Local interactions and non-Abelian quantum loop gases

TL;DR

This paper investigates the properties of two-dimensional quantum loop gases, revealing that non-Abelian variants require non-local interactions for a gapped ground state, unlike Abelian ones which can be local.

Contribution

It demonstrates that gapped non-Abelian quantum loop gases cannot be realized with purely local Hamiltonians, contrasting with Abelian cases.

Findings

Gapped Abelian loop gases are realizable with local Hamiltonians.

Gapped non-Abelian loop gases require non-local interactions.

Local perturbations drive non-Abelian gases into Abelian phases.

Abstract

Two-dimensional quantum loop gases are elementary examples of topological ground states with Abelian or non-Abelian anyonic excitations. While Abelian loop gases appear as ground states of local, gapped Hamiltonians such as the toric code, we show that gapped non-Abelian loop gases require non-local interactions (or non-trivial inner products). Perturbing a local, gapless Hamiltonian with an anticipated ``non-Abelian'' ground-state wavefunction immediately drives the system into the Abelian phase, as can be seen by measuring the Hausdorff dimension of loops. Local quantum critical behavior is found in a loop gas in which all equal-time correlations of local operators decay exponentially.