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

The increasingly vast body of literature on dielectric screening in two-dimensional (2D) crystals has borrowed its founding concepts from the well-known (and long-established) paradigms of the three-dimensional (3D) case: macroscopic potentials are structureless plane waves, and polarizabilities are scalar functions of the wave vector and frequency. Here we show, by performing a rigorous study of the nonanalyticities of the Coulomb kernel in 2D, that neither assumption is a priori valid. In particular, we find that the dielectric functions become $2\times 2$ matrices in quasi-2D crystals, with nonuniform macroscopic potentials that are two-component hyperbolic functions of the out-of-plane coordinate, $z$. This result points to an unexpected increase in dimensionality of the screening problem when the spatial dimensionality of the crystal decreases. We demonstrate our arguments by performing a rigorous derivation of 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, SnS$_2$ and 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.