Expanded Bethe–Peierls approximation for the Blume–Capel model

We have studied the Blume–Capel model by using the expanded Bethe–Peierls approximation. In this approximation, the system is taken as a group of chains composed of a central chain and its nearest-neighbor chains. The nearest-neighbor chains are in an effective field produced by the other spins, which can be determined by the condition that the magnetization of the central chain is equal to that of its nearest-neighbor chains. In the present approximation, the lattice dimensionality can be distinguished in the formulations. With the use of the transfer matrix method, we calculate the free energy, magnetization, specific heat, and transition temperature of the systems on simple cubic (z=6), triangular (z=6) and square (z=4) lattices. It is found that the behavior of the specific heat with temperature and the single-ion anisotropy parameter is similar to that obtained by the effective-field theory. The temperatures at the tricritical point and the phase transition point on simple cubic lattice are higher than those on the triangular lattice. The transition temperatures at zero single-ion anisotropy on simple cubic and square lattices are much closer to those obtained by series expansion method than those obtained by effective-field theory and Bethe lattice approximation, and the value of the single-ion anisotropy at the tricritical point obtained on simple cubic lattice is in agreement with that obtained by the series expansion method.