Magnetoelectric topology: The rope weaving in parameter space

Topology, as a mathematical concept, has been introduced into condensed matter physics since the discovery of quantum Hall effect, which characterizes new physical scenario beyond the Landau theory. The topologically protected physical quantities, such as the dissipationless quantum transport of edge/surface states as well as magnetic/dipole quasi-particles like skyrmions/bimerons, have attracted great research enthusiasms in the past decades. In recent years, another kind of topology in condensed matter was revealed in the magnetoelectric parameter space of multiferroics, which deepens our understanding of magnetoelectric physics. This topical review summarizes recent advances in this area, involving three types of type-II multiferroics. With magnetism-induced ferroelectricity, topological behaviors can be manifested during the magnetoelectric switching processes driven by magnetic/electric fields, such as Roman-surface/Riemann-surface magnetoelectricity and magnetic crankshaft. These exotic topological magnetoelectric behaviors may be helpful to pursue energy-efficient and precise-control devices for spintronics and quantum computing.