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 [1][2][3][4][5][6][7][8]. 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 [9][10][11][12][13][14]. These are reconfigurable structures that can change shape thanks to a collapse mechanism [9] 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 [15]. We show that the vibrational modes of the self-dual mechanical systems exhibit non-Abelian geometric phases [15] that affect the semi-classical propagation of wave packets [16]. 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. * fruchart@uchicago.edu

Non-Abelian generalizations of the Hofstadter model: spin–orbit-coupled butterfly pairs

“…In momentum space, non-Abelian gauge fields manifest as high-dimensional spin-orbit coupling 30,31 in atoms 32,33 , liquid crystals 34 and exciton-polaritons 35,36 . They also affect the wavepacket evolution in the momentum space of mechanical lattices 37 . In addition, non-Abelian topology can also manifest in interconnected nodal lines 38,39 and multiband charges 38,40 .…”

Roton-like acoustical dispersion relations in 3D metamaterials

“…In their work 25 , chirality has been a necessary mechanism, whereas mechanisms based on periodicity, such as ordinary or extraordinary Bragg reflections, are not accounted for in micropolar elasticity theory 24 . More broadly speaking, mechanisms such as ordinary Bragg reflection 26 , 27 , local resonances 28 – 31 , near-ideal joints 32 – 34 introducing soft modes, spatial or temporal symmetry breaking 35 – 38 , topology 39 , 40 , duality 41 , 42 , as well as geometrical nonlinearities 34 , 43 have independently given rise to a wealth of other unusual dispersion relations and quasi-static behaviors of elastic and acoustical metamaterials.…”

Self-Assembly of Isostatic Self-Dual Colloidal Crystals

“…Duality can generate a self-dual point, at which the dual counterpart of the structure is itself, and unique hidden symmetry emerges giving rise to some unusual degeneracy beyond the description of standard space-group theories [6][7][8]. Recently, a self-dual point was discovered in an isostatic mechanical network whose average coordination number is twice the system dimension, which can be used to realize the "mechanical spintronics" [9]. According to the Maxwell rule [10], this kind of structure is at the edge of mechanical stability (isostatic point) [11][12][13][14].…”