Anomalous topological pumping in hyperbolic lattices

Hyperbolic lattices-non-Euclidean regular tilings with constant negative curvature-provide a unique framework to explore curvature-driven topological phenomena inaccessible in flat geometries. While recent advances have focused on static hyperbolic systems, the dynamical interplay between curved space and time-modulated topology remains uncharted. Here, we study the topological pumping in hyperbolic lattices, discovering anomalous phenomena with no Euclidean analogs. Notably, 2D hyperbolic pumping emulates 8D quantum Hall physics, transcending conventional dimensional constraints. We further demonstrate that pumping trajectories are governed by a synergy of Chern numbers (1st to 4th) and periodic boundary condition (PBC) configurations. Remarkably, specific PBCs trigger a periodic topological oscillation, where quantized transport collapses into time-recurrent cycles. Experimentally, time-modulated hyperbolic circuits validate both high-dimensional quantum Hall signatures and PBC-dependent topological dynamics. Our work pioneers the exploration of topological pumping in hyperbolic lattices, showcasing the transformative impact of non-Euclidean geometry on topological phenomena.