate any ground state and also the full sector of fully mobile excitations. All these properties can be captured via a novel type of \textit{non-commutative} and \textit{non-Abelian} fusion category in which the product is associative but does not commute, and can be expressed as a sum of (operator) equivalence classes. Generalized non-invertible symmetries give rise to the feature that the fusion products form a non-unital category without a proper identity. We show that a variant of this model features a deconfined fracton liquid phase and a phase where the dual (magnetic) strings have condensed.