Hyperbolic Matter in topolectric circuits: a novel experimental platform for topological states of matter

We introduce the theoretical foundation and first-ever experimental realization of hyperbolic matter, a novel state of matter in curved space that multifariously contradicts Euclidean geometric intuition. It is made of particles moving in the infinite two-dimensional hyperbolic plane. Curvature of space is emulated through a hyperbolic lattice using topolectric circuit networks relying on a newly developed complex-phase circuit element. The experiment is enabled by the theoretical insights from hyperbolic band theory that momentum space of two-dimensional hyperbolic matter is four-, six- or higher-dimensional. We experimentally realize hyperbolic graphene as an example of topologically nontrivial hyperbolic matter and compare measurements of Dirac particles and Berry curvature to hyperbolic band theory. Our work sets the stage to realize interacting and quantum hyperbolic matter to challenge our established theories of Physics in curved space.