BETA-FUNCTION AND ANOMALY OF THE FERMI-SURFACE FOR A D=1 SYSTEM OF INTERACTING FERMIONS IN A PERIODIC POTENTIAL

We derive a perturbation theory, based on the renormalization group, f or the Fermi surface of a one dimensional system of fermions in a peri odic potential interacting via a short range, spin independent potenti al. The infrared problem is studied by writing the Schwinger functions in terms of running couplings. Their flow is described by a Beta func tion, whose existence and analyticity as a function of the running cou plings is proved. If the fermions are spinless we prove that the Beta function is vanishing and the renormalization flow is bounded for any small interaction. If the fermions are spinning the Beta function is n ot vanishing but, if the conduction band is not filled or half filled and the interaction is repulsive, it is possible again to control the flow proving the partial asymptotic freedom of the theory. This is don e showing that the Beta function is partially vanishing using the exac t solution of the Mattis model, which is the spin analogue of the Lutt inger model. In both these cases Schwinger functions are anomalous so that the system is a ''Lutttinger liquid.'' Our results extend the wor k in [B.G.P.S], where neither spin nor periodic potential were conside red; an explicit proof of some technical results used but not explicit ly proved there is also provided.