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 the small divisors due to the vanishing Fermi velocity.
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