Renormalization-group approach to interacting fermions

Abstract

The stability or lack thereof of nonrelativistic fermionic systems to interactions is studied within the Renormalization Group (RG) framework, in close analogy with the study of critical phenomena using φ 4 scalar field theory. A brief introduction to φ 4 theory in four dimensions and the path integral formulation for fermions is given before turning to the problem at hand. As for the latter, the following procedure is used. First, the modes on either side of the Fermi surface within a cut-off Λ are chosen for study in analogy with the modes near the origin in φ 4 theory and a path integral is written to describe them.Next, an RG transformation which eliminates a part of these modes, but preserves the action of the noninteracting system is identified. Finally the possible perturbations of this free-field fixed point are classified as relevant, irrelevant or marginal. A d = 1 warmup calculation involving a system of fermions shows how , in contrast to mean-field theory, which predicts a charge density wave for arbitrarily weak repulsion, and superconductivity for arb itrarily weak attraction, the renormalization group approach correct ly yields a scale invariant

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