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In recent years, there has been a growing interest in flatband systems whichexhibit macroscopic degeneracies. These systems offer a valuable mathematicalframework for the extreme sensitivity to perturbations and interactions. Thissensitivity unveils a wide variety of exotic and unconventional physicalphenomena. Moreover, the progress in their experimental realization contributesto the expanding landscape of exploration in this field. This thesis aims tosummarize all the works throughout the Ph.D. program. Firstly, an in-depthexploration was conducted on the impact of weak quasiperiodic perturbations onone-dimensional two-band all-bands-flat lattices. These tight-bindingHamiltonian are diagonalized through a sequence of local unitarytransformations. By adjusting the quasiperiodic potential parameters, the keyachievement involves finding a critical-to-insulator transition and identifyingfractality edges in the flatband systems with quasiperiodic perturbations.Next, the investigation delved into the effects of on-site interactions amonghard-core bosons in one- and two-dimensional cross-stitch lattices. One keyfinding is that groundstate energy primarily depends on compact localizedstates. Moreover, the presence of barriers of compact localized states trapbosons, leading to the emergence of non-ergodic excitation and Hilbert spacefragmentation. Lastly, a compact localized eigenstate of the one-dimensionaldiamond chain using an electric circuit was successfully generated via local(linear and non-linear) driving. This achievement opens up a versatile circuitplatform for the generation of flatbands and holds promise for potentialapplications in the field of quantum information. I hope these collectiveefforts have expanded the frontiers of the field and made a meaningfulcontribution to the scientific community.

Quantum Phase Transitions and Dynamics in Perturbed Flatbands