Non-orientable order and non-commutative response in frustrated metamaterials

From atomic crystals to animal flocks, the emergence of order in nature is captured by the concept of spontaneous symmetry breaking1–4. However, this cornerstone of physics is challenged when broken symmetry phases are frustrated by geometrical constraints. Such frustration dictates the behaviour of systems as diverse as spin ices5–8, confined colloidal suspensions9 and crumpled paper sheets10. These systems typically exhibit strongly degenerated and heterogeneous ground states and hence escape the Ginzburg–Landau paradigm of phase ordering. Here, combining experiments, simulations and theory we uncover an unanticipated form of topological order in globally frustrated matter: non-orientable order. We demonstrate this concept by designing globally frustrated metamaterials that spontaneously break a discrete $${{\mathbb{Z}}}_{2}$$ Z 2 symmetry. We observe that their equilibria are necessarily heteregeneous and extensively degenerated. We explain our observations by generalizing the theory of elasticity to non-orientable order-parameter bundles. We show that non-orientable equilibria are extensively degenerated due to the arbitrary location of topologically protected nodes and lines where the order parameter must vanish. We further show that non-orientable order applies more broadly to objects that are non-orientable themselves, such as buckled Möbius strips and Klein bottles. Finally, by applying time-dependent local perturbations to metamaterials with non-orientable order, we engineer topologically protected mechanical memories11–19, achieve non-commutative responses and show that they carry an imprint of the braiding of the loads’ trajectories. Beyond mechanics, we envision non-orientability as a robust design principle for metamaterials that can effectively store information across scales, in fields as diverse as colloidal science8, photonics20, magnetism7 and atomic physics21.