Engineering boundary-dominated topological states in defective hyperbolic lattices

Topological matters with lattice disclinations have been widely investigated in Euclidean space. In recent years, exotic topological phases in hyperbolic lattices, which are regular tessellations in the curved space with a constant negative curvature, have been theoretically proposed and experimentally observed, while the investigation of topologically defective hyperbolic lattices is still lacking. Here, we study topological states in the hyperbolic lattice with polygonal defects. It is shown that the topologically one-way boundary state can still exist in the defective hyperbolic Haldane model. Interestingly, we find that only a single bulk site can induce the formation of boundary-dominated one-way propagations in defective hyperbolic lattices. In experiments, we fabricate the defective hyperbolic circuit with a single bulk site to detect the boundary-dominated topological state. Frequency-dependent impedance responses clearly illustrate the existence of nontrivial band gaps and midgap topological boundary states. Furthermore, the backscattering-immune propagation protected by a single bulk site is observed by measuring the dynamics of voltage packet. Our work suggests a useful platform to study topological phases in defective hyperbolic lattices, and may have potential applications in designing high-efficient topological devices with an extremely small bulk region.