Wilson-Kadanoff Renormalization Group in Higher Orders: One-Dimensional g-ology Model as an Example

TL;DR

This paper applies the Wilson-Kadanoff momentum-space RG scheme to a one-dimensional fermion g-ology model, explicitly deriving two-loop flow equations and analyzing their relation to field-theoretic approaches, with potential for generalizations.

Contribution

The paper demonstrates how to derive RG flow equations at two-loop level within the Wilson-Kadanoff scheme for a 1D fermion model, clarifying the scheme's rules and connections to other methods.

Findings

Derived two-loop RG flow equations for the g-ology model.

Clarified the natural emergence of cascade selection rules from the WK scheme.

Analyzed the relation between WK RG and field-theoretic RG approaches.

Abstract

We apply the standard Wilson-Kadanoff (WK) momentum-space Renormalization Group (RG) scheme for the g-ology model of one-dimensional fermions. By explicitly carrying out calculations at the two-loop level, we show how the RG flow equations can be derived from the summation of the cascades of contractions generated by the effective action's mode elimination at each infinitesimal step of the WK procedure. The rules for selecting these series of cascades appear naturally as a consequence of the WK scheme ``on-shell'' kinematic constraints and conservation laws. The relation between the present RG approach and the field-theoretic schemes used in earlier related studies is analysed. Generalizations for other models and/or higher dimensions are formulated.