Random fields and orientational order in models with a continuous-energy minimum set inqspace and in magnetic systems favoring an incommensurate order

We study the influence of quenched random fields on the ground-state properties of a Ginzburg-Landau-Wilson model with a continuous set of energy minima corresponding to modulated phases with a fundamental wave vector on a ring embedded in d-dimensional reciprocal space. Thus, at low temperatures and in the absence of random fields, there is both orientational and translational order. Arbitrarily weak random-field breaks translational, i.e., long-range modulated order for d<4.5. However, for d<$d^{\mathrm{*}}$, for some $d^{\mathrm{*}}$, long-range orientational order is also unstable. We argue that $d^{\mathrm{*}}$>3. Therefore, in three-dimensional realizations of the model the modulated state breaks into domains having random orientations of their local wave-vectors. Some magnetic systems favoring modulated order can be approximately described by models with a continuous set of energy minima, e.g., magnetic superconductors, such as ${\mathrm{ErRh}}_4$ ${\mathrm{B}}_4$, ${\mathrm{HoMo}}_6$ ${\mathrm{S}}_8$, and ${\mathrm{HoMo}}_6$ ${\mathrm{Se}}_8$. However, spatial anisotropy, due to a crystal lattice, breaks the degeneracy of a continuous energy minimum down to a discrete set of energy minima in q space. Thus, the stability of the orientational order is determined by a competition, which is studied here, between random fields and spatial anisotropy. We speculate that orientationally disordered structures, observed in the coexistence phase of magnetic superconductors, which are composed of domains with different orientations of local wave vectors, could be a random-field effect.