Ground state of the orbitally ordered PbMnBO$_4$ ferromagnet

The ground state of the four-sublattice PbMnBO$_4$ ferromagnet is investigated taking into account the symmetry of the crystal and the spatial distribution of the orbital states of the Mn$^{3+}$ ions. By numerical minimization of the energy of the ground state of the classical model with three types of anisotropy–single-ion, dipole–dipole, and Dzyaloshinskii–Moriya interaction–the anisotropic interaction parameters for magnetic phases with different orientations of the total magnetization in the crystal are obtained. It is demonstrated that the experimentally observed state with the magnetization directed along the a orthorhombic axis becomes the ground state only in the presence of antisymmetric anisotropic exchange.