In the reading group this past week we finished the first chapter of Hewson's The Kondo problem to heavy fermions. There is a bit of a narrative which I will try to highlight.
A key aspect is getting to the Kondo Hamiltonian which describes a single magnetic impurity spin interacting with a band of conduction electrons.
What is the physical origin of the local magnetic moment?
To understand this one needs to look at the Anderson model Hamiltonian.
First, if one solves the model in the mean-field [Hartree-Fock] approximation one finds that a net magnetic moment can occur, i.e. spin rotational symmetry is broken, for certain parameter regimes. This result was half of Anderson's 1978 Nobel Prize.
Second, one can derive the Kondo Hamiltonian as an effective Hamiltonian for the Anderson model in the limit of large U and small hybridisation [the local moment regime]. This is done via a "canonical" transformation due to Schrieffer and Wolff or via van Vleck's quasi-degenerate perturbation theory. On some level one is "integrating out" the charge fluctuations on the impurity. This is analogous to the way one starts with a Hubbard model and derives a Heisenberg model as the effective Hamiltonian for the spin degrees of freedom in a Mott insulator.
This is done in Section 1.7.
The Anderson model with U=0 [Section 1.4] illustrates some key physics.
a. The existence of virtual bound states within the conduction band. These lead to a resonance and a large peak in the density of states (DOS).
It is also pointed out that the non-interaction Anderson model cannot explain the temperature dependence of the spin susceptibility associated with the impurities.
Finally. one introduces the Coqblin-Schrieffer model which is the generalisation of the SU(2) spin symmetry of the spin-1/2 Kondo Hamiltonian to SU(N) symmetry. There are two important reasons for this generalisation.
First, many magnetic impurities involve d and f electrons which have significant orbital degeneracy. For example, Ce3+ has a single electron in the 4f1 configuration which involves (with spin-orbital coupling) total angular momentum j=5/2 and so N=6.
Second, one can consider the large N limit, which turns out to analytically tractable. The corresponding mean-field (slave boson) theory captures a lot of the essential physics [chapters 7 and 8].