We extend the notion of topologically protected semi-metallic band crossings to hyperbolic lattices in negatively curved space. Due to their distinct translation group structure, such lattices support non-Abelian Bloch states which, unlike …

Wave functions on periodic lattices are commonly described by Bloch band theory. Besides Abelian Bloch states labeled by a momentum vector, hyperbolic lattices support non-Abelian Bloch states that have so far eluded analytical treatments. By …

Particles hopping on a two-dimensional hyperbolic lattice feature unconventional energy spectra and wave functions that provide a largely uncharted platform for topological phases of matter beyond the Euclidean paradigm. Using real-space topological …

We introduce and experimentally realize hyperbolic matter as a novel paradigm for topological states, made of particles moving in the hyperbolic plane with negative curvature. Curvature of space is emulated through a hyperbolic lattice using …

Recently, hyperbolic lattices that tile the negatively curved hyperbolic plane emerged as a new paradigm of synthetic matter, and their energy levels were characterized by a band structure in a four- (or higher-) dimensional momentum space. To …

Motivated by the recent experimental realizations of hyperbolic lattices in circuit quantum electrodynamics and in classical electric-circuit networks, we study flat bands and band-touching phenomena in such lattices. We analyze noninteracting …

Curved spaces are usually associated with high-energy physics and cosmology. However, through the possibility of tabletop experiments emulating curved spaces and the interest in related synthetic matter, they have become relevant in condensed matter physics as well.

The Laplace operator encodes the behavior of physical systems at vastly different scales, describing heat flow, fluids, as well as electric, gravitational, and quantum fields. A key input for the Laplace equation is the curvature of space. Here we …

The physics of negatively curved (hyperbolic) geometry has experienced a large resurgence of interest in recent years, both theoretically and experimentally. Although hyperbolic spaces constitute a fundamental ingredient in theoretical studies of …