Combinatorics

Physics

Euclidean geometry

Topological entropy in physics

Topology (electrical circuits)

Acoustics

Geometry

Lattice (music)

Mathematics

Topological order

Quantum mechanics

Topological quantum number

Hyperbolic space

Quantum

Nature Communications

The discovery of novel topological states has served as a major branch in physics and material sciences. To date, most of the established topological states have been employed in Euclidean systems. Recently, the experimental realization of the hyperbolic lattice, which is the regular tessellation in non-Euclidean space with a constant negative curvature, has attracted much attention. Here, we demonstrate both in theory and experiment that exotic topological states can exist in engineered hyperbolic lattices with unique properties compared to their Euclidean counterparts. Based on the extended Haldane model, the boundary-dominated first-order Chern edge state with a nontrivial real-space Chern number is achieved. Furthermore, we show that the fractal-like midgap higher-order zero modes appear in deformed hyperbolic lattices, and the number of zero modes increases exponentially with the lattice size. These novel topological states are observed in designed hyperbolic circuit networks by measuring site-resolved impedance responses and dynamics of voltage packets. Our findings suggest a useful platform to study topological phases beyond Euclidean space, and may have potential applications in the field of high-efficient topological devices, such as topological lasers, with enhanced edge responses. Recent evidence of hyperbolic lattice calls for whether topological states can exist in such non-Euclidean system. Here, the authors evidence firstorder Chern edge states with a nontrivial real-space Chern number and fractal-like midgap higher-order zero modes in hyperbolic circuit networks.

Observation of novel topological states in hyperbolic lattices