Vortex solutions in the Abelian Higgs Model with a neutral scalar

We construct an extension of the Abelian Higgs model, which consists of a complex scalar field by including an additional real, electromagnetically neutral scalar field. We couple this real scalar field to the complex scalar field via a quartic coupling and investigate $U(1)$ vortex solutions in this "extended Abelian Higgs Model". Since this model has two additional homogeneous ground states, the $U(1)$ vortices that can form in this model have a richer structure than in the Abelian Higgs Model. We also find the "phase diagram" of the model showing the parameter space in which the real scalar particle condenses in the vortex state while having a zero vacuum expectation value in the homogeneous ground state.