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Journal of Statistical Physics

Ward Identities and Vanishing of the Beta Function for d = 1 Interacting Fermi Systems

G. Benfatto, Vieri Mastropietro

Published Mar 17, 2004• 25 citations
Publisher
DOI
10.1023/b:joss.0000019825.37968.23

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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 \ensuremath{\beta} 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.
We consider a system of two-dimensional interacting fermions with a flat Fermi surface. The apparent conflict between Luttinger and non-Luttinger liquid behaviors found through different approximations is resolved by showing the existence of a line of nontrivial fixed points, for the renormalization group (RG) flow, corresponding to Luttinger liquid behavior; the presence of marginally relevant operators can cause flow away from the fixed point. The analysis is nonperturbative and based on the implementation, at each RG iteration, of Ward identities obtained from local phase transformations depending on the Fermi surface side, implying the partial vanishing of the beta function.