Spin-orbit coupling that connects a charge current and spin dynamics has been shown to enable hydrodynamics of topological chiral solitons in various materials including magnets and superconductors. In this talk, we will discuss two examples in this exciting direction. The first example is the topological hydrodynamics of magnetic domain walls in easy-plane magnets with additional in-plane anisotropy. We will discuss how we can use spin-orbit coupling to pump domain walls into magnets and, reciprocally, detect a domain-wall current. The second example is spin transport by a vortex liquid in superconductors, in which spin of a magnet is connected to charge of a superconductor via spin-orbit coupling, and the charge of the superconductor is connected to a vortex via charge-vortex relation. We envision that interfaces between two different orders can serve as hosts of a variety of couplings between seemingly non-related physical quantities.

[1] S. K. Kim, S. Takei, and Y. Tserkovnyak, “Topological spin transport by Brownian diffusion of domain walls,” Phys. Rev. B 92, 220409(R) (2015)
[2] S. K. Kim, R. Myers, and Y. Tserkovnyak, "Nonlocal Spin Transport Mediated by a Vortex Liquid in Superconductors," Phys. Rev. Lett. 121, 187203 (2018)