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

Abstract

The mobility edge (ME) that marks the energy separating extended and localized states is a most important 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 the Lieb lattice or extended Lieb lattices. By mapping these models to the 2D Aubry-Andr\'e 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.