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Journal of Physics C Solid State Physics

Three-spin-state generalized Ising model

S Alexander, G. Yuval

Published May 7, 1974• 12 citations
Publisher

Abstract

Second-rank tensor order parameters appear for liquid crystals and crystallographic transitions. The simplest model with a disordered high-temperature state has interactions invariant under cubic spin transformations. It is shown that for spin 1 this leads to the 3-state Potts model. It is suggested that this model is therefore relevant to the understanding of phase transitions with such an order parameter. A formalism for series expansions and a new proof of the Kramers-Wannier symmetry on the square lattice are presented. Pade approximants for the series expansions including a procedure explicitly utilizing the symmetry are discussed. The negative interaction 'orthogonal' model analogous to the Ising antiferromagnet is also discussed and shown to have a highly degenerate ground state. On the square lattice this is the degeneracy of Lieb's ice. Relationships to ferroelectric models and the three-colour problem are also discussed.

DOI
10.1088/0022-3719/7/9/004

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Critical Behaviour in Many Body Systems.

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This work has been concentrated on a detailed study of low temperature (and high field) expansions for the simple Ising model. Methods were considered for their derivation and derived long series; these have been analyzed to provide evidence on critical indices. Also studied were the derivation of series expansions for the Ising model with spin S > 1/2 and the corresponding low temperature behavior. Also reported is a study of the configurational problem presented by the Potts model and the planar Potts model. The former model in particular has attracted particular attention recently in relation to the properties of liquid crystals. There is evidence that its behavior differs significantly from that of the Ising model and may in certain circumstances lead to a first order transition. Work has continued with a general program of tabulation of lattice constants for all standard two and three dimensional lattices. (Modified author abstract)

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Journal of the Physical Society of Japan•Apr 15, 1983•2 citations
Second-rank tensor order parameters appear for liquid crystals and crystallographic transitions. The simplest model with a disordered high-temperature state has interactions invariant under cubic spin transformations. It is shown that for spin 1 this leads to the 3-state Potts model. It is suggested that this model is therefore relevant to the understanding of phase transitions with such an order parameter. A formalism for series expansions and a new proof of the Kramers-Wannier symmetry on the square lattice are presented. Pade approximants for the series expansions including a procedure explicitly utilizing the symmetry are discussed. The negative interaction 'orthogonal' model analogous to the Ising antiferromagnet is also discussed and shown to have a highly degenerate ground state. On the square lattice this is the degeneracy of Lieb's ice. Relationships to ferroelectric models and the three-colour problem are also discussed.
A high-temperature series expansion in the cumulant form of Brout (1959, 1960) is applied to mixed-spin Ising models on simple quadratic, simple cubic and body-centred cubic lattices. Motivation is provided primarily by an investigation of critical phenomena in Ising models with less than the usual translational symmetry and of relevance to the study of ferrimagnetism. Secondary motivation, appropriate to the simple quadratic lattice, is derived from a desire to obtain results that can be compared with those of renormalisation-group calculations performed elsewhere by the authors. Series expansions of the susceptibility and specific heat are obtained to seventh and tenth order, respectively. The derived expansion coefficients are analysed through both ratio and Pade approximant methods and exponent values are found to be in good agreement with those suggested by the universality hypothesis.
We investigate the Ising model with nearest-neighbor ferromagnetic and next-nearest-neighbor antiferromagnetic interactions on the square and body-centered-cubic lattices. For 0<J1J2<320<\frac{{J}_{1}}{{J}_{2}}<\frac{3}{2}0<J2​J1​​<23​ on the bcc lattice the model has two nontrivially degenerate ground states and has been conjectured to have XY\mathrm{XY}XY (n=2n=2n=2) critical exponents. For J1J2=32\frac{{J}_{1}}{{J}_{2}}=\frac{3}{2}J2​J1​​=23​ there are three degenerate ground states and Heisenberg critical exponents (n=3n=3n=3) are expected. We have obtained nine terms in the high-temperature expansion of the staggered susceptibility. For 0<J1J2< 10<\frac{{J}_{1}}{{J}_{2}}<~10<J2​J1​​< 1 we find $\ensuremath{\gamma}=1.25$ (n=1n=1n=1). At J1J2=32\frac{{J}_{1}}{{J}_{2}}=\frac{3}{2}J2​J1​​=23​ the results are not conclusive, but $\ensuremath{\gamma}=1.33$ (n=2n=2n=2) is not ruled out by the series analysis whereas $\ensuremath{\gamma}=1.4$ (n=3n=3n=3) is not probable. Results for the square lattice are inconclusive.
The three-state antiferromagnetic Potts model on the simple cubic lattice is studied by the Monte Carlo method using a new order parameter and employing the histogram technique. It is shown that apart from the low-temperature broken-sublattice-symmetry phase with nonequivalent sublattices, another type of ordering with equivalent sublattices does exist in the model between the broken-sublattice-symmetry and disordered phases. The temperature of the transition between the ordered phases is estimated from the histogram of the order parameter. The value obtained is in good agreement with that obtained from cluster variation circulations.
This work has been concentrated on a detailed study of low temperature (and high field) expansions for the simple Ising model. Methods were considered for their derivation and derived long series; these have been analyzed to provide evidence on critical indices. Also studied were the derivation of series expansions for the Ising model with spin S > 1/2 and the corresponding low temperature behavior. Also reported is a study of the configurational problem presented by the Potts model and the planar Potts model. The former model in particular has attracted particular attention recently in relation to the properties of liquid crystals. There is evidence that its behavior differs significantly from that of the Ising model and may in certain circumstances lead to a first order transition. Work has continued with a general program of tabulation of lattice constants for all standard two and three dimensional lattices. (Modified author abstract)
The three-state antiferromagnetic Potts model on d-dimensional hypercubic lattices is studied, where d is 2, 3, 4, 5 and 6. The Hamiltonian is expressed by the spin Hamiltonian with spin 1. The staggered susceptibility is calculated by the high temperature series expansion. By use of the ratio method and the Padé approximation, the critical temperature and the critical exponent γ are obtained. It is concluded that the three-state antiferromagnetic Potts model exhibits the second-order transition when d ≥3.