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.Comment: 7 pages, 5 figure