Kondo screening and impurity physics

Consider one spin-half impurity and one spin-degenerate metallic screening channel, with a smooth approximately constant local density of states near the Fermi energy. Use dimensionless impurity spin S and conduction spin density s. For this lesson choose an interaction $2J_K\mathbf S\cdot\mathbf s(0)$, explicitly including a factor of two. Positive JK is antiferromagnetic. This normalization differs from the coefficient of an unordered intersite spin bond earlier in the module; comparing bare numbers without the factor would be incorrect.

Let rho be the local density of electronic states per spin for that channel, in inverse energy units, and define $g=\rho J_K$. The band energies are measured relative to the Fermi energy, with initial cutoff D0 representing the half-bandwidth in the flat-band idealization. The local conduction spin uses Pauli matrices divided by two. Changing either the exchange convention or from a per-spin to a two-spin density of states changes the numerical coefficient in the flow equation and the exponent unless g is translated consistently.

Virtual electron and hole processes near the edges of the remaining band renormalize the low-energy spin-flip amplitude. Their denominators produce logarithms when shells of high-energy states are eliminated. To leading logarithmic order the isotropic coupling grows at a rate proportional to its square as the cutoff decreases. An initially positive g therefore increases, whereas a negative isotropic ferromagnetic coupling approaches zero from below. These are weak-coupling statements and cannot be used quantitatively after g becomes order one.

Integrating the leading flow yields a denominator that vanishes at an exponentially reduced cutoff. That formal divergence is the failure of perturbation theory, not an infinite observable or a physical thermodynamic singularity. Its location defines a leading-logarithmic estimate of the Kondo scale. Higher-order scaling and operational definitions introduce algebraic and numerical prefactors, so the exponential estimate should not be treated as a precise experimental temperature without specifying the convention and additional corrections.

For the ordinary single-channel spin-half antiferromagnetic model with a metallic bath, the low-temperature state is screened and has Fermi-liquid behavior. The impurity contribution to entropy crosses from approximately $k_B\ln2$ for a free moment to zero at low temperature. Its susceptibility no longer follows an unlimited Curie divergence. Screening involves correlated conduction states over many scales and positions; the full conduction band is not replaced by one isolated singlet bond.

This evolution is a crossover for one impurity, not a bulk finite-temperature ordering transition. A perturbative logarithmic increase of impurity scattering must therefore not be extrapolated into a literal divergent zero-temperature resistance. The resistance minimum also depends on phonons and other backgrounds and need not equal a uniquely defined Kondo temperature. Larger impurity spins or additional independent screening channels can produce under- or overscreening, and a vanishing or structured bath density of states can change the problem qualitatively. Those extensions are outside the assumptions of the simple flow used here.