Stability of the vison crystal and topological order away from the loop point

Determine whether the 2/3-vison crystal and its toric-code topological properties survive perturbations away from the fine-tuned loop point that make the conserved fluxes dynamical.

Background

The exact loop decomposition, global energy bound, and toric-code mapping rely on the fine-tuned loop point. Perturbations that fail to commute with the plaquette flux operators make fluxes dynamical, and the paper does not establish whether the selected vison crystal or its associated topological properties persist under such perturbations.