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Journal of Mathematical Physics

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

John Stephenson

Published Feb 1, 1970• 166 citations
Publisher

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.

DOI
10.1063/1.1665154

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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.
A class of fourth-order spin correlations whose sum is closely related to the specific heat is calculated exactly by Pfaffian perturbation theory. In the particular case when the four spins lie on a lattice axis, it is shown that the reduced fourth-order correlation function has the asymptotic form 〈σ0, 0σ1, 1σk, kσk+1, k+1〉−〈σ0, 0σ1, 1〉 〈σk, kσk+1, k+1〉≏A′(e−2βk/k2),where β−1 is the (positive) mean range of order. The decay of correlations with spin separation k is symmetric about the critical point β = 0. The amplitude A′ is 1/π2 at the critical point and is 1/(2π) at all other temperatures. Correlations on ferro- and antiferromagnetic triangular and quadratic lattices are discussed in some detail.
We study in detail the asymptotic behavior, for large separations, of the correlation between two spins in the case of the two-dimensional Ising model without magnetic field. In the limit of infinite separation, this correlation is equal to the square of the spontaneous magnetization per spin. This paper is devoted to answering the question how, for fixed temperature, the correlation approaches this limiting value. The investigation is restricted to the situation where the two spins under consideration lie on the same row of the two-dimensional lattice. There are three distinct cases where the fixed temperature is above, below, or equal to the critical temperature or the Curie temperature. In the first two cases, the limiting value is approached exponentially as a function of the separation, while at the critical temperature the correlation behaves asymptotically as the inverse fourth root of the separation. In this paper, we evaluate exactly the first four terms of the asymptotic expansion in the first case, the first three terms in the second case, and the coefficients of the terms proportional to the -\textonequarter{} and the $\ensuremath{-}\frac{9}{4}$ powers of the separation in the third case. All these results are obtained by first expressing the correlation as a Toeplitz determinant, and the method used is capable of giving in principle the entire asymptotic expansion for large classes of such determinants, when the size of the determinant approaches infinity. The explicit results for the two-dimensional Ising model also serve as an example where the prescription of summing the leading terms, or the most divergent terms, fails.
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.
We study the asymptotic behavior, for large separations, of the spin-spin correlation function $〈{\ensuremath{\sigma}}_{0,0}{\ensuremath{\sigma}}_{M,N}〉$ in the two-dimensional Ising model, where the two spins are not necessarily on the same row. Besides the limiting value for infinite separation, which is the square of the spontaneous magnetization, we evaluate the two leading terms in the asymptotic expression in each of the two cases T<TcT<{T}_{c}T<Tc​ and T>TcT>{T}_{c}T>Tc​. It is found that the nearest singularity of the generating function for the correlation is quite simple in the case T>TcT>{T}_{c}T>Tc​, but much more complicated for T<TcT<{T}_{c}T<Tc​. In an Appendix, we also give exactly in a very simple form the correlation $〈{\ensuremath{\sigma}}_{0,0}{\ensuremath{\sigma}}_{N,N}〉$ for symmetrical Ising lattice at the critical temperature Tc{T}_{c}Tc​.