TY - JOUR
T1 - Krein-Schrödinger formalism of bosonic Bogoliubov-de Gennes and certain classical systems and their topological classification
AU - Lein, Max
AU - Sato, Koji
N1 - Funding Information:
The authors would like to thank Kei Yamamoto for many enjoyable discussions on this topic, which kick started this project. Both authors have been supported by JSPS (Grants No. 16K17761 and No. JP17H06460, respectively) and by a Fusion Grant from the WPI-AIMR. Moreover, we would like to thank Hosho Katsura for friendly discussions on an early draft of this work. Lastly, the comments by the two referees helped us squash typos and improve the conclusion.
Publisher Copyright:
© 2019 American Physical Society.
PY - 2019/8/7
Y1 - 2019/8/7
N2 - 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.
AB - 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.
UR - http://www.scopus.com/inward/record.url?scp=85070657449&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=85070657449&partnerID=8YFLogxK
U2 - 10.1103/PhysRevB.100.075414
DO - 10.1103/PhysRevB.100.075414
M3 - Article
AN - SCOPUS:85070657449
VL - 100
JO - Physical Review B
JF - Physical Review B
SN - 2469-9950
IS - 7
M1 - 075414
ER -