Flat bands and band touching in hyperbolic lattices

Recent years have witnessed much interest in condensed matter systems that exhibit flat bands, i.e., energy bands of a tight-binding Hamiltonian that are independent of the crystal momentum. Since the ratio of interaction potential energy to kinetic energy diverges for a dispersionless band, flat-band systems are fertile grounds to engineer exotic many-body states, including ferromagnetism, superconductivity, Wigner crystallization, and the fractional quantum Hall effect. Hopping models on geometrically frustrated lattices, such as the kagome and dice lattices, exhibit perfectly flat energy bands that result from eigenstates localized along contractible and/or noncontractible loops in real space. The key properties of those flat bands, such as their degeneracy and possible touchings with other dispersive bands, follow from an intriguing interplay between real-space topology and momentum-space band theory. In this talk, I will discuss how to generalize this paradigm to frustrated hyperbolic lattices, such as those realized in recent circuit quantum electrodynamics experiments, which may open new avenues for many-body physics with synthetic quantum systems. I will explain how hyperbolic flat-band characteristics can similarly be understood by combining real-space topology arguments with the recently developed hyperbolic band theory.