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We consider a Bogolibov-de Geenes (BdG) Hamiltonian, which is a non-HermitianHamiltonian with pseudo-Hermiticity, for a system of (pseudo) spin-$1/2$ bosonsin a honeycomb lattice under the condition that the population differencebetween the two spin components, i.e., magnetization, is a constant. Such asystem is capable of acting as a topological amplifier, under time-reversalsymmetry, with stable bulk bands but unstable edge modes which can be populatedat an exponentially fast rate. We quantitatively study the topologicalproperties of this model within the framework of the 38-fold way fornon-Hermitian systems. We find, through the symmetry analysis of the BlochHamiltonian, that this model is classified either as two copies of symmetryclass AIII+$\eta_-$ or two copies of symmetry class A+$\eta$ depending onwhether the (total) system is time-reversal-symmetric, where $\eta$ is thematrix representing pseudo-Hermiticity and $\eta_-$ indicates thatpseudo-Hermiticity and chiral symmetry operators anticommute. We prove, withinthe context of non-Hermitian physics where eigenstates obey thebi-orthonormality relation, that a stable bulk is characterized by a singletopological invariant, the Chern number for the Haldane model, independent ofpairing interactions. We construct a convenient analytical description for theedge modes of the Haldane model in semi-infinite planes, which is expected tobe useful for models built upon copies of the Haldane model across a broadarray of disciplines. We adapt the theorem in our recent work [Phys. Rev. A104, 013305 (2021)] to pseudo-Hermitian Hamiltonians that are less restrictivethan BdG Hamiltonians and apply it to highlight that the vanishing of anunconventional commutator between number-conserving and number-nonconservingparts of the Hamiltonian indicates whether a system can be made to act as atopological amplifier.

Topological study of a Bogoliubov-de Gennes system of pseudo spin-$1/2$ bosons with conserved magnetization in a honeycomb lattice