Duality, Hidden Symmetry, and Dynamic Isomerism in 2D Hinge Structures

Recently, a new type of duality was reported in some deformable mechanical networks that exhibit Kramers-like degeneracy in phononic spectrum at the self-dual point. In this work, we clarify the origin of this duality and propose a design principle of 2D self-dual structures with arbitrary complexity. We find that this duality originates from the partial central inversion (PCI) symmetry of the hinge, which belongs to a more general end-fixed scaling transformation. This symmetry gives the structure an extra degree of freedom without modifying its dynamics. This results in dynamic isomers, i.e., dissimilar 2D mechanical structures, either periodic or aperiodic, having identical dynamic modes, based on which we demonstrate a new type of wave guide without reflection or loss. Moreover, the PCI symmetry allows us to design various 2D periodic isostatic networks with hinge duality. At last, by further studying a 2D nonmechanical magnonic system, we show that the duality and the associated hidden symmetry should exist in a broad range of Hamiltonian systems.

Figure 1

(a)–(c) Dual transformation for a single hinge, which is composed by PCI (a)–(b) and a global 90° rotation (b)–(c), with respect to the hinge point. The arrows indicate the vibrational degrees of freedom. (d)–(g) Dynamic isomers of hinge chain with different hinge sequences.

Figure 2

(a)–(b) Directly applying the dual transformation ${\widehat{V}}_0$ to the unit cell of the periodic hinge chain reverses the wave propagation direction. (c) Phononic spectrums of the 2D periodic hinge chain at three different open angles, where $\vartheta =90°$ is the self-dual point at which the hidden symmetry induces the double degeneracy in the whole BZ.

Figure 3

By constructing different hinges in the unit cell, one can obtain various 2D periodic structures with hinge duality. (a) A minimal 2D hinge network, (b) generalized twisted Kagome, (c) dynamic isomer of twisted kagome with two hinge arms having different size, (d) $p2g$ twisted kagome, (e) snub square lattice, (f) $pmg$ lattice [14]. The corresponding bond pairs are drawn in the same color in each figure, which have the same spring constant.

Figure 4

Magnonic spectrums of $p31m$ twisted Kagome lattice at three different open angles. Note that spectrums at $\vartheta =60°$ and 120° are numerically indistinguishable.