-quantization and quaternionic Berry connection with Kramers degeneracy
Abstract
As for a generic parameter-dependent Hamiltonian with time reversal (TR) invariance, a non-Abelian Berry connection with Kramers (KR) degeneracy is introduced by using a quaternionic Berry connection. This quaternionic structure naturally extends to the many-body system with KR degeneracy. Its topological structure is explicitly discussed in comparison with the one without KR degeneracy. Natural dimensions to have nontrivial topological structures are discussed by presenting explicit gauge fixing. Minimum models to have accidental degeneracies are given with/without 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 KR degeneracy are introduced by the complex/quaternionic Berry connections. As for the symmetry-protected 
-quantization of these general Berry phases, a sufficient condition of the 
-quantization is given as the inversion/reflection equivalence.
Topological charges of the SO(3) and SO(5) nonlinear σ-models are discussed in relation to the Chern numbers of the CP1 and HP1 models as well.
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GENERAL SCIENTIFIC SUMMARY Introduction and background. The quantum spin Hall effect (QSHE) is a time-reversal (TR) invariant analogue of the quantum Hall effect (QHE). In both cases, the bulk is characterized by topological quantities and the system with edges is characterized by edge states using the bulk-edge correspondence, which is a key feature of topological insulators. Since Dyson's historic work, it has also been well established that the quaternion (extension of the complex number) is essential in the TR invariant system.
Main results. The paper presents a quaternionic Berry connection for the TR invariant system with Kramers (KR) degeneracy. This additional degeneracy is intrinsic and fundamental when the number of fermions (or spins) is odd. By using an explicit gauge fixing of the Berry connections, it clarifies that the natural dimensions of the parameter spaces without/with KR degeneracy are 2/4. As for the minimum models, it establishes a mapping (more than an analogy) between topological and geometrical quantities without/with KR degeneracy as the Dirac/Yang monopoles (and strings) in 3/5 dimensional parameter spaces, the first/second Chern numbers defined as integrals over 2/4 dimensional spheres (S2/S4) and the 
-quantization of generic Berry phases defined over integrals over S1/S3 with chiral symmetry. Also their relations to the SO(3)/SO(5) non-linear σ-models are given.
Wider implications. The results make topological objects in QSHE clear as an extension of those in QHE. Since the gauge structure of the Berry connection with KR degeneracy is Sp(1), which is equivalent to SU(2), it may give some physical insight to the SU(2) gauge theory.
Figure. Topological objects and strings (singularities) for the Dirac monopole and the Yang monopole.