Ward identities and the β function in the Luttinger liquid

Abstract

One-dimensional metals have a particular symmetry associated with the discrete structure of the Fermi surface: separate charge conservation in low-energy-scattering processes for particles near the left and right Fermi points, respectively. The field-theoretical renormalization group allows for an efficient exploitation of the Ward identities following from this symmetry. As a first application we prove that the β function of the Luttinger model vanishes identically. The same symmetry ensures the finiteness of the compressibility, thus making possible the existence of stable metallic phases with anomalous dimensions in d=1.