The Topological Origin of Boundary Charges at Edges of One-Dimensional Crystals without Inversion Symmetry

We report the edge states and non-zero boundary charges in one-dimensional photonic crystals (1D PhCs) without inversion symmetry. In contrast to common 1D systems, we show that edge states corresponding to non-zero boundary charges do exist in these asymmetric 1D PhCs even if we cannot obtain non-integral topological invariants. Moreover, an edge state could be observed in the interface between the PhC without inversion symmetry and the well-defined trivial PhC. Finally, the origin of the non-quantized boundary charges is unveiled by the non-central Wannier center. Not only exact solutions of photonic systems, but the above topological phenomena can also be found in the tight-binding models. This work proposes a way to study the 1D symmetries-broken systems and provides models to show the topological origin of boundary charges, which is suitable for both classic systems and quantum systems.