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Canadian Journal of Physics

Some exact results for the anisotropic triangular lattice

R. W. Gibberd

Published Nov 15, 1969• 4 citations
Publisher

Abstract

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.

DOI
10.1139/p69-302

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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 problem of N interacting spins on a lattice is equivalent to one of N clusters linked in a specific manner. The energy of any configuration of spins can be expressed in terms of the energy levels of this cluster. A new expression is obtained for the probability of occurrence of any configuration. A closed form expression is obtained for the partition function per spin in terms of the energy levels of this cluster with the degeneracies being a function of temperature. On physical grounds it is suggested to be the form of the exact partition function per spin. The partition functions of Ising-like systems all have a common form. For the 3D Ising model seven functions need to be determined to describe the partition function completely. The key to understanding second order phase transitions and critical phenomena lies in the temperature dependence of various degeneracies. It is necessary to develop new techniques to determine the partition function that account for this temperature dependence, as it would represent the underlying physics correctly.