  We investigate the validity of the non-Hermitian Hamiltonian approach in
describing quantum transport in disordered tight-binding networks connected to
external environments, acting as sinks. Usually, non-Hermitian terms are added,
on a phenomenological basis, to such networks to summarize the effects of the
coupling to the sinks. Here we consider a paradigmatic model of open quantum
network for which we derive a non-Hermitian effective model, discussing its
limit of validity by a comparison with the analysis of the full Hermitian
model. Specifically, we consider a ring of sites connected to a central
one-dimensional lead. The lead acts as a sink which absorbs the excitation
initially present in the ring. The coupling strength to the lead controls the
opening of the ring system. This model has been widely discussed in literature
in the context of light-harvesting systems. We analyze the effectiveness of the
non-Hermitian description both in absence and in presence of static disorder on
the ring. In both cases, the non-Hermitian model is valid when the energy range
determined by the eigenvalues of the ring Hamiltonian is smaller than the
energy band in the lead. Under such condition, we show that results about the
interplay of opening and disorder, previously obtained within the non-Hermitian
Hamiltonian approach, remain valid when the full Hermitian model in presence of
disorder is considered. The results of our analysis can be extended to generic
networks with sinks, leading to the conclusion that the non-Hermitian approach
is valid when the energy dependence of the coupling to the external
environments is sufficiently smooth in the energy range spanned by the
eigenstates of the network.
