Effective Abelian Higgs models
for gluodynamics and high-$T_c$ superconductors
Simulations of 3D Abelian Higgs (toy) models are discussed with emphasis on confinement and string breaking, monopole pairing and the nonperturbative photon propagator in the $Q=1$ Higgs model. The $Q=2$ Higgs model is interesting because of the nontrivial dynamics relating monopole and vortices in a way very similar to QCD. Finally, an Abelian two-Higgs model with $Q=1$ and $Q=2$ matter fields (proposed by Lee, Nagaosa and Wen for overdoped high-$T_c$ superconductors) is presently under investigation. Using thermodynamic and topological observables we present the phase diagram in the parameter space of the gauge and two matter couplings. First results are presented.