English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Tools annual

Global: Zendy Free Plan

Global: Zendy Tools monthly

Global: Zendy Plus monthly

To understand recent works on classical and quantum spin equations and theirtopological classification, we develop a unified mathematical framework forbosonic BdG systems and associated classical wave equations; it applies notjust to equations that describe quantized spin excitations in magnonic crystalsbut more broadly to other systems that are described by a BdG hamiltonian.Because here the generator of dynamics, the analog of the hamiltonian, ispara-, aka Krein-, hermitian but not hermitian, the theory of Krein spacesplays a crucial role. For systems which are thermodynamically stable, theclassical equations can be expressed as a Schr\"odinger equation with ahermitian hamiltonian. We then proceed to apply the Cartan-Altland-Zirnbauerclassification scheme: to properly understand what topological class theseequations belong to, we need to conceptually distinguish between symmetries andconstraints. Complex conjugation enters as a particle-hole constraint (asopposed to a symmetry), since classical waves are necessarily real-valued.Because of this distinction only commuting symmetries enter in the topologicalclassification. Our arguments show that the equations for spin waves inmagnonic crystals are a system of class A, the same topological class asquantum hamiltonians describing the Integer Quantum Hall Effect. Consequently,the magnonic edge modes first predicted by Shindou et al. are indeed analogs ofthe Quantum Hall Effect, and their net number is topologically protected.

The Krein-Schr\"odinger Formalism of Bosonic BdG and Certain Classical Systems and Their Topological Classification