Dualities and non-Abelian mechanics

Dualities-mathematical mappings between different systems-can act as hidden symmetries that enable materials design beyond that suggested by crystallographic space groups. Dualities are mathematical mappings that reveal links between apparently unrelated systems in virtually every branch of physics(1-8). Systems mapped onto themselves by a duality transformation are called self-dual and exhibit remarkable properties, as exemplified by the scale invariance of an Ising magnet at the critical point. Here we show how dualities can enhance the symmetries of a dynamical matrix (or Hamiltonian), enabling the design of metamaterials with emergent properties that escape a standard group theory analysis. As an illustration, we consider twisted kagome lattices(9-15), reconfigurable mechanical structures that change shape by means of a collapse mechanism(9). We observe that pairs of distinct configurations along the mechanism exhibit the same vibrational spectrum and related elastic moduli. We show that these puzzling properties arise from a duality between pairs of configurations on either side of a mechanical critical point. The critical point corresponds to a self-dual structure with isotropic elasticity even in the absence of spatial symmetries and a twofold-degenerate spectrum over the entire Brillouin zone. The spectral degeneracy originates from a version of Kramers' theorem(16,17) in which fermionic time-reversal invariance is replaced by a hidden symmetry emerging at the self-dual point. The normal modes of the self-dual systems exhibit non-Abelian geometric phases(18,19) that affect the semiclassical propagation of wavepackets(20), leading to non-commuting mechanical responses. Our results hold promise for holonomic computation(21) and mechanical spintronics by allowing on-the-fly manipulation of synthetic spins carried by phonons.