Solution of quantum double exchange on the complete graph from spl(2,1) dynamical supersymmetry

The exact and analytical low energy spectrum of the ferromagnetic Kondo lattice model on a complete graph extended with on-site repulsion $U$ is obtained in terms of a dynamical spl(2,1) supersymmetry in the limit of infinitely strong Hund's coupling (double exchange). Furthermore, we show that for the particular value of $U=J_H/2$ the supersymmetry is not constrained to infinite $J_H/t$ only and we calculate the energy including the $t^2/J_H$ corrections analytically and we give numerical evidence which suggest that the Kondo Hamiltonian is itself supersymmetric for any $J_H/t$. On a $N$ site graph, the ferromagnetic ground state is realized for 1 and $N$+1 electrons only. In the leading order in the value of the core spin, the quantum and semiclassical spectra are identical.