Fragile topological phase on the triangular kagome lattice and its bulk-boundary correspondence

We predict and examine various topological states on a two-dimensional (2D) triangular kagome lattice (TKL) using the tight-binding (TB) models theory of quantum chemistry (TQC). Firstly, basis TQC, we diagnose band structures with fragile topology calculate Wilson-loop spectra Hofstadter butterfly to confirm their non-trivial nature. Secondly, bulk-boundary correspondence find that an obstructed-atomic-limit (OAL) insulator hosts fractional corner without being accompanied by structures, which implies presence OALs is not sufficient condition topology. Last but least, phase transition from second-order first-order can be realized in TKL under action magnetic field.