Anderson Model of Localized Magnetic Moments. I. High-Temperature Behavior

Abstract

In this, the first of a series of papers on the nondegenerate Anderson model, there is presented a graphical representation of the equations of motion of the $d$-electron Green's function such that the intra-atomic Coulomb energy $U{\hat{n}}_{+}{\hat{n}}_{-}$ is treated exactly. In this paper, the high-$T$ behavior of the system is studied. It is found that the magnetic susceptibility is given by $\chi ={\chi }_P+\frac{g^2}{T}\left(1-\frac{2\Delta }{\pi U\xi (1-\xi )}-{\left(\frac{2\Delta }{\pi U\xi (1-\xi )}\right)}^2\ln \left|\frac{2U(1-2\xi )\gamma }{\pi T}\right|+\cdots \right),$ where ${\chi }_P$ is the usual temperature-independent Pauli paramagnetism of the host metal; this expression agrees (except for the replacement $W\to \frac{2U|1-2\xi |\gamma }{\pi }$ with that obtained by Scalapino. The method of derivation makes it clear that the existence of a Curie-law susceptibility at high $T$ is intimately connected with the Kondo anomaly present in this model. The resistivity is found to be given by $\rho ={\rho }_0\left(1+\frac{9\Delta }{4\pi ({\epsilon }_F-{\epsilon }_d)}\ln \left|\frac{2U(1-2\xi )\gamma }{\pi T}\right|+\cdots \right),$ the coefficient of the logarithmic term differing from the value $\frac{3\Delta }{\pi ({\epsilon }_F-{\epsilon }_d)}$ obtained in the $s-d$ model. This discrepancy is due to the finite lifetime of the $d$ electron, an important feature of the Anderson model, contrary to the remarks of Schrieffer and Wolf. An even more important lifetime effect (at low $T$ is the replacement of $\ln \left(\frac{W}{T}\right)$ by $\ln \left[\frac{W}{\Gamma \mathrm{}\left(T\right)\mathrm{}}\right]$, where $\Gamma \mathrm{}\left(T\right)\mathrm{}$ is a nonanalytic (in $\Delta$) function of $T$ such that $\Gamma \mathrm{}\left(0\right)\mathrm{}$ is of order $T_c$ and $\Gamma \mathrm{}\left(T\right)\mathrm{}\to 0$ with increasing $T$ [although $\Gamma \mathrm{}\left(T\right)\mathrm{}\ne 0$ for all $T$].