Dualities and non-Abelian mechanics

Michel Fruchart (), Yujie Zhou and Vincenzo Vitelli ()
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Michel Fruchart: James Franck Institute, University of Chicago
Yujie Zhou: Universiteit Leiden
Vincenzo Vitelli: James Franck Institute, University of Chicago

Nature, 2020, vol. 577, issue 7792, 636-640

Abstract: Abstract Dualities are mathematical mappings that reveal links between apparently unrelated systems in virtually every branch of physics1–8. 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 lattices9–15, reconfigurable mechanical structures that change shape by means of a collapse mechanism9. 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’ theorem16,17 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 phases18,19 that affect the semiclassical propagation of wavepackets20, leading to non-commuting mechanical responses. Our results hold promise for holonomic computation21 and mechanical spintronics by allowing on-the-fly manipulation of synthetic spins carried by phonons.

Date: 2020
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DOI: 10.1038/s41586-020-1932-6

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