The phases of two-dimensional spin and four-dimensional gauge systems with Z (N) symmetry

Using a generalised duality transformation, symmetry considerations and assuming criticality to be continuous in the system parameters, the phase diagram for two-dimensional Z (N) spin models (four-dimensional gauge Z (N) models) are obtained. Besides the phases characterised by the spontaneous breakdown of Z (N) symmetries for spin systems (the behaviour of the Wilson loop for gauge systems), the existence of a soft phase characterised by the vanishing of all powers of order and disorder parameters (a Wilson and 't Hooft loop decaying together with all powers like the perimeter) is predicted. For the spin system phases with non-vanishing order and disorder parameters are forbidden when those parameters obey non-trivial commutation relations. For gauge systems all combinations of Wilson and 't Hooft loops decaying as the area and the perimeter are allowed. Duality relations for three-dimensional gauge plus Higgs system are given. (author)