Electric polarization and discrete shift from boundary and corner charge in crystalline Chern insulators

TL;DR

This paper develops a unified theoretical framework to extract topological invariants like electric polarization and discrete shift from boundary and corner charges in crystalline Chern insulators, even with non-zero Chern number.

Contribution

It introduces a general formula linking boundary and corner charge responses to topological invariants, unifying various charge response phenomena in Chern insulators.

Findings

Derived a formula for total charge in terms of topological invariants.

Showed invariants can be obtained from boundary and corner charge measurements.

Unified boundary, corner, disclination, and dislocation charge responses.

Abstract

Recently, it has been shown how topological phases of matter with crystalline symmetry and $U(1)$ charge conservation can be partially characterized by a set of many-body invariants, the discrete shift $\mathscr{S}_{\text{o}}$ and electric polarization $\vec{\mathscr{P}}_{\text{o}}$, where $\text{o}$ labels a high symmetry point. Crucially, these can be defined even with non-zero Chern number and/or magnetic field. One manifestation of these invariants is through quantized fractional contributions to the charge in the vicinity of a lattice disclination or dislocation. In this paper, we show that these invariants can also be extracted from the length and corner dependence of the total charge (mod 1) on the boundary of the system. We provide a general formula in terms of $\mathscr{S}_{\text{o}}$ and $\vec{\mathscr{P}}_{\text{o}}$ for the total charge of any subregion of the system which can include full boundaries or bulk lattice defects, unifying boundary, corner, disclination, and dislocation charge responses into a single general theory. These results hold for Chern insulators, despite their gapless chiral edge modes, and for which an unambiguous definition of an intrinsically two-dimensional electric polarization has been unclear until recently. We also discuss how our theory can fully characterize the topological response of quadrupole insulators.