Nonlinear Dynamics of Hopfion for Frequency Multiplication

Hopfions, associated with higher-dimensional topology through the Hopf fibration, exhibit {exotic} features like {complex knot} and improved stability compared to skyrmions, enhancing their appeal for innovative applications. In this paper, we study the nonlinear response of magnetic Hopfion to microwave fields. {We observe the emergence of higher-order harmonics of the driving microwave field as it interacts with the Hopfion}. By carefully selecting the {driving frequency, the corresponding harmonic can efficiently excite localized magnon state of a Hopfion. Our results demonstrate the promising potential of hopfions in nonlinear magnonics.