Sign freedom of non-abelian topological charges in phononic and photonic topological semimetals

The topological nature of nodal lines in three-band systems can be described by non-abelian topological charges called quaternion numbers. Due to the gauge freedom of the eigenstates, the sign of quaternion numbers can be flipped by performing a 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 semimetals. Here, we elaborate on the sign freedom of non-abelian topological charges by visualizing numerically calculated topological charges in phononic and photonic topological semimetals. For this, we employ a common reference point method for multiple nodal lines and thus confirm that the sign flipping does not cause any inconsistency in building the quaternion group.