The triangular Ising antiferromagnet with first- and second-neighbour interactions

Abstract

The transition temperature for the triangular-lattice Ising model with first- and second-neighbour interactions has been calculated using the interface method. All interactions are assumed to be antiferromagnetic with an arbitrary interaction strength ratio. The authors include the possibility that the second-neighbour interactions are anisotropic and present explicit results, shown in figures 3, 4 and 5, for three special model classes. These special cases correspond to next-nearest-neighbour interactions in one, two or three directions, the nonzero couplings having equal strengths. The results agree with the exact critical temperature in limiting cases where it is known. They call attention to an entanglement problem that one may also encounter in other applications of the interface method if the arrangement is not chosen with care. Going beyond the interface method they discuss the exact limiting behaviour when one of the two interactions is very weak. When the first-neighbour interaction approaches zero, in particular, they argue that the transition temperature should vanish exponentially for the one-direction model, should have an algebraic singularity of order 4/7 for the two-direction model and should vanish linearly for the three-direction model.