Dualities and non-Abelian mechanics

TL;DR

This paper explores the role of dualities in mechanical systems, specifically in reconfigurable twisted Kagome lattices, revealing how self-dual structures exhibit unique vibrational properties and non-Abelian geometric phases.

Contribution

It uncovers a duality-based mechanism explaining vibrational degeneracies in self-dual mechanical lattices and demonstrates the emergence of non-Abelian geometric phases in these systems.

Findings

Pairs of configurations share the same vibrational spectrum due to duality.

Self-dual structures exhibit two-fold degeneracy across the Brillouin zone.

Non-Abelian geometric phases influence wave propagation in these systems.

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.