In recent years, the use of topology has deepened our understanding of collective phases of various waves, from electrons to magnons. I will focus on the mechanics of metamaterials. Can topology be of any use in this context? In return, can we learn general properties of topological phases from their mechanical counterpart?
I will show that, while the use of topology to characterize mechanical metamaterials can be traced back to a seminal work of Maxwell, its recent reformulation allows for a direct experimental measurement of the topological nature of a mechanical configuration [1]. This measurement is unique: it does not require any a priori theoretical description of a metamaterial, yet it allows to predict the location of low energy deformations and self-stresses.
I will then turn to a topological property a priori specific to mechanics, characterizing the deformation of a non-orientable Möbius strip [2]. I will show that the buckling modes of such a strip are constrained to vanish along the ribbon. This property challenges the very concept of bulk-boundary-correspondence of topological phases, by embedding an edge state in the bulk of a topological phase. I will then show that the non-orientable mechanics of this Möbius strip is not restricted to non-orientable metamaterials, but extends to various frustrated phases of matter beyond the realm of mechanics [3].
[1] M. Guzman et al., PNAS 121, e2305287121 (2024)
[2] D. Bartolo and D. Carpentier, Phys. Rev. X 9, 041058 (2019)
[3] X. Guo et al., Nature 618, p.506–512 (2023)
