  The notion of a band gap is ubiquitous in the characterization of matter.
Particularly interesting are pseudo-gaps, which are enigmatic regions of very
low density of states that have been linked to novel phenomena like high
temperature superconductivity. In this work, we discover a new non-Hermitian
mechanism that induces pseudo-gaps when boundaries are introduced in a lattice.
It generically occurs due to the interference between two or more asymmetric
pumping channels, and possess no analog in Hermitian systems. Mathematically,
it can be visualized as being created by divergences of spectral flow in the
complex energy plane, analogous to how sharp edges creates divergent electric
fields near an electrical conductor. A non-Hermitian pseudo-gap can host
symmetry-protected mid-gap modes like ordinary topological gaps, but the
mid-gap modes are extended instead of edge-localized, and exhibit extreme
sensitivity to symmetry-breaking perturbations. Surprisingly, pseudo-gaps can
also host an integer number of edge modes even though the pseudo-bands possess
fractional topological windings, or even no well-defined Chern number at all,
in the marginal case of a phase transition point. Challenging conventional
notions of topological bulk-boundary correspondences and even the very concept
of a band, pseudo-gaps post profound implications that extend to many-body
settings, such as fractional Chern insulators.
