Space and space-time topologies in a type-II hyperbolic lattice.

Recent breakthroughs in hyperbolic lattices have expanded the study of topological phases of matter from Euclidean to non-Euclidean spaces. However, prior work has mostly focused on static spatial topological states at the single outer edge of type-I hyperbolic lattices. The dynamic transfer of hyperbolic topological states across multiple edges, as well as the emergence of spatiotemporal topological phenomena, remains largely unexplored. Here, we establish both space and space-time topologies in a newly discovered type-II hyperbolic lattice possessing outer and inner edges. Using electric circuits, we experimentally realize a type-II hyperbolic Chern insulator and directly observe degenerate chiral edge states (CESs) of opposite chirality at its outer and inner edges. Furthermore, by dynamically modulating the coupled counterpropagating CESs, we demonstrate arbitrary-proportion dynamic transfer of the CESs during modulation from an anti-parity-time-broken phase to a quasi-exceptional point. Finally, we propose a novel paradigm for constructing a (2 + 1)-dimensional hyperbolic space-time crystal that hosts an intertwined topology of spatial Chern and temporal winding numbers, yielding a unique space-time topological string state. Our work expands the frontier of hyperbolic topological physics, paving the way for the spatiotemporal dynamic manipulation of hyperbolic topological states.