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 definitions. 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

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

Therefore

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

Equivalently, if a source uses the total two-spin density of states $\rho_{\mathrm{tot}}=2\rho$, then $j_{\mathrm{tot}}=\rho_{\mathrm{tot}}J=2j$ and the same flow reads $\mathrm dj_{\mathrm{tot}}/\mathrm d\ell=j_{\mathrm{tot}}^2/2$, with $T_K^{(1)}=D_0e^{-2/j_{\mathrm{tot},0}}$. Other texts absorb factors into the exchange vertex itself. The numerical coefficient is meaningful only together with the Hamiltonian and density-of-states definition.

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. Cheng et al. 2017, accepted manuscript, Eqs. (66)–(67), pp. 11–12, PDF displays the same $\boldsymbol\sigma/2$, per-spin-$\rho$ convention and fixes the coefficient above. 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

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

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

$T_K/D_0=e^{-1/j_0}=e^{-12.5}\simeq3.73\times10^{-6}$. 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.
• Cheng, M., Chowdhury, T., Mohammed, A., and Ingersent, K. (2017). “Phase boundaries of power-law Anderson and Kondo models: A poor man’s scaling study.” Physical Review B 96, 045103. doi:10.1103/PhysRevB.96.045103.
• 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.