Krein-unitary Schrieffer-Wolff transformation and band touchings in bosonic Bogoliubov–de Gennes and other Krein-Hermitian Hamiltonians

Krein-Hermitian Hamiltonians, i.e., Hamiltonians Hermitian with respect to an indefinite inner product, have emerged as important class of non-Hermitian in physics, encompassing both single-particle bosonic Bogoliubov--de Gennes (BdG) and so-called ``$PT$-symmetric'' Hamiltonians. In particular, they attracted considerable scrutiny owing the recent surge interest for boson topology. Motivated by these developments, we formulate a perturbative Krein-unitary Schrieffer-Wolff transformation finite-size dynamically stable yielding effective Hamiltonian subspace interest. The is and, sufficiently small perturbations, also stable. As application, use this justify codimension-based analyses band touchings BdG which complement topological characterization. We simple approach based on symmetry codimension revisit known magnon several materials