Dualities and non-Abelian mechanics
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. Dualities—mathematical mappings between different systems—can act as hidden symmetries that enable materials design beyond that suggested by crystallographic space groups.