  Topological materials exhibit edge-localized scattering-free modes protected
by their nontrivial bulk topology through the bulk-edge correspondence in
Hermitian systems. While topological phenomena have recently been much
investigated in non-Hermitian systems with dissipations and injections, the
fundamental principle of their edge modes has not fully been established. Here,
we reveal that in non-Hermitian systems robust gapless edge modes can
ubiquitously appear owing to a mechanism that is distinct from bulk topology,
thus indicating the breakdown of the bulk-edge correspondence. The robustness
of these edge modes originates from yet another topological structure
accompanying the branchpoint singularity around an exceptional point, at which
eigenvectors coalesce and the Hamiltonian becomes nondiagonalizable. Their
characteristic complex eigenenergy spectra are applicable to realize lasing
wave packets that propagate along the edge of the sample. We numerically
confirm the emergence and the robustness of the proposed edge modes in the
prototypical models. Furthermore, we show that these edge modes appear in a
model of chiral active matter based on the hydrodynamic description,
demonstrating that active matter can exhibit an inherently non-Hermitian
topological feature. The proposed general mechanism would serve as an
alternative designing principle to realize scattering-free edge current in
non-Hermitian devices, going beyond the existing frameworks of non-Hermitian
topological phases.
