Topological Edge Modes by Smart Patterning

The research in topological materials and meta-materials reached maturity and is now gradually entering the phase of practical applications and devices. However, scaling down the experimental demonstrations definitely presents a challenge. In this work, we study coupled identical resonators whose collective dynamics is fully determined by the pattern in which the resonators are arranged. We call a pattern topological if boundary resonant modes fully fill all existing spectral gaps whenever the pattern is halved. This is a characteristic of the pattern and is entirely independent of the structure of the resonators and the details of the couplings. Existence of such patterns is proven using $K$-theory and exemplified using a novel experimental platform based on magnetically coupled spinners. Topological meta-materials built on these principles can be easily engineered at any scale, providing a practical platform for applications and devices.