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Australian Journal of Physics

Ising Model on a Triangular Lattice with Three-spin Interactions. I. The Eigenvalue Equation

R. J. Baxter, Fa Yueh Wu

Published Jun 1, 1974• 129 citations
Publisher

Abstract

It is shown that the Ising model with three-spin interactions on a triangular lattice is equivalent to a site-colouring problem on a hexagonal lattice. The transfer matrix method is then used to solve the colouring problem. The colouring of two neighbouri
DOI
10.1071/ph740357

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The use of a transfer matrix method to solve the 3D Ising model is straightforwardly generalized from the 2D case. We follow B.Kaufman's approach. No approximation is made, however the largest eigenvalue cannot be identified. This problem comes from the fact that we follow the choice of directions of 2-dimensional rotations in the direct product space of the 2D Ising model such that all eigenvalue equations reduce miraculously to only one equation. Other choices of directions of 2-dimensional rotations for finding the largest eigenvalue may lose this fascinating feature. Comparing the series expansion of internal energy per site at the high temperature limit with the series obtained from the computer graphic method, we find these two series have very similar structures. A possible correct via a factor Φ(x) is suggested to fit the result of the graphic method.
The authors study a two-dimensional Ising model on a triangular lattice which is divided into two sublattices. One consists of a honeycomb of sites and the other of the remaining interstitial sites. The interaction energies within the honeycomb and between the honeycomb and the interstitial sites differ from each other. The method used consists of a combination of transfer matrix techniques and the phenomenological renormalisation procedure. Results are presented for the critical temperature and exponents in zero field and the phase diagram in the temperature-field plane is obtained.
We study the S=1 Ising model, equivalent to the three-state lattice-gas model, with nearest-neighbor, pairwise interactions on a two-dimensional, triangular lattice. We pay particular attention to the antiferromagnetic phase diagrams. We show its relation to other well-studied models (S=(1/2 Ising, Blume-Capel, Blume-Emery-Griffiths), classify the ground states, and calculate finite-temperature phase diagrams using transfer matrices and finite-size scaling for infinite strips of three and six sites width. The phase diagrams are quite complicated, with surfaces of first- and second-order transitions that intersect along lines of multicritical points of various kinds, providing a rich laboratory for studying a number of first-order phase transitions, critical and multicritical phenomena within the framework of one single model.