Abstract: The modern understanding of quantum materials traces back to Felix Bloch, who first applied quantum mechanics to crystalline lattices in 1928 and developed his widely successful band theory of solids. In ordinary lattices, electrons form (almost) plane waves but with modified notions of momentum and energy. Recently, experiments in circuit quantum electrodynamics, electric and microwave networks, and photonics have demonstrated the coherent propagation of wave-like excitations on hyperbolic lattices – a new form of synthetic matter in which particles hop on a discrete tiling of two-dimensional hyperbolic space, a non-Euclidean space of negative curvature. Because Bloch’s ideas rely on Euclidean geometry, they have been deemed fundamentally incompatible with spatial curvature. In this talk, I will discuss a generalization of band theory and Bloch waves to hyperbolic lattices and use it to quantitatively predict and characterize novel quantum states of matter on hyperbolic lattices.
a joint institute of 
and 
and