  We rigorously analyze the quantum phase transition between a metallic and an
insulating phase in (non solvable) interacting spin chains or one dimensional
fermionic systems. In particular, we prove the persistence of Luttinger liquid
behavior in the presence of an interaction even arbitrarily close to the
critical point, where the Fermi velocity vanishes and the two Fermi points
coalesce. The analysis is based on two different multiscale analysis; the
analysis of the first regime provides gain factors which compensate exactly the
small divisors due to the vanishing Fermi velocity.
