  Non-Hermitian Chern insulators differ from their Hermitian cousins in one key
aspect: their edge spectra are incredibly rich and confounding. For example,
even in the simple case where the bulk spectrum consists of two bands with
Chern number $\pm 1$, the edge spectrum in the slab geometry may have one or
two edge states on both edges, or only at one of the edges, depending on the
model parameters. This blatant violation of the familiar bulk-edge
correspondence casts doubt on whether the bulk Chern number can still be a
useful topological invariant, and demands a working theory that can predict and
explain the myriad of edge spectra from the bulk Hamiltonian to restore the
bulk-edge correspondence. We outline how such a theory can be set up to yield a
thorough understanding of the edge phase diagram based on the notion of the
generalized Brillouin zone (GBZ) and the asymptotic properties of block
Toeplitz matrices. The procedure is illustrated by solving and comparing three
non-Hermitian generalizations of the Qi-Wu-Zhang model, a canonical example of
two-band Chern insulators. We find that, surprisingly, in many cases the phase
boundaries and the number and location of the edge states can be obtained
analytically. Our analysis also reveals a non-Hermitian semimetal phase whose
energy-momentum spectrum forms a continuous membrane with the edge modes
transversing the hole, or genus, of the membrane. Subtleties in defining the
Chern number over GBZ, which in general is not a smooth manifold and may have
singularities, are demonstrated using examples. The approach presented here can
be generalized to more complicated models of non-Hermitian insulators or
semimetals in two or three dimensions.
