Variational method for theZ(2)gauge model

A variational method similar to the one used for the Ising model is applied to the Hamiltonian $Z\left(2\right)\mathrm{}$ lattice gauge theory in three dimensions. It is shown that the proposal of a gauge-invariant ground state leads to a transition in the Wilson loop integral from the area to the perimeter behavior for a value of the coupling constant close to the symmetry point predicted by self-duality. The discontinuity which appears in the variational parameter gives strong evidence in favor of the first-order nature of the transition in contrast to what occurs for the two-dimensional model.