Two dimensional vertex-decorated Lieb lattice with exact mobility edges and robust flat bands

The mobility edge (ME) that marks the energy separating extended and localized states is a central concept in understanding the metal-insulator transition induced by disordered or quasiperiodic potentials. MEs have been extensively studied in three dimensional disorder systems and one-dimensional quasiperiodic systems. However, the studies of MEs in two dimensional (2D) systems are rare. Here we propose a class of 2D vertex-decorated Lieb lattice models with quasiperiodic potentials only acting on the vertices of a (extended) Lieb lattice. By mapping these models to the 2D Aubry-André model, we obtain exact expressions of MEs and the localization lengths of localized states, and further demonstrate that the flat bands remain unaffected by the quasiperiodic potentials. Finally, we propose a highly feasible scheme to experimentally realize our model in a quantum dot array. Our results open the door to studying and realizing exact MEs and robust flat bands in 2D systems.