Kondo Screening and Impurity RG Flow

Antiferromagnetic Kondo exchange grows logarithmically as electronic states are eliminated toward the Fermi surface. In a specified density-of-states convention, leading poor-man’s scaling generates an exponentially small Kondo scale. The divergence marks the failure of weak-coupling RG and a flow toward strong coupling; it is not itself a solution of the screened infrared state.

Required background. Impurity models and local moments supplies the channel and bath contract. Beta functions and momentum-shell RG supply the running-coupling logic.

Poor-man’s scaling in a fixed convention

Take one spin-$1/2$ channel,

$$H_K=\sum_{k\sigma}\varepsilon_kc_{k\sigma}^\dagger c_{k\sigma} +J\,\mathbf S\mathbin{\cdot}\mathbf s(0), \qquad \mathbf s(0)=\frac12c_0^\dagger\boldsymbol\sigma c_0.$$

Let $\rho$ be the density of states per spin and define $j=\rho J$. Eliminating electron and hole shells as the half-bandwidth decreases from $D_0$ to $D$, with $\ell=\log(D_0/D)$, gives

$$\frac{\mathrm dj}{\mathrm d\ell}=2j^2+O(j^3).$$

Therefore

$$j(D)=\frac{j_0}{1-2j_0\log(D_0/D)}, \qquad T_K^{(1)}=D_0e^{-1/(2j_0)}.$$

Some authors define $g=2\rho J$, giving $\mathrm dg/\mathrm d\ell=g^2$ and $T_K=D_0e^{-1/g_0}$. The observable scale is unchanged once the density of states and coupling are translated.

For $J>0$, the coupling grows and the perturbative logarithms $J^n\log^{n-1}(D/E)$ must be resummed. Kondo 1964, pp. 37–43 identified the logarithmic scattering correction, and Anderson 1970, pp. 2438–2440 gives the scaling construction. For $J<0$, $j$ approaches zero from below: the ferromagnetic model is marginally irrelevant rather than screened in the same way.

Kondo temperature is not unique without a definition

Beyond leading logarithms, the prefactor and even the quoted numerical value of $T_K$ depend on the bandwidth scheme and observable definition. Common choices use the zero-temperature impurity susceptibility, the half-width of a spectral resonance, entropy crossover, or conductance scaling. A comparison must state which one is used and the conversion for the model.

The length $\xi_K\sim v_F/T_K$ characterizes the spatial crossover associated with screening. It is not the size of a rigid bound electron orbital; equal-time spin correlations, entanglement, and response reveal different spatial structures. Finite size $L\lesssim\xi_K$, temperature $T\gtrsim T_K$, or a superconducting gap $\Delta_{\mathrm sc}\gtrsim T_K$ cuts off the flow before the ordinary strong-coupling fixed point.

Anisotropic exchange obeys at leading order

$$\frac{\mathrm dj_\perp}{\mathrm d\ell}=2j_\perp j_z, \qquad \frac{\mathrm dj_z}{\mathrm d\ell}=2j_\perp^2.$$

The invariant $j_z^2-j_\perp^2$ organizes the trajectories. A pseudogap bath adds tree-level scaling and can create a finite-coupling critical point; the metallic beta function above cannot be reused unchanged.

Strong-coupling handoff

Weak-coupling RG is reliable while $j(D)\ll1$. Once the running coupling is order one, stop. The screened singlet, $\pi/2$ phase shift, irrelevant operators, and low-temperature observables require Wilson’s numerical RG, Bethe ansatz, boundary field theory, or the local Fermi-liquid description on the strong-coupling page. Wilson 1975 supplies the nonperturbative flow and scale separation.

The structure map shows that the perturbative divergence is a handoff, not an infrared answer.

Poor-man’s scaling generates the Kondo scale and identifies the direction of flow. The divergence only marks loss of perturbative control; it does not compute the infrared fixed point. Original schematic, not to scale.

The impurity claim test matrix records the scheme and stop rule.

Exercise

Integrate the flow. With $j_0=0.08$ in the per-spin convention, estimate $T_K/D_0$.

Solution

$T_K/D_0=e^{-1/(2j_0)}=e^{-6.25}\simeq1.93\times10^{-3}$. This is the leading-log scale. A susceptibility-defined or higher-loop $T_K$ differs by a convention-dependent prefactor, so quoting more digits would be misleading without that definition.

References

• Anderson, P. W. (1970). “A poor man’s derivation of scaling laws for the Kondo problem.” Journal of Physics C 3, 2436–2441. doi:10.1088/0022-3719/3/12/008.
• Kondo, J. (1964). “Resistance minimum in dilute magnetic alloys.” Progress of Theoretical Physics 32, 37–49. doi:10.1143/PTP.32.37.
• Wilson, K. G. (1975). “The renormalization group: Critical phenomena and the Kondo problem.” Reviews of Modern Physics 47, 773–840. doi:10.1103/RevModPhys.47.773.