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Physical Review E

Two-point correlation function for the triangular Ising antiferromagnet

David H. Wojtas, Rick P. Millane

Published Apr 15, 2009• 7 citations
Publisher

Abstract

A number of aspects of the two-point correlation behavior of the nearest-neighbor, triangular Ising antiferromagnet are studied using a combination of numerical evaluation of exact expressions and Monte Carlo simulation. Existing asymptotic results for on-axis correlations at finite temperatures are evaluated and shown to be of limited accuracy. The sublattice structure of the off-axis correlation function is clarified, and rotational invariance is studied as a function of temperature. Separations and temperatures for which the correlation function is significant are identified, and a simple functional expression is developed that allows accurate calculation of the correlation function in this region.

DOI
10.1103/physreve.79.041123

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•Apr 6, 1994•29 citations
A number of aspects of the two-point correlation behavior of the nearest-neighbor, triangular Ising antiferromagnet are studied using a combination of numerical evaluation of exact expressions and Monte Carlo simulation. Existing asymptotic results for on-axis correlations at finite temperatures are evaluated and shown to be of limited accuracy. The sublattice structure of the off-axis correlation function is clarified, and rotational invariance is studied as a function of temperature. Separations and temperatures for which the correlation function is significant are identified, and a simple functional expression is developed that allows accurate calculation of the correlation function in this region.
Monte Carlo simulations of the Ising model on the triangular lattice with the antiferromagnetic nn interaction J and the ferromagnetic nnn interaction J ' are carried out. Several physical quantities (magnetizations, energies, susceptibilities, specific heats of sublattices and of the total system) are calculated. A phase which resembles the partially disordered phase proposed by Mekata is observed in the finite temperature region. But it is shown in this region that the sublattice magnetizations vanish at the thermodynamic limit and the Kosterlitz-Thouless phase transition is predicted by the finite size analysis.
An antiferromagnetic triangular Ising model with ferromagnetic next nearest neighbor interactions is studied by means of a transfer matrix method and a finite size scaling. Using the Roomany-Wyld approximation, it cannot be concluded definitely whether the system has only one critical point or not. By studying decays of the correlation functions of each sublattice, it is found that there is a finite temperature region where the correlation functions of each sublattice show a power law decay, namely, in this region every temperature is a critical temperature. The results by means of the present transfer matrix method are in good agreement with that by Monte Carlo methods, namely, the discrepancy between both results, reported recently, can be got rid of.
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 two-dimensional triangular Ising lattice with the nearest-neighbor antiferromagnetic interaction J 1 and the next-nearest-neighbor ferromagnetic interaction - J 2 is studied by the Monte Carlo simulation. It is shown that the system with a finite J 2 undergoes the successive phase transition, paramagnetic→Kosterlitz-Thouless (KT)-like→ferrimagnetic phases, as temperature decreases. The sub-lattice switching phenomenon observed in our previous work is shown to be an intrinsic property inherent to the present system which has an XY character. The marginal exponent of the order-parameter correlation function at the KT-like transition point is seen to increase from 1/4 as the ratio J 2 / J 1 decreases, namely, it reveals a deviation from the universal behavior of a genuine XY model.
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.