Abstract
The phase diagram of the square-lattice Ising model with first- and second-neighbour interactions in an external magnetic field has been calculated by the interface method. The first-neighbour interactions are assumed to be antiferromagnetic while the interaction ratio, R, assumes arbitrary values. Closed-form second-order phase boundaries are obtained for three ordered phases (here called antiferromagnetic, layered and degenerate phases). This is not in agreement with Monte Carlo (MC) computations which proposed two ordered phases (called antiferro- and superantiferromagnetic phase). A proof is given that the layered and degenerate phase are two separate phases. For R>or=0 all transitions are second-order order-disorder transitions. For R<0 the transition from the ordered (antiferromagnetic) to the disordered phase is first order at low temperatures, and a tricritical-point estimate, which is in good agreement with MC and renormalisation-group calculations, is found. An estimate for the critical chemical potential of the hard-squares lattice gas with first- and second-neighbour exclusion is found. This is about 70% higher than the currently accepted value.