Renormalization group study of interacting electrons

TL;DR

This paper extends the renormalization-group approach to interacting electrons with spin and finite temperature, showing its equivalence to Landau's Fermi liquid theory and analyzing response functions.

Contribution

It generalizes the RG method to include spin and temperature effects, establishing invariance of the Landau function and fixed points for physical scattering vertices.

Findings

Landau interaction function is RG invariant.

Physical forward scattering vertex is a stable fixed point.

RG approach aligns with Landau's Fermi liquid theory in 2D and 3D.

Abstract

The renormalization-group (RG) approach proposed earlier by Shankar for interacting spinless fermions at $T=0$ is extended to the case of non-zero temperature and spin. We study a model with $SU(N)$-invariant short-range effective interaction and rotationally invariant Fermi surface in two and three dimensions. We show that the Landau interaction function of the Fermi liquid, constructed from the bare parameters of the low-energy effective action, is RG invariant. On the other hand, the physical forward scattering vertex is found as a stable fixed point of the RG flow. We demonstrate that in $d=2$ and 3, the RG approach to this model is equivalent to Landau's mean-field treatment of the Fermi liquid. We discuss subtleties associated with the symmetry properties of the scattering amplitude, the Landau function and the low-energy effective action. Applying the RG to response functions, we find the compressibility and the spin susceptibility as fixed points.