  We consider a Bogolibov-de Geenes (BdG) Hamiltonian, which is a non-Hermitian
Hamiltonian with pseudo-Hermiticity, for a system of (pseudo) spin-$1/2$ bosons
in a honeycomb lattice under the condition that the population difference
between the two spin components, i.e., magnetization, is a constant. Such a
system is capable of acting as a topological amplifier, under time-reversal
symmetry, with stable bulk bands but unstable edge modes which can be populated
at an exponentially fast rate. We quantitatively study the topological
properties of this model within the framework of the 38-fold way for
non-Hermitian systems. We find, through the symmetry analysis of the Bloch
Hamiltonian, that this model is classified either as two copies of symmetry
class AIII+$\eta_-$ or two copies of symmetry class A+$\eta$ depending on
whether the (total) system is time-reversal-symmetric, where $\eta$ is the
matrix representing pseudo-Hermiticity and $\eta_-$ indicates that
pseudo-Hermiticity and chiral symmetry operators anticommute. We prove, within
the context of non-Hermitian physics where eigenstates obey the
bi-orthonormality relation, that a stable bulk is characterized by a single
topological invariant, the Chern number for the Haldane model, independent of
pairing interactions. We construct a convenient analytical description for the
edge modes of the Haldane model in semi-infinite planes, which is expected to
be useful for models built upon copies of the Haldane model across a broad
array of disciplines. We adapt the theorem in our recent work [Phys. Rev. A
104, 013305 (2021)] to pseudo-Hermitian Hamiltonians that are less restrictive
than BdG Hamiltonians and apply it to highlight that the vanishing of an
unconventional commutator between number-conserving and number-nonconserving
parts of the Hamiltonian indicates whether a system can be made to act as a
topological amplifier.
