Sign freedom of non-Abelian topological charges in phononic and photonic topological insulators

TL;DR

This paper demonstrates the sign freedom of non-Abelian topological charges in phononic and photonic topological insulators, confirming that sign flipping does not affect the quaternion group structure through numerical visualization.

Contribution

It explicitly shows the sign flipping of non-Abelian topological charges in realistic phononic and photonic systems, confirming the robustness of the quaternion group structure.

Findings

Sign flipping of topological charges is visualized in phononic and photonic systems.

Sign freedom does not affect the quaternion group structure.

Numerical methods confirm the theoretical gauge freedom.

Abstract

The topological nature of nodal lines in a three-band system can be described by non-Abelian topological charges called quaternion numbers. Due to the gauge freedom of the eigenstates, the sign of the non-Abelian topological charges can be flipped by performing the gauge transformation, i.e., choosing a different basis of eigenstates. However, the sign flipping has not been explicitly shown in realistic systems such as phononic and photonic topological insulators. Here, we elaborate the sign freedom by visualizing the numerically calculated topological charges in phononic and photonic topological insulators. In doing so, we employ a common reference point method for multiple nodal lines to confirm that the sign flipping does not cause any issue in building the quaternion group.