Photonic Möbius topological insulator from projective symmetry in multiorbital waveguides

The gauge fields dramatically alter the algebraic structure of spatial symmetries and make them projectively represented, giving rise to novel topological phases. Here, we propose a photonic Möbius topological insulator enabled by projective translation symmetry in multiorbital waveguide arrays, where the artificial π gauge flux is aroused by the inter-orbital coupling between the first (s) and third (d) order modes. In the presence of π flux, the two translation symmetries of rectangular lattices anti-commute with each other. By tuning the spatial spacing between two waveguides to break the translation symmetry, a topological insulator is created with two Möbius twisted edge bands appearing in the bandgap and featuring 4π periodicity. Importantly, the Möbius twists are accompanied by discrete diffraction in beam propagation, which exhibit directional transport by tuning the initial phase of the beam envelope according to the eigenvalues of translation operators. This work manifests the significance of gauge fields in topology and provides an efficient approach to steering the direction of beam transmission.