Phase structure of discrete Abelian spin and gauge systems

It is shown that in a two-dimensional Z₏ spin system for p not too small there exists a massless phase in the middle between the ordered and disordered Ising-type phases. A similar thing happens in a four-dimensional Z₏ gauge theory, where a massless QED-like phase appears between the screened and the confined phases. The existence of the middle phase is deduced logically from the existence of such a phase in the continuous O (2) -invariant models using self-duality and correlation inequalities. For the spin case the transition towards this phase is analyzed using a Kosterlitz type of renormalization group suggesting an essential singularity of the correlation length at both transition points. A Hamiltonian strong-coupling expansion up to ninth order is applied to the Z₏ spin system. The results of the Pad\'e analysis of this expansion are consistent with the phase structure described above. For p4 the analysis suggests two phases with a conventional singularity behavior at the transition. In the nontrivial case of p=3, critical exponents are calculated and found to give good agreement with experiment. For p5 the analysis favors three phases with an essential singularity at the transition.