Monte Carlo Calculations for Lattice Gauge Theories with Discrete Non-Abelian Gauge Groups.

Discrete subgroups of continuous non-abelian gauge groups are examined for possible use as substitutes for the continuous groups in Monte Carlo calculations. All discrete groups have a phase transition due to their discreteness. For values of the coupling constant smaller than the critical point of this phase transition continuous groups diverge. A search is made for discrete subgroups of SU(2) and SU(3) for which the phase transition occurs beyond the region of physical interest. For SU(2) two subgroups are found that have a transition beyond this region. One of these, the double icosahedral group(' )(I), agrees with the group SU(2) well into the asymptotic weak coupling region. No discrete subgroups suitable for approximating the group SU(3) are found to exist. A comparison is made between the group I and the group SU(2)(' ). to see at what order of inverse bare coupling constant squared their strong coupling expansions differ. They are found to first differ at 11th order in this expansion. It is shown that in fact the expansions agree throughout the whole strong coupling region and that the strong coupling expansion breaks down before substantial disagreement is found. Monte Carlo calculations are carried out for three discrete sub-. groups of SU(2). These are the double tetrahedral group (T), the(' ). double octahedral group (O), and the group I. Evidence for first(' ). order phase transitions is found for each of these groups and the. critical value of inverse coupling squared is found. In addition I is(' ). seen to follow SU(2) in exhibiting asymptotic freedom behavior until. its phase transition occurs. Further the results for I are numerically compared with those for SU(2) and agree very closely over a wide range of coupling. A technique is introduced by which fermion effects can be included in the calculations. The fermion variables are first integrated out explicitly and the resulting determinant is written as the exponential of a quantity to be used as an effective action for fermions. This effective action is seen to be the sum of all possible gauge field loops on the lattice with certain coefficients. The sum is truncated so as to only include the smallest gauge loop in the effective action and calculations are performed. This procedure is justified because each link added to a loop includes a multiplicative parameter which is small for the cases considered here. The calculations are performed for T(' )and(' )I and the results are compared with a previously proposed method which incorporates complete fermion effects. Although the latter method is more accurate, the former uses much less computer time. The results of the two calculations are found not to differ with statistically meaningful deviations. However these two results do differ from those of pure gauge theory. Because of the particular loop that is chosen for the effective action it is concluded that these differences mainly arise from finite lattice effects.