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

TL;DR

This paper investigates the real-space renormalization group flow in quantum impurity systems, revealing how characteristic length-scales like the Kondo cloud and crossover regions emerge from fixed points, with implications for understanding moment formation and screening.

Contribution

It introduces a real-space RG framework that identifies crossover length-scales between fixed points in quantum impurity models, clarifying the spatial structure of Kondo screening and moment formation.

Findings

Identification of crossover length-scales $\xi_{LM}$ and $\xi_K$ in the Anderson impurity model.

Demonstration that moment formation occurs inside the Kondo cloud, while screening occurs at larger distances.

Extension of the real-space RG analysis to the two-channel Kondo model, showing different screening regimes.

Abstract

The existence of a length-scale $\xi_K\sim 1/T_K$ (with $T_K$ the Kondo temperature) has long been predicted in quantum impurity systems. At low temperatures $T\ll T_K$, the standard interpretation is that a spin-$\tfrac{1}{2}$ impurity is screened by a surrounding `Kondo cloud' of spatial extent $\xi_K$. 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 behaviour 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 $\xi_{\text{LM}}$ and $\xi_K$, 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 $\sim \xi_K$. Generic aspects of the real-space physics are exemplified by the two-channel Kondo model, where $\xi_K$ now separates `local moment' and `overscreening' clouds.