Hidden Topological Edge States in Spinner Lattices

Topological insulators have sparked immense interest in applied physics and engineering communities to design structures with topologically robust responses. Here, we propose three configurations of spinner dimer lattices that comprise an array of spinners with alternating moments of inertia coupled in three different ways with their nearest neighbors. The dynamical matrices for all three configurations lack chiral symmetry if angular displacements are considered as degrees of freedom. However, we reveal that the dynamical matrices obtained by considering certain linear combinations of the angular displacements as degrees of freedom exhibit chiral symmetry. We then establish bulk-boundary correspondence and report edge states that are topologically protected by the hidden chiral symmetry interestingly for free-free boundary conditions where the latter is an aspect of transformed coordinates. Using laser Doppler vibrometry, we experimentally verify our findings through tunable tabletop designs. Our analysis can also be extended to acoustical and electrical counterparts.