Linking extended vector wave fields with momentum space topology
Topology describes properties of physical systems that remain constant under continuous deformations. For infinite vector waves, global topological invariants in position space are typically associated with periodic patterns. We demonstrate that even for aperiodic Helmholtz-decomposable wave fields, possessing only the wave's intrinsic periodicity, a topological invariant can be found in momentum space. This invariant, the linking number, represents a Berry phase. By utilizing electromagnetic and hydrodynamic surface waves, we confirm the robustness of the linking number against deformations, and experimentally observe discrete transitions between distinct topological sectors. The linking number captures the topology of vector wave fields across both continuous and discrete momentum spaces. Our work introduces a unified topological framework for vector wave fields, enabling their classification via a global invariant.