## Tuesday, November 13, 2012

### What did Kondo do?

Chapter 2 of Alex Hewson's The Kondo problem to heavy fermions reviews what Kondo actually did to get his name on the problem. Here is a brief summary of the highlights from last weeks reading group.

He considered the experimental data on the temperature dependence of the resistivity of metals containing magnetic impurity atoms. It was particularly puzzling that there was a minimum. Generally, one expects scattering (and thus resistivity) to increase with increasing temperature.

First, Kondo recognised that the experimental data suggested that it was a single impurity problem, i.e, one could neglect interactions between the impurities.
Second, the effect seemed to scale with magnitude of the local magnetic moments.
This led him to consider the simplest possible model Hamiltonian the s-d model proposed by Zener in 1951, but now known as the Kondo model.

According to Boltzmann/Drude/Kubo at low temperatures the resistivity of a metal is proportional to the rate at which electrons with momentum k are elastically scattered into different states with momentum k'
Here T_kk' is the scattering T matrix.
Considering Feynman diagrams to second order in J, Kondo showed
One then substitutes this in the formula for the conductivity.
Integrating over energy leads to the famous logarithmic temperature dependence.
I am not really clear on what the essential physics is that leads to this logarithmic divergence, except something to do with spin flips in the particle-hole continuum above the Fermi energy.

The Kondo problem is that this leads to a logarithmic divergence at low temperatures. This suggests perturbation theory diverges. It also suggests an infinite scattering cross section which violates the unitarity limit. Somehow, this divergence must be cut off at lower temperatures by different physics.

The new physics turns out to be formation of spin singlets between the impurity spin and the conduction electron spins. These are known as Kondo singlets, although it was actually Yosida, Anderson, Nozieres, and Wilson who introduced/developed/showed this idea.