The Krein-Schroedinger Formalism of Bosonic BdG and Certain Classical Systems and Their Topological Classification
The Krein-Schroedinger Formalism of Bosonic BdG and Certain Classical Systems and Their Topological Classification
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玻色子BdG和某些经典系统的克赖因-薛定谔形式及其拓扑分类
DOI：
10.1103/physrevb.100.075414
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发表时间：
2019
影响因子：
3.7
通讯作者：
Max Lein and Koji Sato
中科院分区：
文献类型：
--
作者：
Giuseppe De Nittis and Max Lein;Giuseppe De Nittis and Max Lein;Max Lein and Koji Sato
为了理解经典和量子自旋方程及其拓扑分类的最新工作，我们为玻色子Bogoliubov-de Gennes（BdG）系统和相关的经典波动方程建立了统一的数学框架;它不仅适用于描述磁振子晶体中量子化自旋激发的方程，而且更广泛地适用于由BdG Hamilton描述的其他系统。因为这里动力学的生成元，类似于哈密顿量，是仿厄米（也称为伪厄米或克莱因厄米）而不是厄米，克莱因空间理论起着至关重要的作用。对于双稳系统，经典方程可以表示为具有厄米哈密顿量的“薛定谔方程”。然后，我们应用Cartan-Altland-Zirnbauer分类方案：为了正确理解这些方程属于什么拓扑类，我们需要在概念上区分对称性和约束。复共轭作为一种粒子-空穴约束（与对称性相反）进入，因为经典波必然是实值的。由于这种区别，只有交换对称进入拓扑分类。我们的论证表明，磁振子晶体中的自旋波方程是一个A类系统，与描述整数量子霍尔效应的量子哈密顿算子具有相同的拓扑类。因此，磁振边缘模式首先预测Shindouet等人。[Phys. Rev. B 87，174427（2013）] PRBMDO 1098 -012110.1103/PhysRevB.87.174427确实是量子霍尔效应的类似物，并且它们的净数量受到拓扑保护。
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-holeconstraint(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 Shindouet 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.