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 ${\widehat{S}}_x,{\widehat{S}}_y$ and ${\widehat{S}}_z$ in three-dimensions. The considered Hamiltonian consists of bilinear exchange interaction parameters $(J_x,J_y,J_z)$, Dzyaloshinskii-Moriya interactions $({\mathrm{\Delta }}_x,{\mathrm{\Delta }}_y,{\mathrm{\Delta }}_z)$ 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 $J_x\mathrm{\ne }{J}_y\mathrm{\ne }{J}_z$ 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.