Exact Ground States of the Lattice Gas and the Ising Model on the Square Lattice

Abstract

The ground states of the lattice gas and the equivalent Ising spin system in an external field on the square lattice with the interaction range extending to the third neighbors are rigorously determined for arbitrary values of interaction constants and in the whole range of particle density or external field. The difficulty encountered in the previous analysis made by one of the present authors (M.K.) is removed by deriving two new geometrical inequalities which reflect the lattice geometry of a wider region than the previous ones. The underlying theory is applicable to other cases than the square lattice.