Let us investigate in more detail the origin of the logarithmic term that we found in the previous chapter. We examine the s–d Hamiltonian from this point of view. We first consider the two total wavefunctions of conduction electrons moving in two different local potentials. The overlap integral and matrix elements of these two wavefunctions contain logarithmic terms. In particular, the overlap integral vanishes at zero temperature in general. Second, we investigate the development of the wavefunction under a time-dependent perturbation where one potential changes into the other one abruptly at a particular time. This problem is closely related to the physics of the s-d interaction, since the potential changes quite abruptly as the spin-flips occur due to the s-d interaction. This problem is solved using the Nozières-de Dominicis method, and we apply these general considerations to the s–d Hamiltonian to obtain the perturbation expansion of the partition function and other physical quantities. The behavior of the localized spin at low temperatures is clarified using the scaling method applied to the s–d Hamiltonian.

The Anderson orthogonality theorem

This section is based mainly on Anderson (1967a,b).

In the previous chapter, we showed that the logarithmic singularity arises as a function of temperature in several physical quantities for systems governed by the s–d Hamiltonian due to flips of the localized spin. It is essential in this phenomenon that the localized spin has an internal degree of freedom.