It is shown that in a two-dimensional $Z_p$ 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 $Z_p$ 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 $Z_p$ spin system. The results of the Padé analysis of this expansion are consistent with the phase structure described above. For $p\le 4$ the analysis suggests two phases with a conventional singularity behavior at the transition. In the nontrivial case of $p=3\mathrm{}$, critical exponents are calculated and found to give good agreement with experiment. For $p\ge 5$ the analysis favors three phases with an essential singularity at the transition.

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