How to fold a magnet: distorted kagome antiferromagnets as topologically frustrated origami sheets*

Frustration in condensed matter, the hard decision of choosing one ground state among many, is both delicate and robust. In frustrated antiferromagnets, its delicate side of lifting this degeneracy has been used to find exotic states of matter from nearly flat-band spin wave dispersions to quantum spin liquids of anyons. But we have only begun to understand its robust side. A recent discovery links their Hamiltonians to those of balls and springs networks now known to host topological and geometrical invariants. So the science of the robust frustration is waiting to be discovered just by mapping frustrated magnets onto metamaterials and back.

In this talk, I will present a theory of distorted kagome lattice antiferromagnets by mapping their ground states onto triangulated kirigami or origami. This map associates spin moment vectors to directed edges of triangle faces. We illustrate the theory with two ``spin-origami'' materials, ${\rm Cs}_2{\rm ZrCu}_3{\rm F}_{12}$ with flattenable (coplanar) origami ground states and ${\rm Cs}_2{\rm CeCu}_3{\rm F}_{12}$ with crumpled (non-flattenable, non-coplanar) origami. Remarkably, the zero modes are demanded by topological invariants, are exotic with a flat band in the former and a doubly degenerate topological ``Dirac