Numerical Study of thetMatrix in the Kondo Problem

Abstract

A numerical study of the Kondo problem is presented. The calculations are based on the Suhl-Abrikosov-Nagaoka integral equation for the scattering amplitude $t(\omega ,T)$ of the $s-d$ exchange Hamiltonian. Use is made of the exact analytic solution first given in detail by Zittartz and Müller-Hartmann. It is shown that, because of the resonance in $\mathrm{Im}t(\omega ,T)$ which occurs at the Fermi energy $\omega =0\mathrm{}$ at low temperatures, the tunneling density of states of a dilute paramagnetic alloy is very slightly reduced at zero bias voltage. There is, however, a possibility of detecting this change by studying the derivative of the conductance. The details of the Zittartz-Müller-Hartmann expression are found to be unimportant for the low-temperature behavior of the transport coefficients with the exception of the thermoelectric power. A reinvestigation of the thermoelectric power shows that some care is necessary in the evaluation of the integrals $K_n=\int {-\infty }^{\infty }d\omega {\omega }^n{n}_0\left(\omega \right)\tau (\omega ,T)\frac{\partial f}{\partial \omega }$, because of the strong $\omega$ dependence of the electronic lifetime $\tau (\omega ,T)\propto {\left[\mathrm{Im}t\right(\omega ,T\left)\right]}^{-1\mathrm{}}$ at low temperatures. While the transport coefficients reflect the behavior of the scattering amplitude in a small energy interval about the Fermi energy, the specific-heat anomaly is found to be related to the temperature derivative of $t(\omega ,T)$ at large values of the energy variable $\omega$ comparable to the bandwidth. We also point out that the quasiparticle approximation is not valid for electrons interacting with impurity spins due to the rapid variation of $t(\omega ,T)$ near the Fermi energy.