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Journal of Mathematical Physics

Triangular Ising model: Complete expressions for even spin correlations in an exact evaluation of P(h,T)

T. C. Choy

Published Dec 1, 1984• 6 citations
Publisher

Abstract

Complete expressions of all even correlations for the triangular Ising model in a recent exact evaluation of P(h,T), the local magnetic field distribution, are obtained from the Pfaffian representation derived by Stephenson. The results for the ferromagnetic J<0 and antiferromagnetic J>0 cases contain a number of interesting features previously not pointed out in the Ising literature. As a bonus, some new inequalities relating odd and even correlations are also obtained.

DOI
10.1063/1.526096

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Complete expressions of all even correlations for the triangular Ising model in a recent exact evaluation of P(h,T), the local magnetic field distribution, are obtained from the Pfaffian representation derived by Stephenson. The results for the ferromagnetic J<0 and antiferromagnetic J>0 cases contain a number of interesting features previously not pointed out in the Ising literature. As a bonus, some new inequalities relating odd and even correlations are also obtained.
The combinatorial approach to the triangular Ising model is shown to be considerably simplified at a given temperature TD. This enables the partition function and spin-correlation functions to be evaluated without the use of pfaffians, thus providing a simple derivation of a few of the results obtained previously by Stephenson. We confirm the results of Stephenson, that at the temperature TD, the correlation functions have the same structure as the correlations of a one-dimensional Ising lattice. The only new result is a general expression for some of the spin–spin correlation functions when the spins do not lie on the axes of the lattice.
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.
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.
The "star-triangle" and "decoration" transformations are generalized so as to apply to arbitrary mechanical systems coupled to the spins of a standard Ising net. This leads to exact solutions for further plane Ising nets and also for lattices in which the spins on alternate sites have a magnitude greater than S=12S=\frac{1}{2}S=21​. A general class of antiferromagnetic Ising models is constructed; exact closed expressions can be derived for all the thermodynamic and magnetic properties of these models in an arbitrary magnetic field.The magnetizations and susceptibilities of Ising nets in which different spins have different magnetic moments are investigated and a valuable relation between the susceptibilities of the honeycomb and triangular lattices is derived. It is shown how correlation functions involving a given spin can be expressed in terms of correlations involving the nearest-neighbor spins instead.