Topological properties of pseudo spin-1/2 bosonic Bogoliubov-de Gennes systems with conserved magnetization in a honeycomb lattice

We carry out, within the 38-fold way for non-Hermitian systems, a quantitative study of the topological properties of a pseudo spin-1/2 bosonic Bogoliubov-de Gennes (BdG) system with conserved magnetization in a honeycomb lattice, which can be made to act as a topological amplifier with stable bulk bands but unstable edge modes.  We find it either as two copies of symmetry class AIII + η_- or two copies of symmetry class A + η depending on whether the (total) system is time-reversal-symmetric, where η is the matrix representing pseudo-Hermiticity and η_- indicates that pseudo-Hermiticity anticommutes chiral symmetry. We prove that a stable bulk is characterized by the Chern number for the Haldane model, independent of pairing interactions. We construct a simple analytical edge mode description for the Haldane model in semi-infinite planes, which is expected to be useful for any models built upon copies of the Haldane model. We adapt the theorem in our recent work [Phys. Rev. A 104, 013305 (2021)] to pseudo-Hermitian (but non-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.