Mean-plaquette approximation for lattice gauge theories at finite temperature

We extend the mean-plaquette approximation for lattice gauge theories to finite temperature. We show that it is necessary to consider exactly the plaquette variables along a chain in the Euclidean time direction. We apply the technique to a $Z\left(2\right)\mathrm{}$ and $Z\left(3\right)\mathrm{}$ lattice gauge theory in four dimensions with encouraging results.