Critical state generators from perturbed flatbands

One-dimensional all-bands-flat lattices are networks with all bands being flat and highly degenerate. They can always be diagonalized by a finite sequence of local unitary transformations parameterized by a set of angles θi. In a previous work, we demonstrated that quasiperiodic perturbations of a specific one-dimensional all-bands-flat lattice give rise to a critical-to-insulator transition and fractality edges separating critical from localized states. In this study, we generalize these studies and results to the entire manifold of all-bands-flat models and study the effect of the quasiperiodic perturbation on the entire manifold. For weak perturbation, we derive an effective Hamiltonian and we identify the sets of manifold parameters for which the effective model maps to extended or off diagonal Harper models and hosts critical states. For all the other parameter values, the spectrum is localized. Upon increasing the perturbation strength, the extended Harper model evolves into a system with energy dependent critical-to-insulator transitions, which we dub fractality edges. Additionally, the fractality edges are perturbation-independent, i.e., remain constant as the perturbation strength varies. The case where the effective model maps onto the off diagonal Harper model features a tunable critical-to-insulator transition at a finite disorder strength.