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We develop a fundamental theory of the long-range electrostatic interactionsin two-dimensional crystals by performing a rigorous study of thenonanalyticities of the Coulomb kernel. We find that the dielectric functionsare best represented by $2\times 2$ matrices, with nonuniform macroscopicpotentials that are two-component hyperbolic functions of the out-of-planecoordinate, $z$. We demonstrate our arguments by deriving the long-rangeinteratomic forces in the adiabatic regime, where we identify a formerlyoverlooked dipolar coupling involving the out-of-plane components of thedynamical charges. The resulting formula is exact up to an arbitrary multipolarorder, which we illustrate in practice via the explicit inclusion of dynamicalquadrupoles. By performing numerical tests on monolayer BN, SnS$_2$ andBaTiO$_3$ membranes, we show that our method allows for a drastic improvementin the description of the long-range electrostatic interactions, withcomparable benefits to the quality of the interpolated phonon band structure.

Exact long-range dielectric screening and interatomic force constants in quasi-2D crystals