Criterion for Distinguishing between Continuous and Discontinuous Transitions in Spin and Gauge Systems

A conjecture, based on a comparison of different mean-field-like approximations, is proposed for determining the first- or second-order character of phase transitions. The q-state Potts model and several gauge theories [Z₂, U(N), SU(N), SO(3)] on a lattice are considered. For Z₂ gauge theory, in particular, a critical dimensionality dc=3.66 is predicted, whereas for the Potts model qc=2.75 in $d=3$, qc=2.4 in $d=4$, and qc∝exp[1(d-1)] for d→1 are obtained.