  The Ashkin-Teller (AT) model is a generalization of Ising 2-d to a four
states spin model; it can be written in the form of two Ising layers (in
general with different couplings) interacting via a four-spin interaction. It
was conjectured long ago (by Kadanoff and Wegner, Wu and Lin, Baxter and
others) that AT has in general two critical points, and that universality
holds, in the sense that the critical exponents are the same as in the Ising
model, except when the couplings of the two Ising layers are equal (isotropic
case). We obtain an explicit expression for the specific heat from which we
prove this conjecture in the weakly interacting case and we locate precisely
the critical points. We find the somewhat unexpected feature that, despite
universality holds for the specific heat, nevertheless nonuniversal critical
indexes appear: for instance the distance between the critical points rescales
with an anomalous exponent as we let the couplings of the two Ising layers
coincide (isotropic limit); and so does the constant in front of the logarithm
in the specific heat. Our result also explains how the crossover from universal
to nonuniversal behaviour is realized.
