Finetuning localization in interacting flatband networks

Linear wave equations on translationally invariant flatband networks exhibit one or more dispersionless bands in their Bloch spectrum, and host compact localized eigenstates (CLS) with nonzero amplitudes restricted to a finite volume. While yielding remarkable single particle localization features, CLSs are typically highly sensitive to perturbations (e.g. disorder, external fields, interaction). However, the impact of such perturbations strongly depend on the chosen flatband network. In this talk, we focus on the case of interacting flatband networks, and overview finetuning protocols involving the interaction terms (both classical and quantum) and the network geometry which yield diverse localization features. We show how these finetuning protocols result in spatially compact time periodic breather solutions in the classical nonlinear case, and in the emergence of an extensive number of local integrals of motions in the quantum case, yielding signatures of disorder-free many-body localization. We also discuss the case of lattices lacking dispersion (i.e. all bands are flat) where finetuning leads to the complete suppression of transport.