A nonmagnetic effective Hamiltonian of a frustrated tetrahedron Kondo model

We investigate a four-site impurity Kondo model with 1/2 spins locating on vertices of a tetrahedron, when the inter site superexchange interaction J is much larger than band width D of conduction electrons and magnetic Kondo coupling. In this case, four impurity spins form doubly-degenerate ground states and we derive their effective Hamiltonian induced by coupling with conduction electrons. Quadrupole moments of the ground states couple to those of conduction electrons. For particle-hole asymmetric case, we find that the model can be mapped to that with a pseudospin-1/2 impurity coupled with three-orbital conduction electrons with keeping U(1) charge conservation for each of the orbital.