Progress in numerical simulations of systems with a $\theta-$vacuum like   term: The two and three-dimensional Ising model within an imaginary magnetic   field
  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. 
Vicente Azcoiti, Eduardo Follana and Alejandro Vaquero
hep-lat cond-mat.stat-mech