The Connection of Topology between Systems with Different Dimensions: 1D Zak Phases to 2D Chern Number, Weyl Point as the Jumping Channel for One Singularity and Nodal Line to Merge All Singularities

Abstract

The topology in different dimensions has attracted enormous interests, e.g. the Zak phase in 1D systems, the Chern number in 2D systems and the Weyl points or nodal lines in the systems with higher dimensions. It would be fantastic to find the connection of different topology in different dimensions from one simple model and reveal the deep physical picture behind them. In this work, we propose a new model which starts from a binary-layered 1D photonic crystal, and by introducing synthetic dimensions the topology of higher dimension systems could appear. From this model, we find that the topology of band gap and the Chern number of the 2D systems can be predicted by the parity-switching types and the Zak phases of the 1D systems with spatial inversion symmetry(SIS), respectively. The chiral edge state is confirmed by the winding number of the reflection phase in the topological nontrivial gap. Different types of the topological transition in higher dimensions are found, where two bands degenerated as Weyl point or nodal line. Surprisingly, we find that the topological connection between different dimensions and the topological transition types in this model can be explained by the evolving of two singularities which give rise to nonzero Zak phase of the 1D systems with SIS. When transporting one singularity between adjacent bands, the Weyl point takes the role as the instantaneous jumping channel of the singularity in the parameter space, and then both the Zak phases of 1D systems with SIS and the Chern number of 2D systems are changed. While both singularities moves to band-gap edges from two adjacent bands, they will merge into the nodal line. The theory for such model is also constructed. We propose that such topology connection between different dimensions could be quite universal for other systems.