  Recently, it was established that there exists a direct relation between the
non-Hermitian skin effects, -strong dependence of spectra on boundary
conditions for non-Hermitian Hamiltonians-, and boundary zero modes for
Hermitian topological insulators. On the other hand, in terms of the spectral
theory, the skin effects can also be interpreted as instability of spectra for
nonnormal (non-Hermitian) Hamiltonians. Applying the latter interpretation to
the former relation, we develop a theory of zero modes with quantum anomaly for
general Hermitian lattice systems. Our theory is applicable to a wide range of
systems: Majorana chains, non-periodic lattices, and long-range hopping
systems. We relate exact zero modes and quasi-zero modes of a Hermitian system
to spectra and pseudospectra of a non-Hermitian system, respectively. These
zero and quasi-zero modes of a Hermitian system are robust against a class of
perturbations even if there is no topological protection. The robustness is
measured by nonnormality of the corresponding non-Hermitian system. We also
present explicit construction of such zero modes by using a graphical
representation of lattice systems. Our theory reveals the presence of
nonnormality-protected zero modes, as well as the usefulness of the
nonnormality and pseudospectra as tools for topological and/or non-Hermitian
physics.
