Rayleigh waves and cyclotron surface modes of gyroscopic metamaterials

We investigate the elastic normal modes of two-dimensional media with broken time-reversal and parity symmetries due to a Lorentz term. Our starting point is an elasticity theory that captures the low-energy physics of a diverse range of systems such as gyroscopic metamaterials, skyrmion lattices in thin-film chiral magnets, and certain Wigner crystals. By focusing on a circular disk geometry, we analyze finite-size effects and study the low-frequency shape oscillations of the disk. We demonstrate the emergence of the Rayleigh surface waves from the bottom of the excitation spectrum and investigate how the curvature of the disk-boundary modifies their propagation at long wavelengths. Moreover, we discover a near-cyclotron-frequency wave that is almost completely localized at the boundary of the disk but is distinct from the Rayleigh wave. It can be distinguished from the latter by a characteristic excitation pattern in a small region near the center of the disk.