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Using the approach developed in \cite{REFVIC2}, we succeeded inreconstructing the behaviour of the antiferromagnetic Ising model withimaginary magnetic field $i\theta$ for two and three dimensions in the lowtemperature regime. A mean-field calculation, expected to work well for highdimensions, is also carried out, and the mean-field results coincidequalitatively with those of the two- and three-dimensional Ising model. Themean field analysis reveals also a phase structure more complex than the oneexpected for $QCD$ with a topological $\theta-$therm.

nuclear physics b

Progress in numerical simulations of systems with a $\theta-$vacuum like  term: The two and three-dimensional Ising model within an imaginary magnetic  field