Topology reconstruction for asymmetric systems by homomorphic mapping or perturbation approximation

The systems without symmetries, e.g., the spatial and chiral symmetries, are generally thought to be improper for topological study and no conventional integral topological invariant can be well defined. In this work, with multiband asymmetric Rice-Mele--type systems as examples, we show that the topology of all gaps can be reconstructed by two general methods and the topological origin of many phenomena are revealed. An integral topological invariant, i.e., the renormalized real-space winding number, can properly characterize the topology and bulk-edge correspondence of such systems. For the first method, a homomorphic mapping relationship between a Rice-Mele--type system and its chiral counterpart is set up, which accounts for the topology reconstruction in the half-filling gaps. For the second method, the Hilbert space of asymmetric systems could be reduced into degenerate subspaces by perturbation approximation, so that the topology in subspaces accounts for the topology reconstruction in the fractional-filling gaps. Surprisingly, the topology reconstructed by perturbation approximation exhibits extraordinary robustness since the topological edge states even exist far beyond the weak perturbation limit. We also show that both methods can be widely used for other asymmetric systems, e.g., the two-dimensional Rice-Mele systems and the superconductor systems. At last, for the asymmetric photonic systems, we predict different topological edge states by our topology-reconstruction theory and experimentally observe them in the laboratory, which agrees with each other very well. Our findings show the possibility that for some types of asymmetric systems the topology can be reconstructed by two methods and the related phenomena can be observed. Various reconstruction methods might be found for other types of asymmetric systems and the category of topology study could be expanded.