We develop a fundamental theory of the long-range electrostatic interactions in two-dimensional crystals by performing a rigorous study of the nonanalyticities of the Coulomb kernel. We find that the dielectric functions are best represented by $2\times 2$ matrices, with nonuniform macroscopic potentials that are two-component hyperbolic functions of the out-of-plane coordinate $z$. We demonstrate our arguments by deriving the long-range interatomic forces in the adiabatic regime, where we identify a formerly overlooked dipolar coupling involving the out-of-plane components of the dynamical charges. The resulting formula is exact up to an arbitrary multipolar order, which we illustrate in practice via the explicit inclusion of dynamical quadrupoles. By performing numerical tests on monolayer BN, ${\mathrm{SnS}}_2$, and ${\mathrm{BaTiO}}_3$ membranes, we show that our method allows for a drastic improvement in the description of the long-range electrostatic interactions, with comparable benefits to the quality of the interpolated phonon band structure.

中文翻译：

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准二维晶体中精确的长程介电屏蔽和原子间力常数

我们通过对库仑核的非解析性进行严格研究，开发了二维晶体中长程静电相互作用的基本理论。我们发现介电函数最好用下式表示$2\times 2$ 矩阵，具有非均匀宏观势，是平面外坐标的二分量双曲函数 $z$. 我们通过推导绝热状态中的长程原子间力来证明我们的论点，在那里我们确定了以前被忽视的偶极耦合，涉及动态电荷的平面外分量。得到的公式精确到任意多极阶，我们在实践中通过明确包含动态四极杆来说明。通过对单层 BN 进行数值测试，${锡}_2$， 和 ${\mathrm{钛酸钡}}_3$ 膜，我们表明我们的方法允许在长程静电相互作用的描述中得到显着改进，具有与内插声子带结构的质量相当的好处。