Symmetry constraints on topological invariants and irreducible band representations

EBR is considered the building block in TQC and fundamental concept in SI methods. One of the hypophysis is that a fully occupied EBR has zero Berry-Wilczek-Zee phase and those occupied corresponds to trivial topology. Associated with it are the concepts of atomic limit and equivalence between BRs. In this manuscript, an explicit link between the BWZ phase of connected bands and that of its EBR or irreducible band representation (IBR) basis is established. When gapped system occurs under the TB model, the relation between the BWZ phase of a set of connected bands and its BR basis only persist if the later are IBRs. Thus the BWZ phase can be evaluated in terms of the IBRs. Equivalent segments of path integral of BWZ connection with respect to IBRs are established as representation of the space group and selection rule for corresponding BWZ phase established where possible. The occurrence of IBRs is rooted in real space symmetry but dependent on dynamic interaction and band topology. Three gapped systems in honeycomb lattices are discussed. Two spin-less cases are shown to be topologically trivial, whereas the selection rule cannot be developed for the spin-full pz orbital as in graphene. Two necessary conditions for topologically trivial phase are established, namely 1.Connected bands having the same closed set of IBR basis for all k and, 2.The reduced tensor element for the path integral of BWZ connection for such basis is symmetry forbidden due to contractable close loop having zero BWZ phase. Thus the IBRs are the building block of topologically trivial phase and symmetry constraint on BWZ phase are obtained through IBRs and selection rules via Wigner-Eckart theorem. Some examples demonstrate that the basic hypothesis of SI method is false. The analysis here advocate a paradigm shift from EBR to IBR as building block of topologically trivial phase.