Heavy fermions and Kondo lattices

Use a local moment of spin one half coupled antiferromagnetically to the conduction spin density, with the latter defined as one half of the electron Pauli bilinear. The exchange parameter is then an energy. Sources that omit the one-half factor use a different exchange convention, so quoted screening exponents must be translated consistently. The weak-coupling Kondo logarithm describes repeated spin-exchange scattering and grows toward strong coupling as the running energy cutoff is reduced.

In a stated one-loop convention, define a dimensionless coupling using the conduction density of states per spin. Its running grows quadratically, producing an exponentially small screening scale. Prefactors and the operational definition of Kondo temperature are model dependent, and the one-loop flow does not solve the strong-coupling state itself. In a lattice, conduction electrons also mediate intermoment interactions of algebraic order exchange squared times density of states. Their sign and wavevector depend on band structure and separation, rather than being universally antiferromagnetic.

For one impurity, low-energy screening removes the isolated local-moment entropy and produces a many-body resonance. A dense lattice is not obtained by adding independent impurity resistivities at every site. Coherent scattering from a periodic array can generate new bands, while intermoment correlations compete with screening. The simple comparison of exponential and algebraic scales is the Doniach organizing picture, not a quantitatively universal critical-coupling formula.

A useful low-energy mean-field representation mixes a conduction level with a renormalized narrow level through a hybridization amplitude. Diagonalizing that two-band matrix creates an avoided crossing, which can strongly reduce the slope of a branch near the Fermi level. The auxiliary narrow level and hybridization summarize many-body physics within a chosen approximation. A bare two-level diagonalization does not prove Kondo screening, determine a self-consistent constraint or include all incoherent spectral weight.

In a conventional translation-invariant Kondo-screened Fermi liquid, a localized spin-one-half moment per cell contributes to the large Fermi-volume count along with conduction electrons. This conditional theorem does not establish that a given Kondo lattice is in that phase. Magnetic order can enlarge the unit cell, a Kondo insulator can have no Fermi surface, and fractionalized topologically ordered phases can change the conventional counting framework.

A large specific-heat coefficient is evidence for a large low-energy density of states, but alone it does not identify a particular band or uniquely determine a quasiparticle residue. Thermodynamics, momentum-resolved dispersion, quantum oscillations and magnetic response provide complementary constraints. The pure two-level lab can illustrate eigenbasis overlap at an avoided crossing in a limited analogy; it cannot solve a Kondo impurity, impose a many-site constraint or generate a heavy-fermion band from an interacting lattice Hamiltonian.