  We compute the Drude weight and the critical exponents as functions of the
density in non-integrable generalizations of XXZ or Hubbard chains, in the
critical zero temperature regime where Luttinger liquid description breaks down
and Bethe ansatz cannot be used. Even in the regions where irrelevant terms
dominate, no difference between integrable and non integrable models appear in
exponents and conductivity. Our results are based on a fully rigorous
two-regime multiscale analysis and a recently introduced partially solvable
model.
