The generalized method to calculate the real-space winding number for one-dimensional systems with complex...

The topological study of the complicated one-dimensional (1D) systems with multi-band-gap structures, including quasi-crystals (QCs), is very hard since the lack of effective topological invariants to describe the non-triviality of gaps. A generalized method, based on the contracted wave-function, is proposed in this work to calculate the real-space winding number for the complicated 1D systems with multi-band-gap structures. First, the effectiveness of the generalized method is demonstrated to obtain the quantized real-space winding number for the gaps and correctly predict the topological phase transition and the existing fractional charge on the edges for the periodic 4-atoms SSH model (4A-SSH model). Then, we apply the generalized method to more complicated 1D Thue-Morse (TM) systems, which is one kind of QCs. The quantized real-space winding number is obtained for two traditional gaps and two fractal gaps for the TM systems and can also correctly predict the existence of topological edge-states and fractional charge on the ends. Several new phenomena are observed, e.g. the topological phase transition and the edge-states for the gaps in multi-band-gap structures, the1/4fractional charge for the 4A-SSH model, the fluctuation of local charge and the asymmetric (but still with a quantized difference) fractional charge at the ends of TM system. The generalized method could be a powerful tool to study the topology of gaps in the complicated periodic systems or QCs.