Problem: Anderson Model

      The Anderson Hamiltonian describes electrons in a transition metal, including the interactions between the electrons:

 

where and indicate the spin state, , and ; this is similar for ``down" spins. The first term in the Hamiltonian corresponds to electrons in a broad conduction band with energy , where is the wavenumber. Assume that is known and that the corresponding density of states, , is constant. The second and third terms describe electrons in the narrow d band and the transfer between the two bands, respectively. There are states in the narrow band for each spin. The last term represents the interaction between the electrons on the d levels; only electrons with opposite spins are allowed to occupy these levels.

The bandwidth corresponding to is much larger than the transfer matrix element . The Fermi energy , , and are all well within the conduction band.

(a) What kind of magnetic behavior is expected for , if the number of d electrons is ? Discuss what happens to the magnetic properties if is less than or greater than , and is less or greater than .

  (b) For , the exact and full solution to the problem is not known. However, within the framework of the mean field approximation, we can have some ideas about the behavior of the system. Replace the Anderson Hamiltonian with

 

and a similar one for the spins. Here is the expectation value of the spin occupation of the d state (the sum is over the occupied states).

  Use the self-consistency condition to calculate the magnitude of the localized moment, as a function of and , where .

Hint: Search for the solution in the form of with the normalization condition . The expectation value can be converted to an energy integral: , where g(E) is the density of states, determined by solving Eq. 9.12.