# Spin-anisotropic magnetic impurity in a Fermi gas: poor man's scaling equation integration

## TL;DR

This paper develops and solves a scaling equation for a spin-anisotropic magnetic impurity in a Fermi gas with a power-law density of states, revealing phase behaviors and critical surfaces.

## Contribution

It introduces a second-order elliptic function solution to the scaling equations for spin-anisotropic Kondo models with non-constant DOS.

## Key findings

- Identifies phases with infinite isotropic antiferromagnetic exchange and decoupled impurity spin.
- Derives conditions for phases with infinite Ising exchange.
- Analyzes critical surfaces for finite isotropic antiferromagnetic exchange in pseudogap DOS.

## Abstract

We consider a single magnetic impurity described by the spin--anisotropic s-d(f) exchange (Kondo) model and formulate scaling equation for the spin-anisotropic model when the density of states (DOS) of electrons is a power law function of energy (measured relative to the Fermi energy). We solve this equation containing terms up to the second order in coupling constants in terms of elliptic functions. From the obtained solution we find the phases corresponding to the infinite isotropic antiferromagnetic Heisenberg exchange, to the impurity spin decoupled from the electron environment (only for the pseudogap DOS), and to the infinite Ising exchange (only for the diverging DOS). We analyze the critical surfaces, corresponding to the finite isotropic antiferromagnetic Heisenberg exchange for the pseudogap DOS.