  The discovery of novel topological states has served as a major branch in
physics and material science. However, to date, most of the established
topological states of matter have been employed in Euclidean systems, where the
interplay between unique geometrical characteristics of curved spaces and
exotic topological phases is less explored, especially on the experimental
perspective. Recently, the experimental realization of the hyperbolic lattice,
which is the regular tessellation in non-Euclidean spaces with a constant
negative curvature, has attracted much attention in the field of simulating
exotic phenomena from quantum physics in curved spaces to the general
relativity. The question is whether there are novel topological states in such
a non-Euclidean system without analogues in Euclidean spaces. Here, we
demonstrate both in theory and experiment that novel topological states
possessing unique properties compared with their Euclidean counterparts can
exist in engineered hyperbolic lattices. Specially, based on the extended
Haldane model, the boundary-dominated first-order Chern edge state with a
nontrivial real-space Chern number is achieved, and the associated one-way
propagation is proven. Furthermore, we show that fractal-like midgap
higher-order zero modes appear in deformed hyperbolic lattices, where the
number of zero modes increases exponentially with the increase of lattice size.
These novel topological states are observed in designed hyperbolic circuit
networks by measuring site-resolved impendence responses and dynamics of
voltage packets. Our findings suggest a novel platform to study topological
phases beyond Euclidean space and may have potential applications in the field
of designing high-efficient topological devices, such as topological lasers,
with extremely fewer trivial regions.
