The sheet of constant positive curvature – a sphere – arises prominently in the physical world around us. For example, standing waves on a spherical membrane directly relate to quantum numbers of atomic orbitals, impacting the structure of the periodic table of elements. In contrast, the sheet of constant negative curvature, called hyperbolic plane, is hard to realize experimentally. The underlying reason, proved by Hilbert over a century ago, is that the hyperbolic plane cannot be embedded as a smooth manifold in the three-dimensional Euclidean space. Accordingly, models of dynamics in negatively curved spaces have long remained an abstract topic with little relevance for concrete experiments.
Recent experimental works with coupled microwave resonators and with electric circuits overcame the above challenges, and successfully realized models that are naturally interpreted as residing inside the hyperbolic plane. In this seminar, I will first illuminate how we utilized electric-circuit network, patterned according to a regular triangulation of the hyperbolic plane, to simulate a “hyperbolic drum”, i.e., a disk with constant negative curvature [1]. The negative curvature of the set-up is evidenced both in static and in dynamical experiments. I will subsequently discuss how the recently formulated hyperbolic band theory [2] can be utilized to formulate elementary models of topological band insulators on hyperbolic lattices [3], ushering a way towards topological hyperbolic matter.
[1] P. M. Lenggenhager, A. Stegmaier, et al., Electric-circuit realization of a hyperbolic drum, arXiv: 2109.01148 (2021)
[2] J. Maciejko and S. Rayan, Hyperbolic band theory, Sci. Adv. 7, abe9170 (2021)
[2] D. M. Urwyler, P. M. Lenggenhager, I. Boettcher, R. Thomale, T. Neupert, and T. Bzdušek, Hyperbolic topological band insulators, arXiv:2203.07292 (2022)
