Subwavelength topological edge states in a mechanical analogy of nanoparticle chain

Abstract Recent emerge of dielectric nanoparticle chains featuring subwavelength topological states has opened unprecedented avenues for light. 
Here, we demonstrate a mechanical analogy of zigzag nanoparticle chain that supports vibrational and rotational localizations in the form of subwavelength topological edge states at extremely low frequency (near zero). We elaborate analytical methodology to thoroughly analyze the wave dynamics in the near zero-frequency regime. 
Due to weak rotational couplings, we find that motion can be efficiently confined on the boundaries of the chains. Interestingly, the vibration-rotation coupled property enables the granular chain for exotic near zero-frequency waves with spreading rotation inside the chain but localized vibration on the boundaries.
We characterize the propagation properties of elastic waves in the chain, and exhibit the fingerprints of topological edge states on the boundaries. Our study provides the possibilities for vibration control techniques using granular media at extremely low frequency.