Topological state switches in hard-magnetic meta-structures

We propose a metamaterial design principle that enables the remote switching of topological states. Dynamic breaking of space-inversion symmetry is achieved through the intricate design of magnetic spring structures within the metamaterial building blocks, whose stiffness can be remotely altered using an external magnetic field. We develop a mathematical model to predict the magnetic field-induced deformation and tangential stiffness of the spring structure with hard-magnetic constituent phase. Building on the predictive model, we explore the necessary conditions – including the magnetization distribution and the direction of the actuating magnetic field – that enable magnetically tunable stiffness. To demonstrate the functionality of topological state switching, we apply the proposed magnetic spring to the topological metamaterial design where a tunable stiffness landscape is essential for reversible topological phase transition. Our mathematical modeling indicates that we can remotely modulate both the dispersion properties and the topological invariants (including Zak phase and winding number) of the underlying bands in the proposed metamaterial system. Finally, we show that this tunable capability extends to controlling topologically protected edge and interface states within the finite-sized metamaterial lattice. Our design strategy for the switching of topological state paves the way for the realization of smart and intelligent metamaterials featuring tunable and active wave dynamics. It also highlights the potential of magneto-mechanical coupling in the design of advanced functional materials.