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Physical Review Letters

Rigorous solution of a two-dimensional Blume-Emery-Griffiths model

Anders Rosengren, Roland Häggkvist

Published Aug 7, 1989• 29 citations
Publisher

Abstract

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.

DOI
10.1103/physrevlett.63.660

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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.
I study the one-dimensional spin-1 Blume-Emery-Griffiths model with bilinear and biquadratic exchange interactions and single-ion crystal field under an applied magnetic field. This model can be exactly mapped into a tight-binding Hubbard model - extended to include intersite interactions - provided one renormalizes the chemical and the on-site potentials, which become temperature dependent. After this transformation, I provide the exact solution of the Blume-Emery-Griffiths model in one dimension by means of the Green's functions and equations of motion formalism. I investigate the magnetic variations of physical quantities - such as magnetization, quadrupolar moment, susceptibility - for different values of the interaction parameters and of the applied field, focusing on the role played by the biquadratic interaction in the breakdown of the magnetization plateaus.
Abstract We use the Monte Carlo simulation technique to study the critical behavior of a three-state spin model, with bilinear and biquadratic nearest-neighbor interactions, known as the Blume–Emery–Griffiths model (BEG), in a square lattice. In order to characterize this model, we study the phase diagram, in which we identify three different phases: ferromagnetic, paramagnetic and quadrupolar, the later with one sublattice filled with spins and the other with vacancies. We perform our studies by using two algorithms: Metropolis update (MU) and Wang–Landau (WL). The critical scaling behavior of the model is complementary studied by applying results obtained by using both algorithms, while tricritical points and the tricritical scaling behavior is analyzed by means of WL measuring the joint density of states and using the method of field mixing in conjunction with finite-size scaling. Furthermore, motivated by the decoupling between spins observed within the quadrupolar phase, we further generalize the BEG model in order to study the behavior of the system by adding a next nearest neighbour (NNN) interaction between spins. We found that by increasing the strength of the (ferromagnetic) NNN interaction, a new ferromagnetic phase takes over that contains both quadrupolar and ferromagnetic order.
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.
The spin-1 Ising model on the square lattice with nearest-neighbor ferromagnetic exchange interactions [both bilinear (JJJ) and biquadratic (KKK) and crystal-field interaction ($\ensuremath{\Delta}$) is studied via a renormalization-group transformation in position space. The phase diagram in JJJ, KKK, $\ensuremath{\Delta}$ space is found to have one surface of critical phase transitions and two surfaces of first-order phase transitions. These surfaces are variously bounded by an ordinary tricritical line, an isolated critical line, and a line of critical end points. These three lines joint at a special tricritical point corresponding to the transition of the three-state Potts model. The over-all phase diagram is qualitatively similar to that obtained with the mean-field approximation, except in the vicinity of the Potts transition where a four-phase coexistence line in mean-field theory shrinks into a special tricritical point in renormalization-group theory. Symmetry considerations guide the construction of our truncated renormalization-group transformation. The global connectivity and local exponents of the thirteen separate fixed points underlying this quite complicated structure are determined. Local analysis with respect to magnetic field (HHH) and another odd interaction (LLL) is performed. A one-adjusted-parameter version of our transformation yields remarkably quantitative results, predicting the Potts transition temperature, for example, within 0.3% of the exact value.