Renormalization Group Approach to Interacting Fermions

The stability of nonrelativistic fermionic systems to interactions is studied within the Renormalization Group framework. A brief introduction to $\phi^4$ theory in four dimensions and the path integral formulation for fermions is given. The strategy is as follows. First, the modes on either side of the Fermi surface within a cut-off $\Lambda$ are chosen and a path integral is written to describe them. 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, the RG correctly yields a scale invariant system (Luttinger liquid) In $d=2$ and 3, for rotationally invariant Fermi surfaces, {\em automatically} leads to Landau's Fermi liquid theory, which appears as a fixed point characterized by an effective mass and a Landau function $F$, with the only relevant perturbations being of the superconducting (BCS) type The functional flow equations for the BCS couplings are derived and separated into an infinite number of flows, one for each angular momentum. It is shown that similar results hold for rotationally non-invariant (but time-reversal invariant) Fermi surfaces also, A study of a nested Fermi surface shows an additional relevant flow leading to charge density wave formation. For small $\Lambda / K_F$, a 1/N expansion emerges, with $N = K_F/ \Lambda$, which explains why one is able to solve the narrow cut-off theory. The search for non-Fermi liquids in $d=2$