Classical magnetization of a four-dimensional Platonic solid

Abstract

The 600-cell is a regular 4-polytope that is a four-dimensional analog of a Platonic solid. Three-dimensional Platonic solids with icosahedral $I_h$-symmetry have been shown to have a discontinuous ground-state magnetization response in an external field at the classical and quantum level, when spins mounted on their vertices interact according to the antiferromagnetic Heisenberg model. The discontinuities are not due to anisotropy in spin space, but rather to the special connectivity of the molecules. Here the nearest-neighbor antiferromagnetic XX and Heisenberg models in a magnetic field are considered for classical spins mounted on the 120 vertices of the 600-cell. The ground-state magnetization response is rich, characterized by six magnetization discontinuities in the XX case, and six magnetization and three susceptibility discontinuities in the Heisenberg case. This demonstrates that going from three to four spatial dimensions enriches the ground-state magnetization response.