Exploring structural self-dual twisted kagome metamaterials

Fragile topological states of matter are known to lack the protective features usually associated with topology, resulting in relatively weak manifestations. The strength of these states is often influenced by the system's symmetry and boundaries, making them challenging to observe across various symmetries in phononics. These states are generally limited to specific symmetry classes and are not widely studied in the context of phononic media. However, in this study, we present a theoretical prediction of the emergence of fragile topological bands in a twisted kagome lattice in the self-dual configuration. We assume that the hinges are elastic, finite-thickness ligaments that can store bending energy. The interplay between the edge modes in the bandgaps that bound the fragile topological states leads to the emergence of localized non-trivial corner modes at certain corners of a finite domain. This property qualifies the lattice as a second-order topological insulator. To validate our findings, we conducted a series of experiments on a physical prototype using 3D Scanning Laser Doppler Vibrometry. Our results corroborate the theoretical predictions of the emergence of fragile topological states in this specific system, highlighting the potential of these states to manifest in unconventional configurations.