Ising-Model Spin Correlations on the Triangular Lattice

TLDR

A Pfaffian representation of the triangular lattice partition function is used to derive expressions for two, four, and six spin correlations in terms of Pfaffians. The study finds that diagonal pair correlations form a Toeplitz determinant yielding spontaneous magnetization, decay as 1/r^12 at the ferromagnetic critical point with radial symmetry, and at the antiferromagnetic zero point exhibit a degenerate ground state with finite entropy and ε/r^12 decay on sublattices, while the perpendicular susceptibility behaves like other lattices ferromagnetically but diverges as 1/T at low temperatures in the antiferromagnetic case.

Abstract

A Pfaffian representation, of the partition function of the triangular lattice is used to derive expressions for various two, four, and six spin correlations in terms of Pfaffians. The pair correlations along a diagonal are expressed as a Toeplitz determinant whose limiting form yields the spontaneous magnetization. At the ferromagnetic critical point the correlations decay as 1/r12 with approximately radial symmetry. At the antiferromagnetic zero point the ground state is highly degenerate—it has finite entropy—and on a given sublattice the pair correlations along a row decay as ε/r12, where ε=+ε0 on the sublattice containing the origin spin and ε≃−12ε0 on the other two sublattices. Finally, the perpendicular susceptibility, χ⊥, which depends on a finite number of correlations, is calculated; its ferromagnetic behavior is similar to that of the perpendicular susceptibilities of the quadratic and honeycomb lattices, but for an antiferromagnet χ⊥ diverges as 1/T at low temperatures.