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 P⃗αedge, where the integer α enumerates the edges. We introduce a particular manner of cutting a crystal, dubbed "Wannier cut," which allows us to compute P⃗αedge. We find that P⃗αedge 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 2 n independent components of P⃗αedge, only 2 n -1 are independent of the choice of Wannier cut and correspond to physical observables: corner charges and edge dipoles.
- Publication:
-
Physical Review Research
- Pub Date:
- October 2020
- DOI:
- 10.1103/PhysRevResearch.2.043012
- arXiv:
- arXiv:2006.16090
- Bibcode:
- 2020PhRvR...2d3012T
- Keywords:
-
- Condensed Matter - Mesoscale and Nanoscale Physics;
- Condensed Matter - Materials Science;
- Physics - Computational Physics
- E-Print:
- published version. v2: application to Benalcazar-Bernevig-Hughes model