Path Integral Theory of Magnetic Alloys

The theory of dilute magnetic alloys is studied using Anderson's model. The Coulomb interaction is represented by the fluctuating potential acting on single electrons at the impurity site, and the partition function is rigorously formulated as a path integral over all possible time histories of this potential. For any particular path, the response of the electron gas is calculated using a method introduced by Nozi\`eres and De Dominicis, which is exact in the limit that the potential fluctuations are slow. From this, the contribution of any particular path to the partition function is obtained as an explicit and rather simple functional. When the Coulomb interaction is large compared to the width of the virtual bound state, a particular group of paths are singled out on the basis that they make the largest contributions. The functional is evaluated for this set of paths, and gives an expression which can be interpreted as the grand partition function for a one-dimensional gas of classical particles interacting through a logarithmic pair potential. This is identical to the result of a recent study of the s-d exchange model by Anderson and Yuval. An analysis of this result has been given earlier, and it yields a satisfactory description of the Kondo effect. The resistivity is estimated, and found to approach the unitarity limit below the Kondo temperature and the Hartree-Fock value above the Kondo temperature. The correspondence between the Anderson and s-d exchange models is shown to break down when the former is only weakly magnetic.