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

Abstract

Using a gauge-invariant variational formalism we find in D = 4 dimensions a first-order phase transition for Z 2,3,4 and two second ones for Z N with N⩽ 5. At D = 3.2 the transitions of Z 2 and Z 4 become of second order, whereas that of Z 3 for D ⩽ 3 remains of first order. For D > 4 for all N the transition in Z 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.