Two classes of singularities and novel topology in a specially designed synthetic photonic crystals.

Zak phase and topological protected edge state are usually studied in one-dimensional (1D) photonic systems with spatial inversion symmetry (SIS). In this work, we find that specific classes of 1D structure without SIS can be mapped to a system with SIS and also exhibit novel topology, which manifest as phase cut lines (PCLs) in our specially designed synthetic photonic crystals (SPCs). Zak phase defined in SIS is extended to depict the topology of PCLs after redefinition, and a topological protected edge state is also achieved in our 1D structure without SIS. In our SPCs, the relationship between Chern numbers in two dimensions (2D) and the extended Zak phases of 1D PCLs is given, which are bound by the first type singularities. Higher Chern numbers and multi-chiral edge states are achieved utilizing the concept of synthetic dimensions. The effective Hamiltonian is given, based on which we find that the band edges of each PCL play a role analogous to the valley pseudospin, and our SPC is actually a new type of valley photonic crystal that is usually studied in graphene-like honeycomb lattice. The chiral valley edge transport is also demonstrated. In higher dimensions, the shift of the first type singularity in expanded parameter space will lead to the Weyl point topological transition, which we proposed in our previous work. In this paper, we find a second type of singularity that manifests as a singular surface in our expanded parameter space. The shift of the singular surface will lead to the nodal line topological transition. We find the states on the singular surface possess extremely high robustness against certain randomness, based on which a topological wave filter is constructed.