Perturbative studies on the Anderson Hamiltonian

Giuseppe Ernesto Santoro, Purdue University

Abstract

The thesis is a theoretical study of the physics of magnetic impurities in metals, as modeled by the non degenerate Anderson Hamiltonian. The method is an order-by-order diagrammatic expansion in powers of the Coulomb interaction parameter U, which has been shown to provide a converging power expansion for the known exact solution of the one-impurity Anderson model. Chapter 2 presents our results concerning the one-impurity Anderson model, and compares them with known exact results. Specifically, results concerning the impurity self-energy and density of states, the impurity resistivity, the impurity dynamical spin-spin correlation function and spin susceptibility, and the spatial dependence of the instantaneous correlation function between the impurity and conduction-electron spins are presented. In Chapter 3 we give a diagrammatic derivation of some general Fermi liquid Ward identities which hold true to all orders in perturbation theory. The analysis is not restricted to the one-impurity model. The differences between the one and two-impurity models are investigated. The single-parameter universal behavior occurring for the one-impurity model is spoiled, in the two-impurity case, by the appearance of a new non-vanishing vertex function. In Chapter 4 we deal with the specific case of the two-impurity Anderson Hamiltonian, showing how the reflection symmetry of the model can be used to establish exact results for the conduction-electron phase shifts at the Fermi level. Such phase shifts can never be equal to 0 or $\pi$, and they are always equal to ${\pi\over2}$ for the symmetric model. In Chapter 5, explicit second order perturbative calculations for the two-impurity Anderson model are presented. A reasonably complete picture of the properties of the two-impurity model in the small and intermediate U regime emerges from these results. Moreover, evidence is given which supports the conjecture that, in the large U limit, the physics of the two-impurity problem could be very different from the known behavior of two isolated impurities. The details of the wave-vector dependence of the coupling between the impurities and the surrounding electron gas, neglected in most of the previous investigations, can be important in determining the correlations between the impurities. (Abstract shortened with permission of author.)

Degree

Ph.D.

Advisors

Giuliani, Purdue University.

Subject Area

Condensation

COinS