  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.
