The recent discovery of the ferroelectric nematic phase $N_F$ resurrects a question about the stability of the uniform $N_F$ state with respect to the formation of either a standard for the solid ferroelectric domain structure or for the often occurring liquid crystal space modulation of the polarization vector $\mathbf{P}$ (and naturally coupled to $\mathbf{P}$ nematic director $\mathbf{n}$). In this work, within Landau mean-field theory, we investigate the linear stability of the minimal model admitting the conventional paraelectric nematic $N$ and $N_F$ phases. Our minimal model (in addition to the standard terms of the expansion over the $\mathbf{P}$ and director gradients) includes the standard for liquid crystals, the director flexoelectric coupling term ($f$), and, often overlooked in the literature (although similar by its symmetry to the director flexoelectric coupling), the flexodipolar coupling ($\beta$). We find that in the easy-plane anisotropy case (when the configuration with $\mathbf{P}$ orthogonal to $\mathbf{n}$ is energetically favorable), the uniform $N_F$ state loses its stability with respect to one-dimensional (1D) or two-dimensional (2D) modulation. If $f\ne 0$, the 2D modulation threshold (${\beta }_{c2}$ value) is always higher than its 1D counterpart value ${\beta }_{c1}$. There is no instability at all if one neglects the flexodipolar coupling ($\beta =0$). In the easy-axis case (when $\mathbf{n}$ prefers to align along $\mathbf{P}$), both instability (1D and 2D) thresholds are the same, and the instability can occur even at $\beta =0$. We speculate that the phases with 1D or 2D modulations can be identified as discussed in the literature [see M. P. Rosseto and J. V. Selinger, Phys. Rev. E 101, 052707 (2020)] for single-splay or double-splay nematics.

中文翻译：

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均匀铁电向列相的稳定性

铁电向列相的最新发现 ${ñ}_F$ 复活了关于制服稳定性的问题 ${ñ}_F$ 固态铁电畴结构的标准或偏振矢量的经常发生的液晶空间调制的状态 $\mathbf{P}$ （自然而然地 $\mathbf{P}$ 向列导演 $\mathbf{ñ}$）。在这项工作中，在Landau平均场理论下，我们研究了接受常规顺电向列的最小模型的线性稳定性$ñ$ 和 ${ñ}_F$阶段。我们的最小模型（除了扩展的标准条款$\mathbf{P}$ 和指向矢梯度）包括液晶标准，指向矢挠电耦合项（$F$），并且在文献中经常被忽略（尽管其对称性与指向矢挠性电耦合类似），而柔二极耦合（$\beta$）。我们发现在易平面各向异性的情况下（当$\mathbf{P}$ 正交于 $\mathbf{ñ}$ 在能量上有利） ${ñ}_F$状态会相对于一维（1D）或二维（2D）调制失去稳定性。如果$F\ne 0$，即2D调制阈值（${\beta }_{C2}$ 值）始终高于其一维对应值 ${\beta }_{C\mathrm{1个}}$。如果忽略挠性双极耦合，则根本没有不稳定（$\beta =0$）。在易轴情况下（当$\mathbf{ñ}$ 喜欢沿着 $\mathbf{P}$），两个不稳定性（1D和2D）阈值都相同，甚至在 $\beta =0$。我们推测，可以按照文献中的讨论确定具有1D或2D调制的相位[参见M. P. Rosseto和J. V. Selinger，Phys。启ë 101，052707（2020）]为单斜展或双喷射向列相。