Bulk-and-edge to corner correspondence

Abstract

We show that two-dimensional band insulators, with vanishing bulk polarization, obey bulk-and-edge to corner charge correspondence, stating that the knowledge of the bulk and the two corresponding ribbon band structures uniquely determines a fractional part of the corner charge irrespective of the corner termination. Moreover, physical observables related to macroscopic charge density of a terminated crystal can be obtained by representing the crystal as collection of polarized edge regions with polarizations →Pedgeα, where the integer α enumerates the edges. We introduce a particular manner of cutting a crystal, dubbed “Wannier cut,” which allows us to compute →Pedgeα. We find that →Pedgeα consists of two pieces: the bulk piece expressed via quadrupole tensor of the bulk Wannier functions' charge density and the edge piece corresponding to the Wannier edge polarization—the polarization of the edge subsystem obtained by Wannier cut. For a crystal with n edges, out of 2n independent components of →Pedgeα, only 2n−1 are independent of the choice of Wannier cut and correspond to physical observables: corner charges and edge dipoles.