The wealth of a poor man's scaling

Chapter 3 of Hewson's The Kondo Problem to Heavy fermions reviews Anderson's poor man's scaling treatment of the Kondo model.

Starting with the anisotropic Kondo Hamiltonian

one rescales the electronic bandwidth D and see how the interactions J_z and J_ and J+ rescale.

To lowest order in perturbation theory this leads to the renormalisation group equations

Solving these gives the flow diagram below

A few important consequences

1. Antiferromagnetic (AFM) interactions flow to strong coupling.

2. The Kondo energy/temperature is invariant to the flow.

3. This is an example of asymptotic freedom [interactions can weaker at higher energies].

It is impressive that Anderson did this before Wilson and Fisher used renormalisation group ideas to describe critical phenomena in classical phase transitions.

It is fascinating that the same flow equations and flows describe the Kosterlitz-Thouless phase transition associated with topological order [vortex pair unbinding] in a classical two dimensional superfluid.

The spin boson model which describes the quantum decoherence of a single qubit in an ohmic environment can be mapped to the anisotropic Kondo model and so is also described by the same flow equations [See this famous (and rather dense) review by Leggett et al.]