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.