The topological invariants of rotationally symmetric crystals

TL;DR

This paper develops a method to calculate new topological invariants in 2D rotationally symmetric insulators, revealing higher-order topological features and corner charges, extending the understanding of crystalline topological phases.

Contribution

It introduces a practical approach using Wilson loops and line invariants to determine all topological invariants in 2D systems with rotation symmetry, including new invariants related to higher-order topology.

Findings

New invariants related to higher-order topology identified

Method demonstrated on models with two-fold and three-fold rotation symmetry

Link between invariants and corner charges established

Abstract

Recent formal classifications of crystalline topological insulators predict that the combination of time-reversal and rotational symmetry gives rise to topological invariants beyond the ones known for other lattice symmetries. Although the classification proves their existence, it does not indicate a way of calculating the values of those invariants. Here, we show that a specific set of concentric Wilson loops and line invariants yields the values of all topological invariants in two-dimensional systems with pure rotation symmetry in class AII. Examples of this analysis are given for specific models with two-fold and three-fold rotational symmetry. We find new invariants that relate to the presence of higher-order topology and corner charges.