Gauge-invariant variational study of the Hamiltonian U(1) and Z$_{N}$ model and critical space-time dimensionality

Using a gauge-invariant variational formalism the author finds in D=4 dimensions a first-order phase transition for Z/sub 2,3,4/ and two second ones for Z/sub N/ with N>or=5. At D=3.2 the transitions of Z /sub 2/ and Z/sub 4/ become of second order, whereas that of Z/sub 3/ for D>or=3 remains of first order. For D>4 for all N the transition in Z/sub N/ is first order. In the U(1) case for 3<D<4 there is a second- order transition so that the passage to the continuum limit is allowed, whereas for D>4 there is only a first-order transition without any long-range correlation length; D=4 appears therefore as the critical space-time dimensionality under which the theory exists in the continuum in agreement with the usual criterion of renormalizability. For D<3 the U(1) model is always in the confining phase.