Fermi liquids beyond the forward scattering limit: the role of non-forward scatterings for scale invariance and instabilities

Abstract

Landau Fermi liquid theory is a fixed point theory of metals that includes the forward scattering amplitudes as exact marginal couplings. However, the fixed point theory that only includes the strict forward scatterings is non-local in real space. In this paper, we revisit the Fermi liquid theory using the field-theoretic functional renormalization group formalism and show how the scale invariant fixed point emerges as a local theory. The local low-energy effective field theory for Fermi liquids includes not only the forward scatterings but also non-forward scatterings with small but non-zero momentum transfers. In the low-energy limit, the non-forward scattering amplitude takes a scale invariant form if the momentum transfer is scaled along with the energy. If the bare coupling is attractive beyond a critical strength, the coupling function exhibits a run-away flow, signifying potential instabilities in particle-hole channels.What drives those instabilities is the non-trivial renormalization group flow of the non-forward scattering amplitudes. The pairing interaction also obeys a scaling relation if the center of mass momentum of Cooper pairs is comparable with energy.