对称保护Z2-quantization与Kramers简并的四元Berry联络
Symmetry protected Z2-quantization and quaternionic Berry connection with Kramers degeneracy
摘要：
对于具有时间反演（TR）不变性的一般参数相关哈密顿量，利用四元数Berry联络，引入了具有Kramers（KR）简并的非Abel Berry联络。这种四元数结构自然地扩展到具有KR简并度的多体系统。它的拓扑结构进行了明确的讨论相比，没有KR简并。通过给出明确的规范固定，讨论了具有非平凡拓扑结构的自然维数。最小模型有偶然的退化与/不KR退化，描述狄拉克和杨的单极。我们已经证明了杨幂实际上是一个四元数狄拉克幂。通过复/四元数Berry联系引入了具有/不具有KR简并的通用Berry相位。对于这些广义Berry相位的对称保护$\mathbb{Z}_2$量子化，给出了作为逆/反射等价的$\mathbb{Z}_2$量子化的一个充分条件.讨论了SO（3）和SO（5）非线性$σ$模型的拓扑荷与$CP^1 $和$HP^1$模型的Chern数的关系.
英文摘要：
As for a generic parameter dependent hamiltonian with the time reversal (TR) invariance, a non Abelian Berry connection with the Kramers (KR) degeneracy are introduced by using a quaternionic Berry connection. This quaternionic structure naturally extends to the many body system with the KR degeneracy. Its topological structure is explicitly discussed in comparison with the one without the KR degeneracy. Natural dimensions to have non trivial topological structures are discussed by presenting explicit gauge fixing. Minimum models to have accidental degeneracies are given with/without the KR degeneracy, which describe the monopoles of Dirac and Yang. We have shown that the Yang monopole is literally a quaternionic Dirac monopole. The generic Berry phases with/without the KR degeneracy are introduced by the complex/quaternionic Berry connections. As for the symmetry protected $\mathbb{Z}_2$ quantization of these general Berry phases, a sufficient condition of the $\mathbb{Z}_2$-quantization is given as the inversion/reflection equivalence. Topological charges of the SO(3) and SO(5) nonlinear $σ$-models are discussed in their relation to the Chern numbers of the $CP^1 $ and $HP^1$ models as well.
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