  Using the approach developed in \cite{REFVIC2}, we succeeded in
reconstructing the behaviour of the antiferromagnetic Ising model with
imaginary magnetic field $i\theta$ for two and three dimensions in the low
temperature regime. A mean-field calculation, expected to work well for high
dimensions, is also carried out, and the mean-field results coincide
qualitatively with those of the two- and three-dimensional Ising model. The
mean field analysis reveals also a phase structure more complex than the one
expected for $QCD$ with a topological $\theta-$therm.
