Macroscopic descriptions of ferroelectrics have an obvious appeal in terms of efficiency and physical intuition. Their predictive power, however, has often been thwarted by the lack of a systematic procedure to extract the relevant materials parameters from the microscopics. Here we address this limitation by establishing an unambiguous two-way mapping between spatially inhomogeneous fields and discrete lattice modes. This yields a natural treatment of gradient couplings in the macroscopic regime via a long-wavelength expansion of the crystal Hamiltonian. Our analysis reveals an inherent arbitrariness in both the flexoelectric and polarization gradient coefficients, which we ascribe to a translational freedom in the definition of the polar distortion pattern. Remarkably, such arbitrariness cancels out in all physically measurable properties (relaxed atomic structure and energetics) derived from the model, pointing to a generalized translational covariance in the continuum description of inhomogeneous ferroelectric structures. We demonstrate our claims with extensive numerical tests on 180° domain walls in common ferroelectric perovskites, finding excellent agreement between the continuum model and direct first-principles calculations.

中文翻译：

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铁电畴壁处挠曲电的平移协方差

铁电体的宏观描述在效率和物理直觉方面具有明显的吸引力。然而，由于缺乏从显微镜中提取相关材料参数的系统程序，他们的预测能力经常受到阻碍。在这里，我们通过在空间不均匀场和离散晶格模式之间建立明确的双向映射来解决这一限制。这通过晶体哈密顿量的长波长扩展产生了宏观范围内梯度耦合的自然处理。我们的分析揭示了挠曲电和极化梯度系数固有的任意性，我们将其归因于极性畸变模式定义中的平移自由度。值得注意的是，这种任意性抵消了从模型中导出的所有物理可测量特性（松弛原子结构和能量学），表明非均匀铁电结构的连续体描述中存在广义平移协方差。我们通过对常见铁电钙钛矿中的 180° 畴壁进行广泛的数值测试来证明我们的主张，发现连续体模型和直接第一性原理计算之间非常吻合。