A Method of Determining the Orderings of the Ising Model with Several Neighbor Interactions under the Magnetic Field and Applications to Hexagonal Lattices

Abstract

A method is presented to investigate spin orderings in the ground states of Ising models with several neighbor interactions under the external magnetic field (equivalent to determining ordered structures of binary alloys with arbitrary compositions). We make a configuration polyhedron in the space spanned by r₁, r₂, … r_n, m, in which linear inequalities are satisfied, where r_k is the number of the k-th neighbor down-down spin pairs and m the number of down spins. Spin orderings with values of r_k and m which are coordinates of the vertices of the configuration polyhedron are those in the ground states of Ising models. The method has a transparent outlook and is applied to the hexagonal close-packed (HCP) lattice with up to second neighbor interactions and to the plane hexagonal (PH) lattice with up to third neighbor interactions. Nine spin orderings in the ground state for HCP, which have one-to-one correspondences to those in FCC, and thirteen for PH are found and phase diagrams are obtained.