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

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-$\frac{1}{2}$ 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 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 ${\xi }_{\text{LM}}$ 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.

Figure 1

Three physical regions arise at $T=0$ in the Anderson impurity model, corresponding to the FO, LM, and SC FPs (a). The crossover length scale ${\xi }_K$ associated with the flow to the SC FP diverges as the Kondo effect is destroyed on increasing temperature through $T_K$ (b), while ${\xi }_{\text{LM}}$ diverges for $T\gg T_{\text{LM}}$ (c). The spin-$\frac{1}{2}$ moment forming at ${\xi }_{\text{LM}}$ is screened on flowing to the SC FP at ${\xi }_K$.

Figure 2

Comparison of the spectrum vs inverse frequency (upper panel) and normalized excess charge density vs distance (lower panel) for the Anderson model at $T=0$. Plotted for common ${10}^{3}V=3$ and ${10}^3\epsilon =-0.8$, varying ${10}^{3}U=1$, $1.125,$ and $1.25$ (solid, dotted, and dashed lines), chosen to give exaggerated energy- and length-scale separations for clarity.

Figure 3

Effects of temperature on the Anderson model (${10}^{3}V=3$, ${10}^3\epsilon =-0.8$, ${10}^{3}U=1$). Panels (a) and (b): comparison of impurity entropy with length-scales ${\xi }_K$ (solid lines) and ${\xi }_{\text{LM}}$ (dashed) as they evolve with $T$. (c): $\left|\Delta n\right(r,T\left)\right|$ vs $r$ for $T/T_K=0.1$, $0.5$, $0.75,$ and $0.85$. (d): ${\xi }_K\left(T\right)/{\xi }_K\left(0\right)$ vs $T/T_K$ for the same system (circles) and scaling collapse of the divergence to common universal curve for $V=0.05$, $\epsilon =-0.1$, $U=0.3$ (crosses). Dashed line is a fit to the data.

Figure 4

$T=0$ spectrum vs inverse frequency (upper panel) and normalized density vs distance (lower panel) for the 2CK model with $(J_1+J_2)=0.2$. Symmetric case $J_1=J_2$ shown as solid lines, while results for channel 1(2) shown as the dashed (dot-dashed) lines in the asymmetric case $(J_1-J_2)={10}^{-7}$. Common asymptotic behavior for $r\ll {\xi }_{2CK}$ is $\Delta n_i\left(r\right)/\Delta n_i^{ps}\left(r\right)\sim c/{\ln }^2(r/{\xi }_{2CK})-1$ while $\sim$${(r/{\xi }_{2CK})}^{-1/2}$ behavior is observed for $r\gg {\xi }_{2CK}$ ($\ll$${\xi }^{*}$). In the asymmetric case, excellent agreement is seen with the exact result in the vicinity of the FL crossover, Eq. (6) (points).

Figure 5

$T=0$ phase diagram for the 2CK model, showing LM, critical 2CK, and FL regions of space. Direct crossover from LM to FL occurs for large asymmetry (diamond points), but a quantum critical region of space appears for smaller asymmetry. Two distinct scales are observed in this regime: ${\xi }_{2CK}$ (crosses) and ${\xi }^{*}$ (circles). Solid line is a fit to expected behavior ${\xi }_{2CK}\equiv 1/T_{2CK}$, while dashed line is ${\xi }^{*}\equiv 1/T^{*}$.