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.
Highlights
Topological numbers have been important in physics especially in quantum phenomena
Its fundamental physical meaning has become clear by introducing an idea of the geometrical concept which is known as the Berry connection today[8]
We are proposing an idea of the bulk-edge correspondence, which says that the bulk is gapped and only characterized by the topological quantities, there exist characteristic boundary states which reflect the topologically non trivial bulk for the system with boundaries[7, 20]
Summary
Topological numbers have been important in physics especially in quantum phenomena. They give a conceptual foundation of quantizations for various elementary degrees of freedom such as charges, fluxes, vortices and monopoles[1, 2]. We are proposing an idea of the bulk-edge correspondence, which says that the bulk is gapped and only characterized by the topological quantities, there exist characteristic boundary states which reflect the topologically non trivial bulk for the system with boundaries[7, 20] This ”bulkedge correspondence” seems to be a universal feature of the topological ordered states such as the QH states, quantum spins[13, 21, 17], graphene[22], photonic crystals[23], cold atoms[24], characterization of localizations[25] and quantum spin Hall (QSH) systems[26, 27, 28]. As for the application of gauge invariant description of the Chern numbers, a relation between the Chern numbers and the topological charges of the SO(3) and SO(5) nonlinear σ-models are presented shortly
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