## Wednesday, November 28, 2012

### What did Wilson do?

Last week we struggled through chapter 4, "Renormalisation group calculations" of Hewson's book, The Kondo Problem to heavy fermions.

The focus is on Kenneth Wilson's numerical treatment of the Kondo problem, mentioned in his Nobel prize citation. Much of it still remains a mystery to me...
Here are a few key aspects. Please correct me where I am wrong or at least confused...

First, he mapped the three-dimensional Kondo model Hamiltonian into a one dimensional tight binding chain (half-line) with single impurity spin at the boundary. This simplification makes the problem more numerically tractable.

Next, he used a logarithmic discretization (in energy) of the states in the conduction band. This important step is motivated by the logarithmic divergences found by Kondo's perturbative calculation and Anderson's poor man's scaling arguments.

He then numerically diagonalises the Hamiltonian with a discrete set of states for a finite chain. One then rescales the Hamiltonian, truncates the Hilbert space, and adds an extra lattice site.

Eventually, one converges to the strong coupling fixed point and one observes an almost equally spaced excitation spectrum, characteristic of a Fermi liquid.

A surprising thing is that the rescaling parameter Lambda is set to a relatively large value of 2, compared to a value close to one, that one might expect to be needed. Wilson was clever to realise/find that such coarse graining would work so well.

Wilson extracted a large amount of information from his calculations. Here are a few important findings.

1. The impurity specific heat and impurity susceptibility had a Fermi liquid temperature dependence. The latter was given by
[This] shows that there is no residual local moment, and that the impurity spin is fully compensated. The numerical factor 0.4128 is a universal number for the s-d model, and is known as the Wilson number, w. It relates two quite different energy scales for the s-d model, T_K, which is determined from the high temperature perturbative regime, and chi_imp(0), the low temperature susceptibility associated with the strong coupling regime.
2. The Sommerfeld-(Wilson) ratio had a universal value
3. Over an intermediate temperature range (one and half decades) the temperature dependence can be fit to
This Curie- Weiss form corresponds to a reduced moment compared to the free spin form. Thus the impurity moment, even for T ~ T_K, is only of the order of 30% that of the free moment. The residual effects of the screening of the conduction [electrons] persist to very high temperatures because of the logarithmic dependence on T/T_K.
4.  The complete universal dependence with a logarithmic temperature scale is shown below