Polyhedra as ±J closed Ising lattices

Ising lattices are defined for regular polyhedra with spin occupying vertices and interactions laying along edges. Mixed ferromagnetic and antiferromagnetic interactions are considered, cursive Greek chi being the concentration of the former. Competition among local fields brings in frustration making non trivial to solve for physical properties of such lattices. Here, we characterize the most important ground state properties of these systems such as energy, remanent entropy, average frustration segment, diluted lattice (including unfrustrated domains), and site order parameter. The functional dependence on cursive Greek chi is established in each case, comparing among the 6 different polyhedra studied here. The rôle plaid by topology through aspects such as shape of faces and coordination number is brought out. When possible, a comparison with similar two-dimensional flat lattices is performed.