Inverse design of topological diatomic lattices based on complex phase locus

Topological phononics and acoustics have recently garnered significant attention due to their promise of a wide range of advanced wave-controlling applications, including mechanical computing, energy harvesting, and noise isolation. Topological states are vibrational modes emerging inside frequency bandgaps, and typically follow the bulk-boundary correspondence, meaning that the topological features observed at boundaries are determined by the bulk properties—the unit cell. Traditionally, topological states are characterized by analyzing the eigenvectors of the effective Hamiltonian of a given unit cell. However, this approach presents challenges when a rapid and accurate design is needed to achieve desirable topological characteristics as it often involves trial and error to obtain the ideal unit cell parameters. In this study, we propose a rigorous methodology to inversely design one-dimensional diatomic lattices based on the topological properties of complex phase loci, derived from the off-diagonal elements of the effective Hamiltonian. We discuss three representative shapes of complex phase loci: ellipse, epitrochoid, and hypotrochoid. Our methodology can be further expanded to higher dimensions, enabling more complex geometric designs for versatile topological phononic and acoustic features.