Real-space renormalization group flow in quantum impurity systems: Local moment formation and the Kondo screening cloud

Abstract

The existence of a length scale ${\ensuremath{\xi}}_{K}\ensuremath{\sim}1/{T}_{K}$ (with ${T}_{K}$ the Kondo temperature) has long been predicted in quantum impurity systems. At low temperatures $T\ensuremath{\ll}{T}_{K}$, the standard interpretation is that a spin-$\frac{1}{2}$ impurity is screened by a surrounding ``Kondo cloud'' of spatial extent ${\ensuremath{\xi}}_{K}$. We argue that renormalization group (RG) flow between any two fixed points (FPs) results in a characteristic length scale, observed in real space as a crossover between physical behavior typical of each FP. In the simplest example of the Anderson impurity model, three FPs arise, and we show that ``free orbital,'' ``local moment,'' and ``strong coupling'' regions of space can be identified at zero temperature. These regions are separated by two crossover length scales ${\ensuremath{\xi}}_{\text{LM}}$ and ${\ensuremath{\xi}}_{K}$, with the latter diverging as the Kondo effect is destroyed on increasing temperature through ${T}_{K}$. One implication is that moment formation occurs inside the ``Kondo cloud'', while the screening process itself occurs on flowing to the strong coupling FP at distances $\ensuremath{\sim}{\ensuremath{\xi}}_{K}$. Generic aspects of the real-space physics are exemplified by the two-channel Kondo model, where ${\ensuremath{\xi}}_{K}$ now separates local moment and overscreening clouds.