Metastable and unstable states of the Blume–Capel model obtained by the cluster variation method and the path probability method

We investigate the temperature dependence of the order parameters of the Blume–Capel model for zero magnetic field in the lowest approximation of the cluster variation method which is identical to the mean-field approximation. Besides the stable branches of the order parameters, we establish the metastable and unstable parts of these curves for the various values of the coupling parameter, α=D/J. We also study the dynamics of the model by the path probability method, since the metastable behavior is a dynamical behavior. From this study, the “flatness” property of metastable states, “overshooting” phenomenon and as well as the role of the unstable state are seen explicitly. On the other hand, we also investigate how to obtain the metastable phases with two long-range order parameters. Finally, we calculate the phase transitions of the metastable and unstable branches of order parameters besides the stable branches and present the complete phase diagram.