  We identify and investigate two classes of non-Hermitian systems, i.e., one
resulting from Lorentz-symmetry violation (LSV) and the other from a complex
mass (CM) with Lorentz invariance, from the perspective of quantum field
theory. The mechanisms to break, and approaches to restore, the bulk-boundary
correspondence in these two types of non-Hermitian systems are clarified. The
non-Hermitian system with LSV shows a non-Hermitian skin effect, and its
topological phase can be characterized by mapping it to the Hermitian system
via a non-compact $U(1)$ gauge transformation. In contrast, there exists no
non-Hermitian skin effect for the non-Hermitian system with CM. Moreover, the
conventional bulk-boundary correspondence holds in this (CM) system. We also
consider a general non-Hermitian system in the presence of both LSV and CM, and
we generalize its bulk-boundary correspondence.
