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

A detailed discussion of pair correlations ω₂(r) = 〈σ₀σ_r〉 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 T_C 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 T_N (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 T_N and a uniquely located temperature T_D, there is antiferromagnetic short‐range order along the two axes with the strongest interactions, and ferromagnetic short‐range order along the other axis. T_D is named the disorder temperature because it divides the short‐range‐order region T_N < T < T_D from the region T_D < T < ∞, in which the axial pair correlations have exponential decay with temperature‐dependent oscillatory envelope. There is no singularity in the partition function at T_D, so there are only two thermodynamic phases: ordered below the Néel point, and disordered above. Correlations at T_D decay exponentially. Finally, special consideration is given to the anisotropic antiferromagnetic lattice when the two weakest interactions are equal, and T_N = T_D = 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

$${\omega }_2{\text{∼2}}^{\frac{1}{2}}{E}^2{\mathrm{·r}}^{-\frac{1}{2}}\cos (\frac{1}{2}{\text{πr)[1−(8r}}^2{)}^{\text{−1}}\mathrm{+\dots ]}$$

, where

$$2^{\frac{1}{2}}{E}^2\text{=0.588352663…}$$

, and E is a decay constant relating to pair correlations at the Curie point of a square lattice.