Generalized Ising Model in the Absence of Magnetic Field

The Ising model on an one-dimensional monoatomic equidistant lattice with different exchange interactions between atomic spins at the sites of nearest neighbors and different interactions between atomic spins at the sites of second neighbors is investigated. The generalized Kramers–Wannier transfer matrix with translation on two periods of a lattice is derived. The phase diagrams representing all possible magnetic types of ordering in the ground state are constructed. Many triple points of intersection of phases and lines of pair coexistence of phases in strict accordance with the Gibbs phase rule are found and a property similar to supercooling and superheating is detected. It is shown that at triple points the phases are not individualized but significantly frustrated, which corresponds to the phenomenon of critical opalescence. Exact analytical expressions for the Gibbs free energy, as well as for such thermodynamic characteristics of the system as internal energy, heat capacity and entropy including zero-temperature entropy are derived. A variety of particular cases are analyzed and a comparison is made with all known results including on a two-dimensional lattice.