Mechanical Quantum Hall Effect in Time-Modulated Elastic Materials

Floquet topological insulators have inspired analogues in photonics, optics, and acoustics, in which nonreciprocal wave propagation in time-modulated materials is achieved due to the breaking of time-reversal symmetry. We investigate a mechanical wave analogue of Thouless pumping and the quantum Hall effect (QHE) in one(1D) and two-dimensional (2D) periodically time-modulated materials, respectively. In one dimension, wave propagation in the time-modulated system is characterized based on the adiabatic theorem, and topologically protected one-way edge modes are numerically demonstrated by the principle of bulk-edge correspondence. In two dimensions, a time-modulation scheme of a hexagonal lattice is suggested and polarized edge states characteristics of QHE are put into evidence by the plane wave expansion (PWE) method. The transition from the trivial state to the topological one is captured and interpreted by an invariant Chern number. We numerically demonstrate the existence of topologically protected one-way edge states immune to scattering by sharp corners, defects, randomly disordered modulation phases, and dissipation effects.