Phase diagram for the $O(n)$ model with defects of “random local field” type and verity of the Imry–Ma theorem

It is shown that the Imry–Ma theorem stating that in space dimensions $d<$ 4 the introduction of an arbitrarily small concentration of defects of the “random local field” type in a system with continuous symmetry of the $n$-component vector order parameter ($O(n)$ model) leads to long-range order collapse and to the occurrence of a disordered state is not true if the anisotropic distribution of the defect-induced random local field directions in the space of the order parameter gives rise to the effective anisotropy of the “easy axis” type. In the case of a weakly anisotropic field distribution, in space dimensions 2 $\le d<$ 4 there exists some critical defect concentration, above which the inhomogeneous Imry–Ma state can exist as an equilibrium one. At a lower defect concentration, long-range order takes place in the system. In the case of a strongly anisotropic field distribution, the Imry–Ma state is suppressed completely and long-range order state takes place at any defect concentration.