Confined states and topological phases in two-dimensional quasicrystalline π -flux model

Motivated by topological equivalence between an extended Haldane model and a chiral-$\ensuremath{\pi}$-flux model on a square lattice, we apply $\ensuremath{\pi}$-flux models to two-dimensional bipartite quasicrystals with rhombus tiles in order to investigate topological properties in aperiodic systems. Topologically trivial $\ensuremath{\pi}$-flux models in the Ammann-Beenker tiling lead to massively degenerate confined states whose energies and fractions differ from the zero-flux model. This is different from the $\ensuremath{\pi}$-flux models in the Penrose tiling, where confined states only appear at the center of the bands as is the case of a zero-flux model. Additionally, Dirac cones appear in a certain $\ensuremath{\pi}$-flux model of the Ammann-Beenker approximant, which remains even if the size of the approximant increases. Nontrivial topological states with nonzero Bott index are found when staggered tile-dependent hoppings are introduced in the $\ensuremath{\pi}$-flux models. This finding suggests a direction in realizing nontrivial topological states without a uniform magnetic field in aperiodic systems.