Phase transitions in the charged compact abelian lattice Higgs model

We consider the (compact) abelian lattice Higgs model with charge $k\ge 1$ and show, using charged Wilson~loop observables and charged versions of the Marcu–Fredenhagen ratio, that this model exhibits several distinct phase transitions. In particular, we show that if $k=2$, then the Marcu–Fredenhagen ratio and Wilson~loop observables together can distinguish among three distinct phases of the parameter space, and hence both can be used as order parameters for the model.