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Journal of Applied Physics

The spin-one Ising model on a honeycomb lattice

J.W. Tucker

Published Apr 15, 1991• 26 citations
Publisher

Abstract

A finite cluster effective-field theory in which the single-site kinematic relations are treated exactly is applied to a study of the spin-one Ising ferromagnet on a honeycomb lattice. The Ising model studied contains both biquadratic exchange, J′ and single-ion anisotropy D in addition to the usual bilinear exchange interactions J. The second-order phase boundary is studied for a range of interaction strengths and the results compared to those of other methods. The dependence of the results on cluster size is investigated. The magnetization predicted by the four-site cluster theory for a selection of single-ion anisotropy strengths is also compared with exact results that have recently been deduced in the subspace J′/J=−[ln cosh(βJ)]/(βJ). In extending the theory to account for site dilution, a form of configurational averaging is adopted that is superior to that of earlier authors. The variation with site dilution of the tricritical point in the T-D plane is reported.

DOI
10.1063/1.348793

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A finite cluster effective-field theory in which the single-site kinematic relations are treated exactly is applied to a study of the spin-one Ising ferromagnet on a honeycomb lattice. The Ising model studied contains both biquadratic exchange, J′ and single-ion anisotropy D in addition to the usual bilinear exchange interactions J. The second-order phase boundary is studied for a range of interaction strengths and the results compared to those of other methods. The dependence of the results on cluster size is investigated. The magnetization predicted by the four-site cluster theory for a selection of single-ion anisotropy strengths is also compared with exact results that have recently been deduced in the subspace J′/J=−[ln cosh(βJ)]/(βJ). In extending the theory to account for site dilution, a form of configurational averaging is adopted that is superior to that of earlier authors. The variation with site dilution of the tricritical point in the T-D plane is reported.
Site diluted spin-1/2 Ising and spin-1 Blume Capel (BC) models in the presence of transverse field interactions are examined by introducing an effective-field approximation that takes into account the multi-site correlations in the cluster of a considered lattice with an improved configurational averaging technique. The critical concentration below which the transition temperature reduces to zero is determined for both models, and the estimated values are compared with those obtained by the other methods in the literature. It is found that diluting the lattice sites by non magnetic atoms may cause some drastic changes on some of the characteristic features of the model. Particular attention has been paid on the global phase diagrams of a spin-1 BC model, and it has also been shown that the conditions for the occurrence of a second order reentrance in the system is rather complicated, since the existence or extinction of reentrance is rather sensitive to the competing effects between D/J, Ω/J and c.
A cluster mean-field method is introduced and the applications to the Ising and Heisenberg models are demonstrated. We divide the lattice sites into clusters whose size and shape are selected so that the equivalence of all sites in a cluster is preserved. Since the strength of interactions of a cluster with its surrounding clusters is strongly dependent on the spin configuration of the central cluster itself, we include this contribution in the effective fields acting on the spins. The effects of ``correlations'' between clusters can be taken into account beyond the standard mean-field level and as a result our cluster-based method gives qualitatively (and even quantitatively) correct results for the both Ising and Heisenberg models. Especially, for the Ising model on the honeycomb and square lattices, the calculated results of the critical temperature are very close (overestimated by only less than 5%) to the exact values.
In the Weiss molecular-field approximation, the S=1S=1S=1 Ising-model ferromagnet possessing additional single-ion-type uniaxial anisotropy is known to exhibit (depending upon the ratio of anisotropy and interaction constants) both first-order (discontinuous) and second-order (continuous) magnetic phase transitions which join at a tricritical point. We further investigate the above model system in the spin-pair approximation using the cluster-variation method. Within this improved approximation, results are found for the temperature and anisotropy dependences of the thermal-average quantities 〈Si〉〈{S}_{i}〉〈Si​〉, 〈Si2〉〈S_{i}^{}{}_{}{}^{2}〉〈Si​​2〉, $〈{S}_{i}{S}_{{i}^{\ensuremath{'}}}〉$, $〈{S}_{i}S_{{i}^{\ensuremath{'}}}^{}{}_{}{}^{2}〉$, $〈S_{i}^{}{}_{}{}^{2}S_{{i}^{\ensuremath{'}}}^{}{}_{}{}^{2}〉$ (iii, ${i}^{\ensuremath{'}}$ are nearest-neighbor lattice sites), a better estimate for the location of the tricritical point is achieved, and the specific heat and initial paramagnetic susceptibility are calculated. To illustrate these results, the simple cubic lattice is used as an example of application.
In the light of the effective-field theory, the present paper attempts to discuss the investigation done into the phase diagrams of a random nearest-neighbour interaction spin-1 Ising model with a single-ion anisotropy in an applied transverse field on the honeycomb lattice. The interactions are assumed to be independent random variables with distribution P(J ij )=p delta (J ij -J 1 )+(1-p) delta (J ij -J 2 ), where J 1 >0and mod J 2 /J 1 mod <or=1. We find a number of interesting phenomena, such as two different types of re-entrant phenomenon due to the competition between the random bond and the negative single-ion anisotropy parameter, as well as the usual frustration of J ij . The influence of a transverse field on the behaviours of the tricritical point and re-entrant transition is also discussed in this paper.