Monte Ćarlo calculations and a model of the phase structure for gauge theories on discrete subgroups of SU(2)

A Monte Carlo technique is used to evaluate path integrals for lattice gauge theories with gauge groups given by the SU(2) images of the rotational symmetries of a tetrahedron, a cube, and an icosahedron, called $\overline{T}$, $\overline{O}$, and $\overline{I}$, respectively. The coupling-constant dependence of the mean action per plaquette provides evidence for two phases, in each of these theories, separated by a first-order phase transition. The critical gauge coupling constant moves toward zero as the order of the group is increased. The mean plaquette action, the expectations of square gauge loops, and an estimate of the string tension for the largest group $\overline{I}$ agree with Creutz's results for SU(2) over a wide range of coupling constants. A model for the phase transitions in $\overline{T}$, $\overline{O}$, $\overline{I}$, and $Z_N$ is discussed which predicts critical coupling constants close to the observed values and suggests, as expected, that these transitions are a special property of gauge theories over discrete groups.