Ising-Model Spin Correlations on the Triangular Lattice. IV. Anisotropic Ferromagnetic and Antiferromagnetic Lattices

Abstract

A detailed discussion of pair correlations ω2(r) = 〈σ0σr〉 between spins at lattice sites 0 and r on the axes of anisotropic triangular lattices is given. The asymptotic behavior of ω2(r) for large spin separation is obtained for ferromagnetic and antiferromagnetic lattices. The axial pair correlation for the ferromagnetic triangular lattice has the same qualitative behavior as that for the ferromagnetic rectangular lattice: There is long-range order below the Curie point TC and short-range order above. It is shown that correlations on the anisotropic antiferromagnetic triangular lattice must be given separate treatment in three different temperature ranges. Below the Néel point TN (antiferromagnetic critical point), the completely anisotropic lattice exhibits antiferromagnetic long-range order along the two lattice axes with the strongest interactions. Spins along the third axis with the weakest interaction are ordered ferromagnetically. Between TN and a uniquely located temperature TD, there is antiferromagnetic short-range order along the two axes with the strongest interactions, and ferromagnetic short-range order along the other axis. TD is named the disorder temperature because it divides the short-range-order region TN < T < TD from the region TD < T < ∞, in which the axial pair correlations have exponential decay with temperature-dependent oscillatory envelope. There is no singularity in the partition function at TD, so there are only two thermodynamic phases: ordered below the Néel point, and disordered above. Correlations at TD decay exponentially. Finally, special consideration is given to the anisotropic antiferromagnetic lattice when the two weakest interactions are equal, and TN = TD = 0. The single disordered phase exhibits exponential correlation decay with oscillatory envelope for T > 0. The exact values of the axial pair correlations at T = 0 are calculated. For large spin separation r along the strong interaction axis, ω2 = (−1)r, and along the weak (equal) interaction axes ω2∼212E2·r−12cos (12πr)[1−(8r2)−1+…],where 212E2=0.588352663…,and E is a decay constant relating to pair correlations at the Curie point of a square lattice.