Non-Abelian topological charge of photons in periodic/non-periodic media and non-Hermitian systems

Multigap topology, which considers topology between more than two bands, generalizes the geometrical phase of a single band to the ones of multiple bands leading to non-Abelian band topology. While this non-Abelian band topology is normally observed for three-dimensional (3D) periodic structures such as electronic crystals, photonic crystals, and metamaterials, one- or two-dimensional (1D/2D) periodic structures can be employed with an additional dimension called ‘synthetic’ dimension. For the demonstration, for example, coupled optical waveguides and coupled ring resonators with phase and amplitude modulators have been employed. To observe non-Abelian nature of photon propagation, one can even employ non-periodic systems such as a finite number of coupled optical waveguides with spatially modulated couplings. Recently, the non-Abelian propagation of photons has been considered in non-Hermitian systems where gain and loss are controlled and the Bloch wave description fails. In this review, we introduce the basic concept of non-Abelian topology in different dimensions and symmetries and review recent development in the realization of non-Abelian topological charges in various physical systems such as 3D photonic crystals, metamaterials, coupled waveguides and cavity systems. Further, we discusshow non-Hermiticity plays a role in the non-Abelian topology of bands or eigenvalues starting from reviewing theoretical models in condensed matter physics.