Topological phases in photonic microring lattices with projective symmetry

Symmetry plays a key role in classifying topological phases, which can be enriched by the projective symmetry group in the presence of artificial gauge fields (AGFs). Here, we utilize two-dimensional (2D) photonic microring lattices to create three different topological states based on projective symmetry. By engineering link rings, we are able to flexibly manipulate the AGFs and coupling magnitude. As each plaquette carries a $\ensuremath{\pi}$ flux, the two translation symmetries of a rectangle microring lattice are projectively represented. By applying different types of dimerization, we tune the spatial space to break translation symmetry and achieve a M\"obius topological insulator with twisted edge bands and a graphenelike topological semimetal with flat bands, which we reflect by unique excitation spectra and field distributions. Additionally, by changing the configuration of gauge flux, the mirror and translation operators become anticommutative, leading to the fractal translation of the Brillouin zone. As a result, the band structure of topological edge modes experiences twice the period in momentum space. All results are confirmed by full-wave simulation. Our study has the potential to construct unprecedented photonic topological insulators benefiting from gauge fields.