  The modern theory of polarization allows for the determination of the
macroscopic end charge of a truncated one-dimensional insulator, modulo the
charge quantum $e$, from a knowledge of bulk properties alone. A more subtle
problem is the determination of the corner charge of a two-dimensional
insulator, modulo $e$, from a knowledge of bulk and edge properties alone.
While previous works have tended to focus on the quantization of corner charge
in the presence of symmetries, 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 elucidate the dependence of the interior quadrupole
and edge dipole contributions upon the gauge used to construct the Wannier
functions, finding that while these are individually gauge-dependent, their sum
is gauge-independent. From this we conclude that the edge polarization is not
by itself a physical observable, and that any Wannier-based method for
computing the corner charge requires the use of a common gauge throughout the
calculation. We satisfy this constraint using two Wannier construction
procedures, one based on projection and another based on a gauge-consistent
nested Wannier construction. We validate our theory by demonstrating the
correct prediction of corner charge for several tight-binding models. We
comment on the relations between our approach and previous ones that have
appeared in the literature.
