  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.
