*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.

Where does this come from?

What is the physical origin of the local magnetic moment?

To understand this one needs to look at the

**Anderson model**Hamiltonian.

Key physics is associated with the Coulomb repulsion U [aka Hubbard] associated with two electrons on the localised orbital associated with the impurity.

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).

b. The energy scale associated with

**hybridisation.**

c. The

**Friedel sum rule**which connects the phase shift associated with scattering from the impurity with the change in the DOS and the total additional number of states that appear in the conduction band.

The above three concepts will all be useful for later understanding

**the Kondo resonance.**

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].

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