Classical spin models with broken symmetry: Random-field-induced order and persistence of spontaneous magnetization in the presence of a random field

We consider classical spin models of two- and three-dimensional spins with continuous symmetry and investigate the effect of a symmetry-breaking unidirectional quenched disorder on the magnetization of the system. We work in the mean-field regime. We show by perturbative calculations and numerical simulations that although the continuous symmetry of the magnetization is lost due to disorder, the system still magnetizes in specific directions, albeit with a lower value as compared to the case without disorder. The critical temperature at which the system starts magnetizing and the magnetization at low- and high-temperature limits in the presence of disorder are estimated. Moreover, we treat the SO(n) n-component spin model to obtain the generalized expressions for the near-critical scalings, which suggest that the effect of disorder in magnetization increases with increasing dimension. We also study the behavior of magnetization of the classical XY spin model in the presence of a constant magnetic field, in addition to the quenched disorder. We find that in the presence of the uniform magnetic field, disorder may enhance the component of magnetization in the direction that is transverse to the disordered field.