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Chinese Journal of Physics

Transformation Properties of a Spin-1 System in Statistical Physics

Fa Yueh Wu

Published Apr 1, 1992• 6 citations
OpenAlex

Abstract

In this paper we discuss transformation properties of the spin-1 Blume-Emery-Griffiths (BEG) model arising in statistical physics. It is shown that the BEG model can always be transformed into either a spin-(1/2) Ising model or a 3-state Potts model. It is also shown that there exist other transformations relating the BEG model with spin-(1/2) king models, valid in special subspaces of the parameter space. Physical properties of the BEG model that can be drawn from the equivalence are discussed. We also present a formulation of the BEG model relating it to a 3-state vertex model.

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In this paper we discuss transformation properties of the spin-1 Blume-Emery-Griffiths (BEG) model arising in statistical physics. It is shown that the BEG model can always be transformed into either a spin-(1/2) Ising model or a 3-state Potts model. It is also shown that there exist other transformations relating the BEG model with spin-(1/2) king models, valid in special subspaces of the parameter space. Physical properties of the BEG model that can be drawn from the equivalence are discussed. We also present a formulation of the BEG model relating it to a 3-state vertex model.
We study the phase diagram of the 3-matrix model. This model corresponds to the Blume-Emery-Griffiths (BEG) model on a random surface. The BEG model has parameters J and K which are the spin and the lattice-gas coupling respectively. If K = 0, it is the usual tri-critical Ising model. If K/J = 3, this model becomes the 3-state Potts model. These models are solved in the context of the Coulomb gas representation. The intermediate region K/J = 1 is equivalent to the 3-matrix chain model and we can solve it exactly using the orthogonal polynomial method. We find the 3rd order critical line on which the spin ordering phase transition occurs. This line ends at the 4th order critical point which corresponds to the tri-critical point of the BEG model. The phase diagram is quite similar to that of the BEG model on a regular lattice.
Introduction Among pseudospin systems studied theoretically, a great attention is paid to Ising-type models, that is, to systems with the Hamiltonian containing only z-component of pseudospin. That is so due to relative simplicity with which approximate calculations for these models can be carried out and tested and possibility of their application to a wide class of real objects. Thus, the Blume-Emery-Griffiths model (BEG) corresponding to the Isingtype model with j = 2 (hereafter S z = \\Gammaj; \\Gammaj +2; : : : ; j \\Gamma 2; j) was proposed for the investigation of phase transitions (PT) in He 3 \\Gamma He 4 mixture [1]. A general form of the Ising-type model with j = 2 contains extra terms, as compared to the BEG model, like S z i (S z j<
The spin-1 Ising model with bilinear (J) and biquadratic (K) nearest-neighbor exchange interactions and a single-ion potential (\ensuremath{\Delta}), a model introduced by Blume, Emery, and Griffiths (BEG), is studied on a honeycomb lattice. The exact solution of the BEG model is obtained for a certain relation K=K(J). Also an approximate treatment of the BEG model outside the surface, in the space of coupling constants, given by the relation above is discussed. Finally a digression is made on the critical temperature of the spin-1 Ising model.
The mixed spin-2 and spin-3/2 Blume-Emery-Griffiths (BEG) Ising ferrimagnetic system is studied by the Bethe lattice approach. The ground-state phase diagram is constructed. The influence of the crystal-field and the biquadratic interactions among neighboring spins on the thermal behaviors of the system is singled out. The system displays very rich critical behaviors with the existence of tricritical points. Compensation points where the global magnetization of the system vanishes have been detected for appropriate values of the system parameters.
We consider the Blume-Emergy-Griffiths (BEG) model on the honeycomb lattice and obtain a closed-form expression for the critical surface of second-order transitions. The BEG model is first formulated as a three-state vertex model. Using the fact that the BEG critical surface coincides with that of a general three-state vertex model, we construct critical surfaces by forming polynomial combinations of vertex weights that are invariant under an O(3) gauge transformation. We then carry out a finite-size analysis of the BEG model, and use data so obtained to determine coefficients appearing in the polynomial combination. This procedure leads to a closed-form expression of the critical surface which reproduces all numerical data accurately.