Multicomponent gauge-Higgs models with discrete Abelian gauge groups

We consider a variant of the charge-Q compact Abelian-Higgs model, in which an Nf-dimensional complex vector is coupled with Abelian Z_q gauge field. For Nf=2 and Q=1 we observe several transition lines that belong to O(4), O(3), O(2) universality classes, depending on symmetry breaking pattern at transition. The class independent $q$ as long q>=3. uniquely determined by behavior scalar fields; fields do not play any role. also investigate system for Nf=15 Q=2. In presence U(1) fields, undergoes transitions associated charged fixed points field theory. These continuous turn into first-order ones when are replaced discrete fields: present model appear be very sensitive nature