Hyperbolic Non-Abelian Semimetal.

We extend the notion of topologically protected semi-metallic band crossings to hyperbolic lattices in a negatively curved plane. Because of their distinct translation group structure, such lattices are associated with a high-dimensional reciprocal space. In addition, they support non-Abelian Bloch states which, unlike conventional Bloch states, acquire a matrix-valued Bloch factor under lattice translations. Combining diverse numerical and analytical approaches, we uncover an unconventional scaling in the density of states at low energies, and illuminate a nodal manifold of codimension five in the reciprocal space. The nodal manifold is topologically protected by a nonzero second Chern number, reminiscent of the characterization of Weyl nodes by the first Chern number.

Quantum Holography in a Graphene Flake with an Irregular Boundary.

Electrons in clean macroscopic samples of graphene exhibit an astonishing variety of quantum phases when strong perpendicular magnetic field is applied. These include integer and fractional quantum Hall states as well as symmetry broken phases and quantum Hall ferromagnetism. Here we show that mesoscopic graphene flakes in the regime of strong disorder and magnetic field can exhibit another remarkable quantum phase described by holographic duality to an extremal black hole in two-dimensional anti-de Sitter space. This phase of matter can be characterized as a maximally chaotic non-Fermi liquid since it is described by a complex fermion version of the Sachdev-Ye-Kitaev model known to possess these remarkable properties.

Hyperbolic matter in electrical circuits with tunable complex phases.

Curved spaces play a fundamental role in many areas of modern physics, from cosmological length scales to subatomic structures related to quantum information and quantum gravity. In tabletop experiments, negatively curved spaces can be simulated with hyperbolic lattices. Here we introduce and experimentally realize hyperbolic matter as a paradigm for topological states through topolectrical circuit networks relying on a complex-phase circuit element. The experiment is based on hyperbolic band theory that we confirm here in an unprecedented numerical survey of finite hyperbolic lattices. We implement hyperbolic graphene as an example of topologically nontrivial hyperbolic matter. Our work sets the stage to realize more complex forms of hyperbolic matter to challenge our established theories of physics in curved space, while the tunable complex-phase element developed here can be a key ingredient for future experimental simulation of various Hamiltonians with topological ground states.