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The presence of  π  gauge flux enabled by positive and negative hopping amplitude can lead to Möbius bands, which was recently demonstrated in both realistic acoustic and photonic lattices, twisted at  k  =  π . Here, we show that the artificial gauge flux configuration can be achieved by exploiting the interactions between photonic  s  and  p  orbital-like fundamental modes in circular and peanut-shaped waveguides, respectively. By manipulating the interplay between the gauge fields and the crystal symmetry, we show that breaking the primitive translation symmetry through lattice site dimerization and deformation can cause the original Dirac semimetal phase, characterized by a four-fold Dirac point at the Brillouin zone center, to transform into various topological phases. The designed photonic waveguide array supports topological phases such as Möbius insulator and Weyl-like semimetal phases. Noticeably different to the existing cases, we explicitly show that the twisting Möbius bands cross each other at  k  = 0 due to the lattice gauging with alternating sign, which results in distinct beam dynamics excitation. We also present Weyl-like flat-band edge states in such photonics waveguide arrays. Our results suggest that such  s  −  p  hybridized photonic waveguide array servers as a convenient and flexible platform for studying topological physics, particularly in simulating the effects of gauge field in alternative configuration.

Möbius edge band and Weyl-like semimetal flat-band in topological photonic waveguide array by synthetic gauge flux