Observation of novel topological states in hyperbolic lattices

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.

Automated summary

This study demonstrates the existence of exotic topological states in engineered hyperbolic lattices, which have unique properties compared to their Euclidean counterparts. They achieved the boundary-dominated first-order Chern edge state and observed fractal-like midgap higher-order zero modes in deformed hyperbolic lattices. The findings suggest a useful platform to study topological phases beyond Euclidean space and may have potential applications in high-efficient topological devices.