The Curie temperature of the Ising ferromagnet with higher-order exchange interaction

The effects of the higher-order exchange interactions of the type (S_i.S_j)² and (S_i.S_j)(S_j.S_k) on the Curie temperature of the two-dimensional quadratic Ising ferromagnetic of spin one have been investigated with use of the Bethe-Peierls method. Apart from the dominant bilinear exchange interaction between the nearest-neighbour spins, the three-site four-spin interaction (S_i.S_j)(S_j.S_k) is shown to give a more significant effect on the Curie temperature than the biquadratic exchange interaction and the bilinear exchange interaction between the next-nearest neighbour spins. When the ratio of the three-site four-spin interaction to the bilinear exchange interaction between the nearest-neighbour spins becomes -0.22 the ferromagnetic order is shown to disappear. The calculation has been extended to the case of a simple cubic Ising ferromagnet, and the three-site four-spin interaction is also shown to be significant with the critical ratio -1.17, which is compared with -0.22 of the quadratic Ising ferromagnet. The origin of the three-site four-spin interaction has also been discussed by taking into consideration the magnetoelastic effect and perturbation expansion.