Topological Elasticity of Nonorientable Ribbons

In this article, we unravel an intimate relationship between two seemingly unrelated concepts: elasticity, that defines the local relations between stress and strain of deformable bodies, and topology, that classifies their global shape. Focusing on Mobius strips, we establish that the elastic response of surfaces with nonorientable topology is nonadditive, nonreciprocal, and contingent on stress history. Investigating the elastic instabilities of nonorientable ribbons, we then challenge the very concept of bulk-boundary correspondence of topological phases. We establish a quantitative connection between the modes found at the interface between inequivalent topological insulators and solitonic bending excitations that freely propagate through the bulk nonorientable ribbons. Beyond the specifics of mechanics, we argue that non-orientability offers a versatile platform to tailor the response of systems as diverse as liquid crystals and photonic and electronic matter.