Narrow Results

The equilibrium properties of the Blume–Emery–Griffiths (BEG) model Hamiltonian with the arbitrary bilinear (J), biquadratic (K) and crystal field interaction (D) are studied using the genetic algorithm technique. Results are compared with lowest approximation of the cluster variation method (CVM), which is identical to the mean field approximation. We found the genetic algorithm to be very efficient for fast search at the average fraction of the spins, especially in the early stages as the system is far from the equilibrium state. A combination of the genetic algorithm followed by one of the well-tested simulation techniques seems to be an optimal approach. The curvature of the inverse magnetic susceptibility is also presented for the stable state of the BEG model.

This work focuses on theoretically-obtained state solutions on molecular crystals with order–disorder phenomena in both the orientation and position of molecules. The stable, metastable and unstable solutions of the modified Pople–Karasz theory of molecular crystals are studied using two different methods. The first displays the free energy surfaces in the form of the contour mapping. The second obtains the flow diagrams and the relaxation curves by solving the dynamical equations. At temperatures lower than the lower limit of the stability temperature (T_l), two solutions, namely unstable and stable, are obtained. At these temperatures the system always relaxes to a stable state. Between the lower limit and the upper limit of the stability temperatures (T_u), there are three solutions: stable, metastable and unstable. In this case, the system relaxes to either a stable or a metastable state. The unstable state is a saddle-point which plays an important role in explaining how to get a system frozen-in in the metastable state. Above T_u, there is a single stable solution. The solutions obtained from these two methods are in exact agreement with each other for a given temperature and coupling constant.

The spin-pair correlation functions in the Ising model on the square lattice are obtained in three approximations of the simplified cluster variation method. We find good estimates of the critical exponents ην, μ and ν, by applying the coherent-anomaly method to the obtained spin-pair correlation functions.

The cluster variation method is applied to the ferromagnetic six-state clock model on the triangular lattice up to the hexagonal cluster approximation. The Kosterlitz-Thouless type phase transition of the model is studied by using the coherent anomaly method (CAM).