Polar Hopf rings emerge in antiferroelectrics

Abstract

Hopfions are widely involved in topology, group theory, optics, etc. Recently Hopfion rings have been successfully characterized in a magnet, which makes them promising objects for electronic information applications. Here, we reveal that high-density polar transitions at domain boundaries are a condition for topological domain generation. Based on this, we investigate polar Hopf rings in the incommensurate antiferroelectric, which consist of two intertwined vortices and are mathematically equivalent to a$2_1^2$link in topology. The superposition of three-dimensional high-frequency polar waves with different orientations at domain boundaries is responsible for the stability of polar Hopf rings. Furthermore, we demonstrate the reversible switching between polar Hopf rings and trivial domain by an electron beam. Our findings of atomic-scale Hopf rings may provide direction for the design of next-generation high-density electronic information devices.