Renormalized Perturbation Theory

In chapter 5 where the Fermi liquid theory for magnetic impurity models was developed we considered two approaches. One was the phenomenological approach, based on the work of Landau and Nozières, where we conjectured a specific form for a quasi-particle Hamiltonian with a local interaction term. This Hamiltonian was essentially equivalent to the effective Hamiltonian near the strong coupling fixed point obtained by Wilson in his numerical renormalization group calculation (section 4.5, equation (4.49)). In the later chapters, where we used this approach for the N-fold degenerate Anderson model (section 7.4) and the n-channel Kondo model for n = 2S (section 9.3), the conjectured form for the quasi-particle Hamiltonian was not backed up by any first principles renormalization group calculation. The other approach developed in chapter 5 was the microscopic Fermi liquid theory based largely on the work of Luttinger (1960, 1961) and Yamada & Yosida (1975) using a conventional perturbation expansion in powers of U. This microscopic treatment confirmed all the results based on the conjectured quasi-particle Hamiltonian. Here we develop a synthesis of the two approaches which we will refer to as ‘renormalized perturbation theory’ (Hewson, 1992). It is based on the general idea of renormalization used in quantum field theory. The results at low temperatures correspond essentially to our earlier calculations with the conjectured quasi-particle Hamiltonian. These are obtained from first and second order perturbation theory in powers of the renormalized interaction Ũ.