We present Monte Carlo studies of $Z\left(N\right)\mathrm{}$ and U(1) lattice gauge theories in three space-time dimensions. The $Z\left(N\right)\mathrm{}$ gauge theory is analyzed via its dual spin system. We find that $Z\left(N\right)\mathrm{}$ has one phase transition that moves to zero temperature as $N\to \infty$. The U(1) theory has only a single, confining phase. A study of Wilson loops in $Z\left(8\right)\mathrm{}$ shows that the coefficient of the area law vanishes for all temperatures below the critical temperature. No such zero is seen in the string tension for U(1) gauge theory. We also study the U(1) gauge theory in five dimensions and see a clear signal for a first-order phase transition.

• Received 5 February 1980

DOI:https://doi.org/10.1103/PhysRevD.21.2892