Symmetry of the phononic landscape of twisted kagome lattices across the duality boundary

In this Rapid Communication, we investigate the symmetry of the phonon landscape of twisted kagome lattices across their duality boundary. The study is inspired by recent work by Fruchart et al. [Nature (London) 577, 636 (2020)NATUAS0028-083610.1038/s41586-020-1932-6], who specialized the notion of duality to the mechanistic problem of kagome lattices and linked it to the existence of duality transformations between configurations that are symmetrically located across a critical point in configuration space. Our first goal is to elucidate how the existence of matching phonon spectra between dual configurations manifest in terms of observable wave-field characteristics. To this end, we explore the possibility of aggregating dual kagome configurations into bidomain lattices that are geometrically heterogeneous but retain a dynamically homogeneous response. Our second objective is to extend the analysis to structural lattices of beams, which are representative of realistic cellular metamaterials. We show that, in this case, the symmetry of the phonon landscape across the duality boundary is broken, implying that the conditions for dual behavior do not merely depend on the geometry, but also on the dominant mechanisms of the cell, suggesting a dichotomy between geometric and functional duality.