CMT Forum: Titus Neupert

Hyperbolic lattices tile the negatively curved hyperbolic plane. They recently emerged as a new paradigm of synthetic matter. A particularly appealing feature is that their energy levels are characterized by a band structure in a four-dimensional momentum space, even though the hyperbolic space is two-dimensional. I will give an overview over several results on hyperbolic lattices, including: (i) an experimental realization of a hyperbolic lattice using electrical circuits, (ii) an attempt to explore the topological aspects arising in hyperbolic band theory, by defining the hyperbolic Haldane model and the hyperbolic Kane-Mele model, (iii) a failed attempt to define four-dimensional topological states on a hyperbolic plane. The latter resulted in something way more interesting than the expected result: instead of a topological band insulator, we obtained a topological semimetal. Its semimetalicity is intricately linked to the non-Abelian nature of the translation group in hyperbolic space. These results show that hyperbolic space holds many surprises, of which even more can be expected beyond the non-interacting regime I discuss.