Electronic flat band systems have gained considerable attention as a platform to realize quantum matter phenomena including the fractional quantum Hall effect, topological phases, unconventional superconductivity, and new forms of magnetism. Electrons in a flat band are characterized by a quenched kinetic energy (bandwidth), a diverging effective mass, and localized wave functions, leading to strong correlation physics in the presence of Coulomb interactions. In recent years, significant efforts have been directed at realizing electronic flat bands in special ‘line graph’ lattices, which under certain conditions exhibit eigenstates that are spatially ‘trapped’ by the total destructive interference of hopping pathways (compact localized states). The realization of new flat band systems and their tunability through external parameters like chemical composition, carrier doping, or lattice geometry (dimensionality) is an exciting frontier for engineering quantum phases of matter at potentially high temperatures.
In this talk, I will discuss how flat bands can be realized and tuned in two paradigmatic line graph systems – the kagome and pyrochlore lattice. We use angle-resolved photoemission spectroscopy (ARPES) as a probe of the electronic band structure in crystalline materials hosting these lattices. I will first discuss our observation of two-dimensional topological flat bands in kagome intermetallic compound CoSn  and how their quantum geometric properties can be visualized using ARPES . Next, I will present evidence for one-dimensional flat bands giving rise to heavy fermion-like behavior in kagome metal Ni3In . Last, I will show recent work on Laves phase AB2 compounds hosting a B-site pyrochlore sublattice, showing the realization of multiple three-dimensional flat bands in metallic CaNi2 and superconducting CaRh2 with flat bands near the Fermi energy .
 M. Kang, et al., Topological flat bands in frustrated kagome lattice CoSn. Nature Comm. 11, 4004 (2020).
 M. Kang, et al., Measurements of the quantum geometric tensor in solids, under review.
 L. Ye, et al., A flat band-induced correlated kagome metal, https://arxiv.org/abs/2106.10824.
 J. Wakefield, et al., Three Dimensional Flat Bands in Pyrochlore Metal CaNi2, under review.