  The Ashkin-Teller model can be formulated as a pair of 2D Ising models,
interacting via a four-spin interaction. I consider the case of weak anisotropy
(slight a-symmetry between the two Ising layers) and weak coupling. I show that
the system admits two critical temperatures whose difference varies
continuously with the strength of the coupling, scaling with an anomalous
exponent as one let the a-symmetry parameter go to zero. The specific heat
diverges logarithmically at the critical points (as for Ising) but the constant
in front of the logarithm is renormalized by an anomalous critical exponent.
The logarithmic divergence of the specific heat dominates only in an
exponentially small interval around the critical temperatures and outside it is
modified into an anomalous power law behaviour. The proof is based on an exact
mapping of Ashkin-Teller into a model of (1+1)D interacting fermions and on the
implementation of constructive Renormalization group methods on the fermionic
system. This PhD thesis includes: (1) a review of the exact solution of 2D
Ising in terms of Grassmann functional integrals; (2) a review of the Grassmann
representation for a class of interacting Ising models, including Ashkin-Teller
and the 8 vertex model; (3) a detailed discussion of the multiscale analysis of
the Ashkin-Teller model, based on fermionic Renormalization Group methods,
including the study of the flow of the effective coupling constants (in
particular a proof of vanishing of the beta function for this and similar
Luttinger-like models, based on modified approximate Ward identities, is
included). This thesis is based on joint work with V. Mastropietro.
