  Recently, there has been a lot of activity in the research field of
topological non-Hermitian physics, partly driven by fundamental interests and
partly driven by applications in photonics. However, despite these activities,
a general classification and characterization of non-Hermitian Dirac models
that describe the experimental systems is missing. Here, we present a
systematic investigation of massive Dirac models on periodic lattices,
perturbed by general non-Hermitian terms. We find that there are three
different types of non-Hermitian terms. For each case we determine the bulk
exceptional points, the boundary modes, and the band topology. Our findings
serve as guiding principles for the design of applications, for example, in
photonic lattices. For instance, periodic Dirac systems with non-Hermitian mass
terms can be used as topological lasers. Periodic Dirac systems with
non-Hermitian anti-commuting terms, on the other hand, exhibit exceptional
points at the surface, whose non-trivial topology could be utilized for optical
devices.
