A paradigm for many-body physics

This weeks reading from Philip Phillips' book, Advanced Solid State Physics was Chapter 7, Quenching of Local Moments: The Kondo Problem.

First, the Kondo problem is not just of interest and importance because it exhibits some highly profound many-body physics which is of relevance to not just magnetic impurities in metals. It is the first known example of asymptotic freedom, where an interaction actually becomes stronger at lower energy scales.

Phenomenology. In most metals resistivity decreases with decreasing temperature. However, in the presence of magnetic impurities the resistivity actually increases below some temperature. Kondo sought to explain this. He found that a perturbation theoretical calculation actually led to a resistivity that diverged logarithmically at low temperatures. However, it was found experimentally, that the resistivity saturated. So did the magnetic susceptibility and the specific heat coefficient. Furthermore, the Sommerfeld-Wilson ratio for the impurity contribution was found to be two (in contrast to one for simple metals). The Kondo temperature sets the temperature scale at which the low-temperature saturation occurs.

A key step in resolving the Kondo problem (the crossover from the logarithmic divergence to the low temperature saturation) was the Anderson-Yuval-Hamman renormalisation group treatment (poor man's scaling) of the problem. It is important to appreciate that this was before Wilson introduced RG ideas to condensed matter physics. This led to the notion that the system had a strong coupling fixed point which leads to the formation of a spin singlet state (Kondo singlet) between the impurity spin and a large fraction of the spins of the metal. This state is nicely descibed by Yoshida's variational wave function.

The Kondo Hamiltonian can be derived as a limit of the single impurity Anderson model Hamiltonian (the local moment regime, where only virtual charge fluctuations matter).

Next week we will work through Sections 7.5 and 7.6 in detail.

Kondo physics is also important because it underlies much of the important physics captured by dynamical-mean field theory, particularly the crossover from a bad metal to a Fermi liquid in the Hubbard model.