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We study a robust topological transport carried by vortices in a thin film of an easy-plane magnetic insulator between two metal contacts. A vortex, which is a nonlocal topological spin texture in twodimensional magnets, exhibits some beneficial features as compared to skyrmions, which are local topological defects. In particular, the total topological charge carried by vorticity is robust against local fluctuations of the spin order-parameter magnitude. We show that an electric current in one of the magnetized metal contacts can pump vortices into the insulating bulk. Diffusion and two-dimensional nonlocal Coulomb-like interaction between these vortices will establish a steadystate vortex flow. Vortices leaving the bulk produce an electromotive force at another contact, which is related to the current-induced vorticity pumping by the Onsager reciprocity. The voltage signal decays algebraically with the separation between two contacts, similarly to a superfluid spin transport. Finally, the vorticity and closely related skyrmion type topological hydrodynamics are generalized to arbitrary dimensions, in terms of nonsingular order-parameter vector fields.

We formulate a quantum theory of vorticity (hydro)dynamics on a general two-dimensional bosonic lattice. In the classical limit of a bosonic condensate, it reduces to conserved plasma-like vortex-antivortex dynamics. The nonlocal topological character of the vorticity flows is reflected in the bulk-edge correspondence dictated by the Stokes theorem. This is exploited to establish physical boundary conditions that realize, in the coarse-grained thermodynamic limit, an effective chemical-potential bias of vorticity. A Kubo formula is derived for the vorticity conductivity-which could be measured in a suggested practical device-in terms of quantum vorticity-flux correlators of the original lattice model. As an illustrative example, we discuss the superfluidity of vorticity, exploiting the particle-vortex duality at a bosonic superfluid-insulator transition.

Topological magnetic monopoles, also known as hedgehogs or Bloch points, are threedimensional (3D) nonlocal spin textures that are robust to thermal and quantum fluctuations due to their topology 1-4 . Understanding their properties is of both fundamental interest and practical applications 1-9 . However, it has been difficult to experimentally produce topological magnetic monopoles in a controlled manner and directly observe their 3D magnetization vector field and interactions at the nanoscale.Here, we report the creation of 138 stable topological magnetic monopoles at the specific sites of a ferromagnetic meta-lattice at room temperature. We further develop 3D soft xray vector ptychography to determine the magnetization vector and emergent magnetic field of the topological monopoles with a 3D spatial resolution of 10 nm. This spatial resolution is comparable to the magnetic exchange length of transition metals 10 , enabling us to probe monopole-monopole interactions. We find that the topological monopole pairs with positive and negative charges are separated by 18.3±1.6 nm, while the positively and negatively charged pairs are stabilized at comparatively longer distances of 36.1±2.4 nm and 43.1±2.0 nm, respectively. We also observe virtual topological monopoles created by magnetic voids in the meta-lattice. This work demonstrates that ferromagnetic metalattices could be used as a new platform to create and investigate the interactions and dynamics of topological magnetic monopoles. Furthermore, we expect that soft x-ray vector ptychography can be broadly applied to quantitatively image 3D vector fields in magnetic and anisotropic materials at the nanoscale.