Quadrupole moments, edge polarizations, and corner charges in the Wannier representation

The modern theory of polarization determines the macroscopic end charge of a truncated 1D insulator, modulo e, from a knowledge of bulk properties alone. A more subtle problem is the determination of the corner charge of a 2D insulator from a knowledge of bulk and edge properties. While previous works mainly considered symmetry constraints, here we focus on the case that the only bulk symmetry is inversion, so that the corner charge can take arbitrary values. We develop a Wannier-based formalism that allows the corner charge to be predicted, modulo e, only from calculations on ribbon geometries of two different orientations. We find that the interior quadrupole and edge dipole contributions depend upon the gauge used to construct the Wannier functions, although their sum is gauge-independent. From this we conclude that any Wannier-based method for computing the corner charge requires the use of a common gauge throughout the calculation. We satisfy this constraint by using a projection-based Wannier construction, thereby successfully predicting the corner charge for several tight-binding models.