Fig. 2: Derivation of the Kondo–Heisenberg model.

a Materials-specific microscopic parameters are contained in the multi-orbital periodic Anderson lattice model (MO-PAM), where f-electrons are well localized due to a strong Coulomb repulsion. These states, however, still fluctuate through hybridization with the conduction electron bands. By integrating out the high-energy conduction band states, one arrives at a minimal Kondo + Heisenberg lattice model (KHM). The Kondo interaction contains the spin exchange interaction with the conduction electrons (blue dashed line), while the Heisenberg interaction is a combination of short-range interactions arising from completely filled or completely unoccupied conduction bands (purple p-like orbital between the local moments) and a long-range interaction due to particle-particle processes not captured through the Kondo exchange (right process in the MO-PAM sketch, and pink wavy line in the KHM sketch). This Hamiltonian encapsulates the physics of heavy-fermion materials such as CeIn₃. Finally, the low-energy conduction electrons can be integrated out to yield an effective spin Hamiltonian between local f-moments, which contains both the long-range RKKY interaction from the Kondo exchange (blue wavy line), and the other contributions explained above. b Results derived for CeIn₃ using this approach are shown. The upper panel shows the electronic bandstructure that serves as input for the MO-PAM along the path RΓXMΓ. The inset on the right illustrates the position of these high symmetry points in the Brillouin zone. The cut-off Λ = 0.5 eV around the Fermi-energy E_F allows short-range superexchange and long-ranged interactions to be seperated (see text for details). The lower panel shows the various contributions to the net magnetic interaction \({\tilde{I}}_{{{{{{{{\bf{q}}}}}}}}}\) in CeIn₃ shifted by the indicated energies for visibility. The color of each contribution denotes the corresponding process of the same color in a.