Link to pdf version of notes for lecture 23. Those note contain information for this lecture. Additional information is in handwritten notes available from Prof. Martin.

References:
O. E. Gunnarsson and K. Schonhammer, Phys. Rev. 28, 4315 (1983).
A discussion is given in the first edition of Mahan (omitted in the second edition) Mahan, p. 977 ff.

This lecture follows the previous lecture, where the Anderson and Kondo models are defined. Here we continue with a discussion of the solution of the Anderson model. The Hartree -Fock solution of Anderson sets up the problem. Above a critical value of U, the HF solution is a broken symmetry magnetic state with a net spin on the impurity and a tendency of the conduction electrons to have opposite spin. This is a degenerate state that is coupled to a metallic sea of electrons with low energy excitations. The correct solution is a linear combination that lifts the degeneracy. This is an example of the correct solution with no magnetic moment but a large susceptibility (low energy scale for spin flips) whereas the mean-field HF gave a broken symmetry degenerate solution.

The spectra for the Anderson model can be derived analytically for the model in which the local state has degeneracy N, and letting N go to infinity. The original Kondo model was N=2 for spin 1/2. (There are solutions using many methods that demonstrate the nature of the exact ground state and excitations. The present discussion agrees with the exact solutions and gives a simple way to get the main features.) The large N limit has the formal property that mean field theory becomes exact and one can write dow the analytic solution is simple case a numerical solution in any case. Furthermore, real cases like rare earths with f-states where N=14 are reasonably approximated by such a theory.

The resulting spectra is the famous three-peak form. Two peaks are the expected remnants of the
split Hubbard peaks separated b U. The third is at the Fermi energy and is a narrow resonce that is the rement of the U=0 peak in the non-nteracting case. It is required to obey the sum rules and it contains the new information on the new energy scale for spin-flip excitations, i.e., the Kondo temeprature that is due to te many-bod interacting electron problem.

The pdf notes explain more details and Prof.. Martin can provide more detailed hand-written notes to anyone. Homework 4 is designed to illustrate key points.

Understanding this impurity problem is an important step in understanding the ideas of dynamical mean field theory (DMFT).