Title: Topology reconstruction for asymmetric systems by isomorphic mapping or perturbation approximation

Abstract: 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 multi-band asymmetric Rice-Mele-like systems as examples, for the first time we show that the topology of all gaps can be reconstructed by two general methods and topological origin of many phenomena are revealed. A new 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, an isomorphic mapping relationship between a Rice-Mele-like 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 (2D) 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 open a door for investigating new topological phenomena in asymmetric systems by various topological reconstruction methods which should greatly expand the category of topology study.