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Generalized spin density wave state of the Hubbard model on the C12 and C60 clusters

The one-band Hubbard model at half-filling on a truncated tetrahedron (C$_{12}$), and on the C$_{60}$ molecule is studied. Within the Hartree-Fock approximation, we find a magnetic-like instability of the `Fermi sea' towards a spin-ordered phase for an intermediate value of the coupling ${\rm (U/t)_c}\approx 2.6$. The ordered phase presents a spin arrangement similar to that of the classical Heisenberg model defined on the same clusters. We study the linear excitations around the Hartree-Fock ground states using the random-phase approximation. On a finite cluster, we expect that these results signal the presence of a $rapid$ $crossover$ between a paramagnetic region and a regime where spin correlations are important. The relationship with the Heisenberg limit (large ${\rm U/t}$) is discussed. Finally, we comment on implications of our results for purely repulsive models of superconductivity in alkali-metal-doped fullerenes.