# Poor man's scaling and Lie algebras

## TL;DR

This paper derives general scaling equations for quantum impurity models with degenerate levels, analyzing their symmetries and applying to specific Lie algebra-based Hamiltonians, including anisotropic Coqblin-Schrieffer models.

## Contribution

It introduces a unified approach to scaling equations for impurity models with $su(3)$ symmetry and explores their connection to different symmetry groups.

## Key findings

- Scaling equations for $su(3)$-based Hamiltonians are derived.
- The connection between interaction symmetry and scaling behavior is analyzed.
- Application to anisotropic Coqblin-Schrieffer models is demonstrated.

## Abstract

We consider a general model, describing a quantum impurity with degenerate energy levels, interacting with a gas of itinerant electrons, derive general scaling equation for the model, and analyse the connection between its particular forms and the symmetry of interaction. On the basis of this analysis we write down scaling equations for the Hamiltonians which are the direct products of $su(3)$ Lie algebras and have either $SU(2)\times U(1)$ or $SU(2)$ symmetry. We also put into a new context anisotropic Coqblin -- Schrieffer models proposed by us earlier.