Critical-to-Insulator Transitions and Fractality Edges in Perturbed Flatbands

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 $\theta_i$. Without loss of generality, we focus on the case of two bands with bandgap $\Delta$. Weak perturbations lead to an effective Hamiltonian with both on- and off-diagonal quasiperiodic terms that depend on $\theta_i$. For some angle values, the effective model coincides with the extended Harper model. By varying the parameters of the quasiperiodic potentials, \iffalse and the manifold angles $\theta_i$ \fi 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.