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Modern Physics Letters A

PHASE DIAGRAMS OF 3-MATRIX MODEL

Kenji Fukazawa, Kenji Hamada, Haru-Tada Sato

Published Nov 20, 1990• 8 citations
Publisher

Abstract

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.

DOI
10.1142/s0217732390002808

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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.
Abstract We derive the phase diagram of the one-dimensional three-state Potts model with an additional mean-field interaction in the canonical ensemble. The free energy is obtained by mapping the model onto the spin-1 Blume–Emery–Griffiths model and solving it by using a Hubbard–Stratonovich transformation combined with the transfer matrix method. A complex structure with lines of first-order transitions, two triple points and a critical point appears at finite temperature. The phase diagram is two-dimensional, since there are two adjustable parameters, the nearest-neighbour coupling K and the temperature T . We show that the phase diagram does not present second-order phase transition lines, due to the fact that the order parameter is not a symmetry-breaking one. Quite remarkably, we are able to determine analytically one of the first-order phase transition lines. We also prove that, when the nearest-neighbour coupling K is large and negative, the first-order transition temperature becomes asymptotically independent of K .
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.
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.
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.