Scaling theory of the Kondo problem with intersite interactions.

Abstract

We study the ground state of heavy fermions by renormalization-group theory and numerical exact diagonalizations. Our observation is that the ferromagnetic state is unstable compared to the Kondo state and that the ferromagnetic Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction scales to a trivial one close to zero, while in contrast the antiferromagnetic RKKY interaction ${\mathrm{J}}_{\mathrm{ij}}$ grows as scaling proceeds. This explains why we observe no ferromagnetic ordering in heavy-fermion materials in zero field. The antiferromagnetic RKKY interactions produce effects to reduce the \ensuremath{\beta} functions ${\mathrm{\ensuremath{\beta}}}_{\mathrm{J}}$\ensuremath{\equiv}dJ/d lnD and compete with the Kondo effect. Scaling theory and the diagonalizations show that for small ${\mathrm{J}}_{\mathrm{ij}}$\ensuremath{\ll}${\mathrm{T}}_{\mathrm{K}}$ the low-energy scale is given by \ensuremath{\Delta}E\ensuremath{\approx}${\mathrm{T}}_{\mathrm{K}}$${\mathrm{e}}^{\mathrm{\ensuremath{-}}\mathrm{\ensuremath{\alpha}}{\mathrm{J}}_{\mathrm{ij}}}$, and for large ${\mathrm{J}}_{\mathrm{ij}}$ we have a power law \ensuremath{\Delta}E\ensuremath{\approx}(1/${\mathrm{J}}_{\mathrm{ij}}$${)}^{\mathrm{\ensuremath{\alpha}}\ensuremath{'}}$. In the intermediate regime where the Kondo and RKKY effects cancel each other, our results support the existence of a zero of ${\mathrm{\ensuremath{\beta}}}_{\mathrm{J}}$.