Stabilizing semilocal strings by polarization

Semilocal strings are vortices in the extended Abelian-Higgs model with two complex Higgs scalar fields among which a global $SU(2)$ symmetry acts. They are known to be stable (unstable against expansion) in type-I (II) superconductors, in which gauge field is heavier (lighter) than the Higgs scalar field. In this paper, we find that vortices can be stabilized in the whole parameter region including the type-II region by adding a potential term breaking the $SU(2)$ symmetry. We construct numerical solutions in various parameters and determine the vortex phase diagram consisting of six phases. In two phases, a vortex is polarized, that is, split into two half-quantized vortices with a certain distance, to form a vortex molecule, while in the rests a vortex is identical to the conventional Abrikosov-Nielsen-Olesen vortex.