The Fermi liquid as a renormalization group fixed point

Abstract

The renormalization-group (RG) method is applied to study interacting fermions at finite temperature. A model based on the [psi][superscript]4-Grassmann effective action with SU(N )-invariant short-range interaction and a rotationally invariant Fermi surface in spatial dimensions d = 2, 3 is studied. We show how the key results of the Landau Fermi liquid theory can be recovered by this finite-temperature RG technique. Applying the RG to response functions, we find the compressibility and the spin susceptibility as solutions of the RG flow equations. We discuss subtleties associated with the symmetry properties of the four-point vertex (the implications of the Pauli principle). We point out distinctions between three quantities: the bare interaction of the low-energy effective action, the Landau function and the forward scattering vertex.The bare interaction of the effective action is not a RG fixed point, but a common starting point of the flow trajectories of two limiting forms of the four-point vertex. We have derived RG equations for the Landau channel that take into account both contributions of the direct (ZS) and the exchange (ZS' ) particle-hole graphs at one-loop level.The basic quantities of Fermi Liquid theory, the Landau interaction function and the forward scattering vertex, are calculated as fixed points of these flows in terms of the effective action's interaction function. The classic derivations of Fermi Liquid theory applying the Bethe-Salpeter equation and other analogous approaches, tantamount to some sort of RPA-type (decoupled) approximation, neglect the zero-angle singularity in the ZS' graph. As a consequence, the antisymmetry of the forward scattering vertex is not guaranteed in the final result, and the RPA sum rule must be imposed by hand on the components of the Landau function to satisfy the Pauli principle. This sum rule, not indispensable in the original phenomenological formulation of the Landau FLT, is equivalent, from the RG point of view, to a fine tuning of the effective interaction. Our results show that the strong interference of the direct and exchange processes of the particle-hole scattering near zero angle invalidates the RPA (decoupled) approximation in this region, resulting in temperature-dependent narrow-angle anomalies in the Landau function and scattering vertex, revealed by the RG analysis. In the present RG approach the Pauli principle is automatically satisfied. As follows from the RG solution, the amplitude sum rule, being an artefact of the RPA approximation, is not needed to respect statistics and, moreover, is not valid.