Correlations for theS=1/2 antiferromagnet on a truncated tetrahedron

The antiferromagnetic Heisenberg Hamiltonian is investigated on a truncated tetrahedron, which is a closed 12-site system. We find that the ground state has many similarities to that of ${\mathrm{C}}_{60}$. We study two- and four-spin correlations in the classical ground state of the truncated tetrahedron and calculate the same correlations in the exact S=1/2 ground state. We find that the classical correlations survive for a range of bond strengths in the Heisenberg Hamiltonian and that one can construct a good trial wave function based on the classical ground state. This suggests that the correlations present in the classical ground state of ${\mathrm{C}}_{60}$ also survive in the exact ground state of that system, for a range of bond strengths about the physically relevant ${\mathit{J}}_2$≊${\mathit{J}}_1$. We calculate the momentum-space correlation function S(q), which is measurable by neutron scattering, for both ${\mathrm{C}}_{12}$ and ${\mathrm{C}}_{60}$. We also calculate correlations at finite temperature.