The krein-schrödinger formalism of bosonic bdg 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 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 paraaka 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 proceed to 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. [1] are indeed analogs of the Quantum Hall Effect, and their net number is topologically protected.