Ising-Model Spin Correlations on the Triangular Lattice. III. Isotropic Antiferromagnetic Lattice

TLDR

The pair correlation ω2(r)=〈σ0σr〉 on an isotropic antiferromagnetic triangular lattice has been studied using Toeplitz determinant theory. The study investigates its asymptotic behavior at large spin separation and fixed nonzero temperature. The authors reconsider a special class of fourth‑order correlations ω4(r)=〈σ0σδσrσr+δ〉−〈σ0σδ〉〈σrσr+δ〉 between four spins on the lattice axis to analyze the system. They obtain the leading asymptotic terms for ω2(r) at large separation and fixed temperature, show that ω2(r) behaves as ε0 r⁻¹ᐟ² cos(2π⁄3 r) with ε0≈0.6322, and determine the asymptotic form of the fourth‑order correlation ω4(r) for all fixed temperatures.

Abstract

The asymptotic behavior of the pair correlation ω2(r) = 〈σ0σr〉 between two spins at sites 0 and r on an axis of an isotropic antiferromagnetic triangular lattice is investigated with the aid of the theory of Toeplitz determinants as developed by Wu. The leading terms in the asymptotic expansion are obtained for large spin separation at fixed nonzero temperature. Evidence is presented that the zero-point behavior of the correlation is of the form ω2(r) ∼ ε0r−½ cos ⅔πr, where r = |r| is the spin separation and ε0=212(E0T)2=0.632226080…,E0T being the decay amplitude of the pair correlation at the Curie point (critical point) of an isotropic ferromagnetic triangular lattice. A special class of fourth-order correlations ω4(r) = 〈σ0σδσr σr+δ〉 − 〈σ0σδ〉 〈σrσr+δ〉 between the four spins at sites 0, δ, r, and r + δ on the same lattice axis, where δ is a lattice vector, is reconsidered. The asymptotic form of the correlation for large separation of pairs of spins r = |r| is obtained for all fixed temperatures.