Correlated electrons in flat bands: Concepts and Developments

When the electronic dispersion in a material is independent of momentum, it gives rise to strongly correlated flat bands, with the single particle energy, quenched. Though the notion of flat bands had been known since long, their experimental realization is achieved much later with the advent of ultra cold atomic gases, followed by photonic lattices, coordination polymers and more recently solid state materials. By the virtue of their quenched kinetic energy scales the flat band materials provide an ideal platform to engineer, customize and investigate the interplay between topology, geometry and strong electronic correlations; giving rise to exotic quantum phases such as, unconventional superconductivity, Mott insulator, non Fermi liquid metals etc. This review presents a comprehensive overview of the theoretical foundation and material realization of the many body systems with flat electronic bands. We discuss the origin of the flat bands and their mathematical construction in prototypical lattices, particularly focussing on those with Lieb and Kagome geometries. Observations made and inferences drawn based on the recent experimental and theoretical investigations are presented along with the framework for a non perturbative numerical approach to address the quantum phases in the flat band materials. By synthesizing insights from both theory and experiment, this review aims to provide a unifying perspective on the emergent many-body phenomena in flat band systems and to outline future directions for the field.