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Journal of Physics A Mathematical and General

Triplet order parameter of the triangular Ising model

R. J. Baxter

Published Nov 1, 1975• 18 citations
Publisher

Abstract

Considers the usual Ising model with pair interactions between nearest-neighbour spins sigma on the triangular lattice. The author evaluates the correlation M 3 =( sigma 1 sigma 2 sigma 3 ), where sigma 1 , sigma 2 , sigma 3 are the three spins at the vertices of a triangular face. This can be thought of as an order parameter associated with a three-spin 'field'. It vanishes at and above T c , and has a critical exponent of 1 / 8 .

DOI
10.1088/0305-4470/8/11/015

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Journal of the Physical Society of Japan•May 15, 1991•25 citations
Considers the usual Ising model with pair interactions between nearest-neighbour spins sigma on the triangular lattice. The author evaluates the correlation M 3 =( sigma 1 sigma 2 sigma 3 ), where sigma 1 , sigma 2 , sigma 3 are the three spins at the vertices of a triangular face. This can be thought of as an order parameter associated with a three-spin 'field'. It vanishes at and above T c , and has a critical exponent of 1 / 8 .
Reports studies of the critical behaviour of the Ising model with first and second neighbour interactions J 1 , J 2 on the triangular lattice. High-temperature series have been used to determine the locations of the ferromagnetic critical line and the critical line in the region (J 1 <0, J 2 >0). On the latter line the ordering susceptibility appears to diverge algebraically with the exponent gamma approximately 2.4. The existence of a sinusoidal phase and an associated Lifshitz point are conjectured.
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.
Positions of critical temperatures of antiferromagnetic Ising model with ferromagnetic next nearest neighbour interactions on the triangular lattice are investigated by making use of their properties of the Kosterlitz-Thouless type of phase transition. In particular, finite-size scaling method and Monte Carlo Renormalization Group methods (MCRG) with flow diagrams of the anomalous dimension η show the fixed line structure where the correlation length diverges. Another MCRG with flow diagram of the stiffness constant also shows a critical property clearly. The temperature dependence of the long range order is also discussed in order to locate the lower critical point.