Antiferromagnetic Heisenberg model on clusters with icosahedral symmetry

The nearest-neighbor antiferromagnetic Heisenberg model is considered for spins ${s}_{i}=\frac{1}{2}$ and 1 located on the vertices of the dodecahedron and the icosahedron, which belong to the point symmetry group ${I}_{h}$. Taking into account the permutational and spin-inversion symmetries of the Hamiltonian results in a drastic reduction of the dimensionality of the problem, leading to full diagonalization for both clusters. There is a strong signature of the frustration present in the systems in the low-energy spectrum for ${s}_{i}=\frac{1}{2}$, where the first excited states are singlets. Frustration also results in a doubly peaked specific heat as a function of temperature for the dodecahedron. Furthermore, there is a discontinuity in the magnetization as a function of magnetic field for the dodecahedron, where a specific total spin sector never becomes the ground state in a field. This discontinuity is accompanied by a magnetization plateau. For ${s}_{i}=1$ both systems again have singlet excitations. The magnetization of the dodecahedron has now two discontinuities in an external field and also magnetization plateaux, and the specific heat of the icosahedron, a two-peak structure as a function of temperature. The similarities between the two systems suggest that the antiferromagnetic Heisenberg model on a larger cluster with the same symmetry, the 60-site cluster, will have similar properties.