Uniform uniaxial anisotropic systems with coupled vector order parameters

Phase transitions in magnets, described by two coupled, m-component vector order parameters, having uniform uniaxial anisotropies are studied. Using a phenomenological model, it is shown that when both order parameters are anisotropic, phase transitions are always second order, in either the uniaxial or the (m???1)-isotropic phase. This is contrary to the isotropic case of two coupled order parameters, for which phase transitions are fluctuation-induced first order. The transitions are still continuous into the m-isotropic phase even when the only anisotropic order parameter is the one with the lowest mean-field critical temperature. New discontinuous transitions still occur in either the uniaxial or the (m???1)-isotropic phase, when the only anisotropic order parameter has the highest mean-field critical temperature.