Krein-Schrödinger formalism of bosonic Bogoliubov-de Gennes and certain classical systems and their topological classification

To understand recent works on classical and quantum spin equations and their topological classification, we develop a unified mathematical framework for bosonic Bogoliubov-de Gennes (BdG) systems and associated classical wave equations; it applies not just to equations that describe quantized spin excitations in magnonic crystals but more broadly to other systems that are described by a BdG Hamiltonian. Because here the generator of dynamics, the analog of the Hamiltonian, is para-Hermitian (also known as pseudo- or Krein-Hermitian) but not Hermitian, the theory of Krein spaces plays a crucial role. For systems which are thermodynamically stable, the classical equations can be expressed as a "Schrödinger equation" with a Hermitian Hamiltonian. We then apply the Cartan-Altland-Zirnbauer classification scheme: To properly understand what topological class these equations belong to, we need to conceptually distinguish between symmetries and constraints. Complex conjugation enters as a particle-hole constraint (as opposed to a symmetry), since classical waves are necessarily real-valued. Because of this distinction, only commuting symmetries enter in the topological classification. Our arguments show that the equations for spin waves in magnonic crystals are a system of class A, the same topological class as quantum Hamiltonians describing the integer quantum Hall effect. Consequently, the magnonic edge modes first predicted by Shindou et al. [Phys. Rev. B 87, 174427 (2013)]PRBMDO1098-012110.1103/PhysRevB.87.174427 are indeed analogs of the quantum Hall effect, and their net number is topologically protected.