Topological order and the deconfinement transition in the (2+1)   dimensional compact Abelian Higgs model
  We study an Abelian compact gauge theory minimally coupled to bosonic matter with charge q, which may undergo a confinement--deconfinement transition in (2+1)D. The transition is analyzed using a nonlocal order parameter $\tilde W$, which is related to large Wilson loops for fractional charges. We map the model to a dual representation with no gauge field but only a global q-state clock symmetry and show that $\tilde W$ correspond to the domain wall energy of that model. $\tilde W$ is also directly connected to the concept of topological order. We exploit these facts in Monte Carlo simulations to study the detailed nature of the deconfinement transition. 
Anders Vestergren and Jack Lidmar