Quantization of fractional corner charge in $C_n$-symmetric higher-order topological crystalline insulators

TL;DR

This paper demonstrates that in $C_n$-symmetric higher-order topological insulators, fractional corner charges are quantized and can be systematically characterized using topological indices related to symmetry representations.

Contribution

It introduces a systematic method to relate symmetry representations to fractional corner charges and explores the role of additional chiral symmetry in these topological phases.

Findings

Corner charges are quantized in multiples of e/n.

Topological indices relate symmetry representations to fractional charges.

Zero-energy corner states accompany half-integer corner charges when chiral symmetry exists.

Abstract

In the presence of crystalline symmetries, certain topological insulators present a filling anomaly: a mismatch between the number of electrons in an energy band and the number of electrons required for charge neutrality. In this paper, we show that a filling anomaly can arise when corners are introduced in $C_n$-symmetric crystalline insulators with vanishing polarization, having as consequence the existence of corner-localized charges quantized in multiples of $\frac{e}{n}$. We characterize the existence of this charge systematically and build topological indices that relate the symmetry representations of the occupied energy bands of a crystal to the quanta of fractional charge robustly localized at its corners. When an additional chiral symmetry is present, $\frac{e}{2}$ corner charges are accompanied by zero-energy corner-localized states. We show the application of our indices in a number of atomic and fragile topological insulators and discuss the role of fractional charges bound to disclinations as bulk probes for these crystalline phases.