Critical-to-insulator transitions and fractality edges in perturbed flat bands

We study the effect of quasiperiodic perturbations on one-dimensional all-bands-flat lattice models. Such networks can be diagonalized by a finite sequence of local unitary transformations parameterized by angles θᵢ. Without loss of generality, we focus on the case of two bands with bandgap Δ. Weak perturbations lead to an effective Hamiltonian with both on- and off-diagonal quasiperiodic terms that depend on θᵢ. For some angle values, the effective model coincides with the extended Harper model. By varying the parameters of the quasiperiodic potentials, we observe localized insulating states and an entire parameter range hosting critical states with subdiffusive transport. For finite quasiperiodic potential strength, the critical-to-insulating transition becomes energy dependent with what we term fractality edges separating localized from critical states.