Periodic Reporting for period 3 - TopMechMat (Topological Mechanical Metamaterials) Reporting period: 2021-02-01 to 2022-07-31 Summary of the context and overall objectives of the project Vibrational properties of solids are typically fixed by given material constants. However, by shaping the geometry of small building blocks out of which materials are built, one can enrich their behavior beyond the one given by the host material. One is speaking of a meta-(beyond) material. However, we are typically confronted with the problem, that we wish to design a property observable at the large scale of a material, not the properties of the smallest building blocks. Bridging between these small blocks that can be designed, and the large scale functional requirements is a challenge. With our work we address this challenge using concept from quantum mechanical low-temperature electron systems and apply them to the problem of vibrations in materials. By overcoming this challenge, we will be able to provide materials engineers with design templates that allow them to fabricate a large variety of devices. Possible applications are signal filters for WiFi and 5G communications that allow for more bandwidth at lower power levels. Other applications could be enhanced vibration isolation of large industrial facilities to reduce noise emission, or in contrast, intelligent energy harvesting out of environmental noise. Just about anything that requires exquisit control of how waves travel through materials. The overall objective is to further our basic understanding of topological band theory in the context of classical vibrations. Once we harness this design tool, the aforementioned device application could become available. Work performed from the beginning of the project to the end of the period covered by the report and main results achieved so far The main work performed towards the goal of this project was the experimental establishing of two new variants of topological bands. In a topological band, the vibrations of a material are characterized by a property that cannot be changed by small imperfections. Current research into topological condensed matter physics is exploring novel ways how this stability against imperfections is arising. It turns out, that it can always be traced back to a "knot" that is tied into the mathematical objects describing the behavior of the material. Over the last years know types of knots have been theoretically hypothesized. Our work is centered around the concrete experimental implementation of these novel knots. With this we contribute both to the further development of the theory of topological bands as well as towards the engineering of materials with novel properties. Our main results achieved so far on the fundamental side are the implementation of a higher order topological insulator as well as the first observation of a fragile topological bands. Towards applications, our key contributions have been the work on superlattices of colloidal nano-crystals as well as the design of reconfigurable metamaterials based on spiral local resonators. Progress beyond the state of the art and expected potential impact (including the socio-economic impact and the wider societal implications of the project so far) The key places where we went beyond the state of the art is in the type of topology we managed to investigate with mechanical degrees of freedom. After 40 years of research on topology in condensed matter systems, there are still theoretical advances. We have shown that these advances are not merely a theoreticians dream, but can be readily observed in classical systems. This fuels the hope that such novel effects will be transferable to applications and devices useful for everyday technology. Surfaces of an acoustic crystal.