Ferromagnetic Heisenberg model with the Dzyaloshinskii-Moriya interaction

The spin-1/2 Heisenberg model is formulated in terms of a mean-field approximation (MFA) by using the matrix forms of spin operators $\hat{S}_x,\hat{S}_y$ and $\hat{S}_z$ in three-dimensions. The considered Hamiltonian consists of bilinear exchange interaction parameters $(J_x,J_y,J_z)$, Dzyaloshinskii-Moriya interactions $(\Delta_x,\Delta_y,\Delta_z)$ and external magnetic field components $(H_x,H_y,H_z)$. The magnetization and its components are obtained in the MFA with the general anisotropic case with $J_x\neq J_y \neq J_z$ for various values of coordination numbers $q$. Then, the thermal variations of magnetizations are investigated in detail to obtain the phase diagrams of the model for the isotropic case with $J_x=J_y=J_z>0$. It is found that the model exhibits ferromagnetic, paramagnetic, random phase regions and an extra ferromagnetic phase at which the components of magnetizations present branching.