Stability of the uniform ferroelectric nematic phase

Recent discovery of the ferroelectric nematic phase N_F resurrects a question about stability uniform state with respect to formation either standard for solid ferroelectrics domain structure, or often occurring in liquid crystals space modulation polarization vector P (and naturally coupled director. In this work within Landau mean-field theory we investigate linear minimal model admitting conventional paraelectric N and phases. Our model, (besides terms expansion over director gradients) includes, also crystals, flexoelectric coupling term f overlooked literature (although similar by its symmetry coupling) flexo-dipolar coupling. We find that easy-plane anisotropy case loses one-dimensional two-dimensional modulation. For non-zero 2D threshold is always higher than 1D counterpart. No any instability at all if one neglects dipole-flexoelectric easy-axis both thresholds are same, can occur even without flexo-dipolr speculate phases modulations be identified discussed [see M.P.Rosseto, J.V.Selinger, Physical Review E, volume 101, page 052707 (2020)] single splay double nematics.