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 P - αedge, where the integer α enumerates the edges. We introduce a particular manner of cutting a crystal, dubbed "Wannier cut,"which allows us to compute P - αedge. We find that P - αedge 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 P - αedge, only 2n-1 are independent of the choice of Wannier cut and correspond to physical observables: corner charges and edge dipoles.