Cubic anisotropy created by defects of “random local anisotropy” type, and phase diagram of the $O(n)$ model

The expression for the cubic-type-anisotropy constant created by defects of “random local anisotropy” type is derived. 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 anisotropy” type in a system with continuous symmetry of the $n$-component vector order parameter ($O(n)$ model) leads to the long-range order collapse and to occurrence of a disordered state, is not true if an anisotropic distribution of the defectinduced random easy axes directions in the order parameter space creates a global anisotropy of the “easy axis” type. For a weakly anisotropic distribution of the easy axes, in space dimensions 2 $\le d<$ 4 there exists some critical defect concentration, when exceeded, the inhomogeneous Imry–Ma state can exist as an equilibrium one. At the defect concentration lower than the critical one the long-range order takes place in the system. For a strongly anisotropic distribution of the easy axes, the Imry–Ma state is suppressed completely and the long-range order state takes place at any defect concentration.