Symmetry protected Z2-quantization and quaternionic Berry connection with Kramers degeneracy

TL;DR

This paper explores the quaternionic Berry connection in time-reversal invariant systems with Kramers degeneracy, revealing new topological structures and their relation to monopoles and $2$-quantized Berry phases.

Contribution

It introduces a quaternionic Berry connection framework for systems with Kramers degeneracy, extending topological analysis and linking monopoles to quaternionic structures.

Findings

Quaternionic Berry connection naturally extends to many-body systems with Kramers degeneracy.

Yang monopole is shown to be a quaternionic Dirac monopole.

Conditions for $2$-quantization of Berry phases are identified as inversion/reflection symmetry.

Abstract

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 $\sigma $-models are discussed in their relation to the Chern numbers of the $CP^1 $ and $HP^1$ models as well.