Double-leaf Riemann surface topological converse magnetoelectricity

Electric field control of magnetism in solids, i.e. the converse magnetoelectricity, is highly desired for applications of scalable energy-efficient logic devices. However, it is not only a technical challenge but also a scientific paradox, since in principle the electric and magnetic degrees of freedom obey distinct rules of symmetries. Despite the great progresses obtained in the community of multiferroics during the past decades, the success of magnetoelectricity remains on its way and more alternative approaches with conceptual revolution are urgently needed. Here, by introducing the concept of topology into multiferroics, an exotic magnetoelectric double-leaf Riemann-surface is unveiled based on the mechanism of spin-dependent $d-p$ hybridization in a two-dimensional magnet: GdI$_2$ monolayer. Protected by the topology, a $180^\circ$ spin reversal can be precisely achieved by an electric cycle, leading to a robust and dissipationless converse magnetoelectric function. Such a topological magnetoelectricity allows the nontrivial manipulation of magnetization by AC electric field. In this category, more candidate materials with better performance are designed targetedly, which pave the road to the potential applications with topological magnetoelectrics.