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

Shang Ren, Ivo Souza, and David Vanderbilt
Phys. Rev. B 103, 035147 – Published 28 January 2021

Abstract

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.

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  • Received 26 October 2020
  • Revised 31 December 2020
  • Accepted 6 January 2021

DOI:https://doi.org/10.1103/PhysRevB.103.035147

©2021 American Physical Society

Physics Subject Headings (PhySH)

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

Shang Ren1,*, Ivo Souza2,3, and David Vanderbilt1

  • 1Department of Physics & Astronomy, Rutgers University, Piscataway, New Jersey 08854, USA
  • 2Centro de Física de Materiales, Universidad del País Vasco, 20018 San Sebastián, Spain
  • 3Ikerbasque Foundation, 48013 Bilbao, Spain

  • *Corresponding author: shangren@physics.rutgers.edu

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Issue

Vol. 103, Iss. 3 — 15 January 2021

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