Theory of the Two-Impurity Kondo Effect in the Presence of an Impurity-Impurity Exchange Interaction

We examine the two-impurity Kondo effect by deriving the equation of motion of a set of Green's functions using the two-impurity $s-d$ Hamiltonian with an added exchange term of the form $W\left({\overrightarrow{\mathrm{S}}}_0·{\overrightarrow{\mathrm{S}}}_1\right)$, where ${\overrightarrow{\mathrm{S}}}_0$ and ${\overrightarrow{\mathrm{S}}}_1$ are the spin operators of the two impurities. The resulting Green's functions are truncated and solved for self-consistency, keeping the most divergent terms. Our results show that all the $\ln T$ terms arising in the single-impurity Kondo effect are modified and replaced by $\ln {(T^2+{W^{\prime }}^2)}^{\frac{1}{2}}$, where $W^{\prime }$ is an energy approximately equal to $W$. This results in an effective Kondo temperature $T_K^E$, where $T_K^E=T_K^0{[1-{\left(\frac{W^{\prime }}{T_K^0}\right)}^2]}^{\frac{1}{2}}$, and $T_K^0$ is the single-impurity Kondo temperature. Thus the effective Kondo temperature decreases as the impurity-impurity interaction increases, and when $W^{\prime }$ is greater than $T_K^0$ the Kondo divergence is removed by the impurity-impurity interaction. Our results show that the interaction $W$ strongly modifies the spin-compensated state. We also derive expressions for the conduction-electron polarization as a function of $W^{\prime }$ for high values of $W^{\prime }$ or high temperatures.