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Equilibrium properties of a spin-1 Ising system with bilinear, biquadratic and odd interactions

KESKIN, M, Ali Kökçe, Cabir Temırcı

Published Oct 1, 1996• 23 citations
Publisher

Abstract

The equilibrium properties of the spin-1 Ising system Hamiltonian with arbitrary bilinear (J), biquadratic (K) and odd (L), which is also called dipolar-quadrupolar, interactions is studied for zero magnetic field in the lowest approximation of the cluster variation method. The odd interaction is combined with the bilinear (dipolar) and biquadratic (quadrupolar) exchange interactions by the geometric mean. In this system, phase transitions depend on the ratio of the coupling parameters, alpha = J/K; therefore, the dependence of the nature of the phase transition on alpha is investigated extensively and it is found that for alpha less than or equal to 1 and alpha greater than or equal to 2000 a second-order phase transition occurs, and for 1 < alpha < 2000 a first-order phase transition occurs. The critical temperatures in the case of a second-order phase transition and the upper and lower limits of stability temperature in the case of a first-order phase transition are obtained for different values of alpha calculated using the Hessian determinant. The first-order phase transition temperatures are found by using the free energy values while increasing and decreasing the temperature. Besides the stable branches of the order parameters, we establish also the metastable and unstable parts of these curves and the thermal variations of these solutions as a function of the reduced temperature are investigated. The unstable solutions for the first-order phase transitions are obtained by displaying the free energy surfaces in the form of a contour map. Results are compared with the spin-1 Ising system Hamiltonian with the bilinear and biquadratic interactions and it is found that the odd interaction greatly influences the phase transitions.

DOI
10.1016/0378-4371(96)00094-5

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The equilibrium properties of the spin-1 Ising system Hamiltonian with arbitrary bilinear (J), biquadratic (K) and odd (L), which is also called dipolar-quadrupolar, interactions is studied for zero magnetic field in the lowest approximation of the cluster variation method. The odd interaction is combined with the bilinear (dipolar) and biquadratic (quadrupolar) exchange interactions by the geometric mean. In this system, phase transitions depend on the ratio of the coupling parameters, alpha = J/K; therefore, the dependence of the nature of the phase transition on alpha is investigated extensively and it is found that for alpha less than or equal to 1 and alpha greater than or equal to 2000 a second-order phase transition occurs, and for 1 < alpha < 2000 a first-order phase transition occurs. The critical temperatures in the case of a second-order phase transition and the upper and lower limits of stability temperature in the case of a first-order phase transition are obtained for different values of alpha calculated using the Hessian determinant. The first-order phase transition temperatures are found by using the free energy values while increasing and decreasing the temperature. Besides the stable branches of the order parameters, we establish also the metastable and unstable parts of these curves and the thermal variations of these solutions as a function of the reduced temperature are investigated. The unstable solutions for the first-order phase transitions are obtained by displaying the free energy surfaces in the form of a contour map. Results are compared with the spin-1 Ising system Hamiltonian with the bilinear and biquadratic interactions and it is found that the odd interaction greatly influences the phase transitions.
The spin-1 Ising model with bilinear and biquadratic interactions is investigated using a generalized constant-coupling approximation with two parameters. Equations to determine these parameters are derived by minimizing the variational free energy. The averages of the dipolar and quadrupolar moments expressed in terms of these parameters are first ascertained to give exact results for a linear chain, then investigated in detail for the quadrupolar-coupled system in comparison with the molecular-field results. The pair-correlation and the heat capacity of this system are also investigated in connection with the phase transition. For the system with the bilinear and biquadratic interactions the effect of the anisotropy constant to the phase transition is discussed.
A statistical-mechanical study of a solvable spin-one system with bilinear and biquadratic interactions is presented for various values of α, the ratio of the bilinear to biquadratic interaction. First, equations to determine the order-parameters and the free energy in the equilibrium are derived from the partition function to investigate the phase transitions occurring the model. The magnetic susceptibility is then calculated, and an argument about the critical exponents at the second-order phase transition is given for the case of the critical ratio α=2/3, below which the first order phase transition takes place.
Analytical expressions of the order parameters near the transition temperatures in the spin-3/2 Ising system with bilinear (J) and biquadratic (K) interactions are presented for various values of J/K. First, we obtain the free energy expression and the equations to determine order parameters by using the mean-field approximation. Then, the order parameters are expressed in the vicinity of the transition temperatures in which these expressions are very important to study the dynamics of the system by means of Onsager's theory of irreversible thermodynamics. Hence, we investigate the phase transitions occurring in the system and also obtain two tricritical points analytically. Finally, the specific heat and magnetic susceptibility are calculated and an argument about the critical exponents at the second-order phase transitions and tricritical points is given.
In this work, the critical properties of the classical spin-1 Ising Hamiltonian applied to magnetic systems characterized by the first-neighbors biquadratic exchange, the anisotropy and the external magnetic field contributions are theoretically investigated. The first-neighbors bilinear exchange interaction is set equal to zero. For magnetic systems the bicubic exchange interaction must be set equal to zero as it would break the time-reversal invariance of the exchange Hamiltonian. To determine the critical behavior, the spin-1 Ising Hamiltonian is mapped onto the spin-1/2 Ising Hamiltonian by using the Griffith’s variable transformation. The critical surface of a 2D square magnetic lattice is determined in the parameter space as a function of the magnetic parameters and the phase transition occurring across it is quantitatively discussed by calculating, for each spin, the free energy and the magnetization. The free energy of the 2D square magnetic lattice, described via the three-state spin-1 Ising model, is obtained from an empirical expression of the partition function recently proposed for a spin-1/2 Ising model in an external magnetic field and applied to a 2D magnetic lattice. These results could pave the way to numerical simulations and to measurements able to confirm the analytical predictions.
The two-state Ising model with nearest-neighbour bilinear exchange constant J (2) , three-spin coupling constant J (3) and four-spin coupling constant J (4) is studied using the effective-field approximation. The temperature variation in spontaneous magnetization is studied for different coordination numbers z and for various values of the parameters alpha =J (3) /J (2) and alpha '=J (4) /J (2) . It is seen that, in the absence of three-spin coupling, there exists a critical value of alpha ' (say, alpha ' c ) above which the magnetization becomes double valued at and beyond the Curie temperature T c , forming a bulge near T c which indicates the occurrence of a first-order transition. The expressions for alpha ' c for z=3, 4, 6 are derived and it is found that alpha ' c increases as z increases. In the absence of four-spin coupling it is found that first-order transitions appear for alpha >or=0. The magnetization curves are also studied for the case when all the interactions are present. The results are discussed with reference to those obtained from the molecular-field approximation.
The spin-1 quantum Ising systems with three-spin interactions on two-dimensional triangular lattices are studied by mean-field method. The thermal variations of order parameters and phase diagrams are investigated in detail. The stable, metastable and unstable branches of the order parameters are obtained. According to the stable conditions at critical point, we find that the systems exhibit tricritical points. With crystal field and biquadratic interactions, the system has rich phase diagrams with single reentrant or double reentrant phase transitions for appropriate ranges of the both parameters.