# Dualities and non-Abelian mechanics

## Abstract

Dualities are mathematical mappings that reveal links between apparently unrelated systems in virtually every branch of physics1,2,3,4,5,6,7,8. 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 lattices9,10,11,12,13,14,15, 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.

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## Data availability

No external data set was used during the current study.

## Code availability

The code used to compute the band structures and the holonomies, to perform the group-theoretical analysis, to integrate the semiclassical equations of motion and to verify the duality relations is available on Zenodo at https://doi.org/10.5281/zenodo.3417426 under the 2-clause BSD licence.

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## Acknowledgements

We thank B. Bradlyn, V. Cheianov, S. Huber, W. Irvine, P. Lidon, N. Mitchell, S. Ryu, C. Scheibner, D. Son, A. Souslov, P. Wiegmann and B. van Zuiden for discussions. V.V. was supported by the Complex Dynamics and Systems Program of the Army Research Office under grant no. W911NF-19-1-0268. M.F. was primarily supported by the Chicago MRSEC (US NSF grant DMR 1420709) through a Kadanoff–Rice postdoctoral fellowship and acknowledges partial support by the University of Chicago through a Big Ideas Generator (BIG) grant and the Netherlands Organization for Scientific Research (NWO/OCW) as part of the Frontiers of Nanoscience program. LEGO is a trademark of the LEGO Group of companies which does not sponsor, license or endorse its use in this work.

## Author information

Authors

### Contributions

M.F. and V.V. designed the research, performed the research, and wrote the paper. Y.Z. and M.F. fabricated the mechanical kagome lattices. All authors contributed to discussions and manuscript revision.

### Corresponding authors

Correspondence to Michel Fruchart or Vincenzo Vitelli.

## Ethics declarations

### Competing interests

The authors declare no competing interests.

Peer review information Nature thanks Muamer Kadic, Ronny Thomale and the other, anonymous, reviewer(s) for their contribution to the peer review of this work.

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## Supplementary information

### Supplementary Information

Supplementary Notes: contains supplementary information about the system, the derivation, and the numerical simulations.

### Video 1

Video demonstrating the zero-energy mechanism and its relation with the duality.

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Fruchart, M., Zhou, Y. & Vitelli, V. Dualities and non-Abelian mechanics. Nature 577, 636–640 (2020). https://doi.org/10.1038/s41586-020-1932-6

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