Band representation, band connectivity and irreducibility

Abstract: The BR, formulated by Zak in the 80s, is widely used in studies of topological phase of material. EBR are considered the building block of TQC. However, there were debate on whether they admit split bands, or if they contain band invariant subspaces. This manuscript presents a clear definition of the basis and illustrate that the band transformation properties are dependent on the band connectivity. Two different Fourier transform conventions in defining the BR induced from the same set of localised Wannier functions are discussed and identified as related by a simple gauge transformation. The TB BR are used to derive the form of TB Hamiltonian using group theoretical technique. The transformation properties of EBR allow explicit decomposition in terms of irreps of space group, and use of projection operators in determining symmetry permitted band connectivity. It is shown that EBRs arising from Wyckoff positions with multiplicity can have multiple atomic limits and identifies explicitly what occurs at HSPs on the surface of BZ as what determines the band connectivity and the irreducible nature of the connected band. Such multiple configurations arise from multiple ways to block-diagonalising representation matrix at these HSPs. The symmetry permitted connectivity allows the existence of dynamic band invariant subspaces known as irreducible band representations. Such IBR, induced from Wannier functions centred on linear combination of orbits of Wyckoff positions, can also arise from interaction among composite EBRs. They are rooted in the permutation symmetry of crystallographic orbits of Wyckoff positions with multiplicity and can occur even if the parental EBR is classified as indecomposable. When symmetry permitted split EBR occurs, which does not correspond to division between connected IBRs, the topologies of the bands are potentially non-trivial on both sides of the gap.