  In recent years, there has been a growing interest in flatband systems which
exhibit macroscopic degeneracies. These systems offer a valuable mathematical
framework for the extreme sensitivity to perturbations and interactions. This
sensitivity unveils a wide variety of exotic and unconventional physical
phenomena. Moreover, the progress in their experimental realization contributes
to the expanding landscape of exploration in this field. This thesis aims to
summarize all the works throughout the Ph.D. program. Firstly, an in-depth
exploration was conducted on the impact of weak quasiperiodic perturbations on
one-dimensional two-band all-bands-flat lattices. These tight-binding
Hamiltonian are diagonalized through a sequence of local unitary
transformations. By adjusting the quasiperiodic potential parameters, the key
achievement involves finding a critical-to-insulator transition and identifying
fractality edges in the flatband systems with quasiperiodic perturbations.
Next, the investigation delved into the effects of on-site interactions among
hard-core bosons in one- and two-dimensional cross-stitch lattices. One key
finding is that groundstate energy primarily depends on compact localized
states. Moreover, the presence of barriers of compact localized states trap
bosons, leading to the emergence of non-ergodic excitation and Hilbert space
fragmentation. Lastly, a compact localized eigenstate of the one-dimensional
diamond chain using an electric circuit was successfully generated via local
(linear and non-linear) driving. This achievement opens up a versatile circuit
platform for the generation of flatbands and holds promise for potential
applications in the field of quantum information. I hope these collective
efforts have expanded the frontiers of the field and made a meaningful
contribution to the scientific community.
