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Rich topological phenomena, edge states and two types of corner states, areunveiled in a two-dimensional square-lattice dielectric photonic crystalwithout both $C_4$ and $M_{x(y)}$ symmetries. Specifically, non-trivial type-Icorner states, which do not exist in systems with $C_4$ and $M_{x(y)}$ sincethe degeneracy, are protected by non-zero quadrupole moment, no longerquantized to but less than $0.5$. Excellent properties, e.g. sub-wavelengthlocalization and air-concentrated field distribution, are presented. Type-IIcorner states, induced by long-range interactions, are easier realized due toasymmetry. This work broadens the topological physics for the symmetries-brokensystems and provides potential applications.

Topological properties of two-dimensional photonic square lattice without $C_4$ and $M_{x(y)}$ symmetries