Transition regions in the parameter space based on the transverse Ising model with a four-spin interaction

By taking into account the four-spin interaction in the transverse Ising model (TIM), the transition feature of the ferroelectric thin film is investigated within the framework of the Green's function technique. Various transition regions in the parameter space with respect to the ratios of the tunneling frequency and the four-spin interaction to the two-spin interaction are calculated by a higher-order decoupling approximation beyond the usual mean field approximation. The results show that the transition regions are greatly different from the ones obtained within the framework of the mean-field approximation. It is suggested that the usual mean-field approximation exaggerates not only the second-order but also the first-order ferroelectric transition regions for a ferroelectric thin film.