Hermitian topologies originating from non-Hermitian braidings

The complex energy bands of non-Hermitian systems braid in momentum space even in one dimension. Here, we reveal that the non-Hermitian braiding underlies the Hermitian topological physics with chiral symmetry under a general framework that unifies Hermitian and non-Hermitian systems. Particularly, we derive an elegant identity that equates the linking number between the knots of braiding non-Hermitian bands and the zero-energy loop to the topological invariant of chiral-symmetric topological phases in one dimension. Moreover, we find an exotic class of phase transitions arising from the critical point transforming different knot structures of the non-Hermitian braiding, which are not included in the conventional Hermitian topological phase transition theory. Nevertheless, we show the bulk-boundary correspondence between the bulk non-Hermitian braiding and boundary zero modes of the Hermitian topological insulators. Finally, we construct typical topological phases with non-Hermitian braidings, which can be readily realized by artificial crystals.