Dualities and non-Abelian mechanics

Abstract

Dualities are mathematical mappings that reveal unexpected links between apparently unrelated systems or quantities in virtually every branch of physics. Systems that are mapped onto themselves by a duality transformation are called self-dual and they often exhibit remarkable properties, as exemplified by an Ising magnet at the critical point. In this Letter, we unveil the role of dualities in mechanics by considering a family of so-called twisted Kagome lattices. These are reconfigurable structures that can change shape thanks to a collapse mechanism easily illustrated using LEGO. Surprisingly, pairs of distinct configurations along the mechanism exhibit the same spectrum of vibrational modes. We show that this puzzling property arises from the existence of a duality transformation between pairs of configurations on either side of a mechanical critical point. This critical point corresponds to a self-dual structure whose vibrational spectrum is two-fold degenerate over the entire Brillouin zone. The two-fold degeneracy originates from a general version of Kramers theorem that applies to classical waves in addition to quantum systems with fermionic time-reversal invariance. We show that the vibrational modes of the self-dual mechanical systems exhibit non-Abelian geometric phases that affect the semi-classical propagation of wave packets. Our results apply to linear systems beyond mechanics and illustrate how dualities can be harnessed to design metamaterials with anomalous symmetries and non-commuting responses.

arXiv commentSee http://home.uchicago.edu/~vitelli/videos.html for Supplementary Movie

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• Nonlinear Topological Mechanics in Elliptically Geared Isostatic Metamaterials
• Existence of Corner Modes in Elastic Twisted Kagome Lattices
• Experimental realization of stable exceptional chains protected by non-Hermitian latent symmetries unique to mechanical systems
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• Duality of Topological Edge States in a Mechanical Kitaev Chain
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• Some results on the Guest-Hutchinson modes and periodic mechanisms of the Kagome lattice metamaterial
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• Finite-Frequency Topological Maxwell Modes in Mechanical Self-Dual Kagome Lattices
• Fully-Polarized Topological Isostatic Metamaterials in Three Dimensions
• Duality,Hidden Symmetry and Dynamic Isomerism in 2D Hinge Structures
• Electric circuit emulation of topological transitions driven by quantum statistics
• Non-singular and singular flat bands in tunable acoustic metamaterials
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• Symmetry-driven Phononic Metamaterials
• Omnidirectional domain wall modes protected by fragile topological states
• Crystal chemistry criteria of the existence of spin liquids on the kagome lattice
• Breaking Mechanical Holography in Combinatorial Metamaterials
• Cell augmentation framework for topological lattices
• Non-local twist sequences in floppy kagome chains
• Transformable Super-Isostatic Crystals Self-Assembled from Segment Colloidal Rods