The geometric properties of a lattice can have profound consequences on its band spectrum. For example, geometric frustration and symmetry constraints can give rise to dispersionless (flat) and topologically nontrivial bands, respectively. In particular, flat bands indicate a quenching of kinetic energy and diverging effective mass for particles on these lattices, making flat-band lattices a rich playground for strongly interacting many-body physics. Related theoretical work has explored phenomena including ferromagnetism, flat-band many-body localization, unconventional superconductivity, and zero-magnetic-field fractional quantum Hall states. Experimentally, flat bands have been realized in photonics, superconducting circuits, and ultracold atoms, as well as in materials such as magic-angle twisted bilayer graphene.

A family of lattices called line-graph lattices, in particular, arise naturally from superconducting circuits with microwave cavity resonators. In this talk, we will explore properties of these lattices and their flat bands. This will allow us to tackle the question, "Are line-graph-lattice flat bands topological?" and catch a glimpse into the world of topological materials [1]. Our set of theoretical results motivates experimental studies of line-graph-lattices with superconducting circuits. At the same time, new directions in strongly interacting many-body flat-band systems are also within reach, with applications ranging from quantum materials synthesis [2] to quantum devices.

[1] C. S. Chiu, et al. Fragile Topology in Line-Graph Lattices with 2, 3, or 4 Gapped Flat Bands. Physical Review Research 2, 043414 (2020).

[2] C. S. Chiu, et al. Line-graph-lattice crystal structures of stoichiometric materials. arXiv: 2109.12897 (2021)