The Ising model with nearest- and next-nearest-neighbor interactions on the Bethe lattice: The exact recursion relations

The Ising model with nearest- and next-nearest-neighbor (NNN) bilinear interactions is examined on the Bethe lattice (BL) in terms of exact recursion relations (ERR) when the external magnetic H is turned on. The thermal variation of the magnetization belonging to the central spin is investigated to calculate the possible phase diagrams of the model for given coordination numbers. Different phase regions, ferromagnetic (FM), antiferromagnetic (AFM) and paramagnetic (PM), are discovered and the phase lines in terms of first-order or second-order phase transitions are calculated. These lines are found to be order-disorder or order-order phase transition lines. It is also found that they combine at some special points or terminate at some end points for appropriate values of the model parameters.