Nonabelian elastic metamaterials using holonomies acquired by crossing degeneracies

Abstract

Embedding nonabelian features into elastic metamaterials promises remarkable opportunities for wave control in many practical applications such as surface acoustic wave devices, mode multiplexers, and on-material computation. Nevertheless, current realizations are limited to arrangements of coupled resonators with fine-tuned interactions, limiting their applicability to continuous media. This theoretical and numerical study introduces a design principle for continuous nonabelian elastic metamaterial waveguides. The basic configuration consists of a composite waveguide made of multiple cylindrical waveguides coupled by spatially varying elements. These elements are engineered to follow geometrically-controlled parameter variations that cross selected degeneracies and produce a targeted nonabelian holonomy. The strategy based on crossing degeneracies fundamentally differs from abelian geometric phases, where parameters avoid and encircle degeneracies, or nonabelian Wilczek-Zee phases, where parameters are fine-tuned to maintain degeneracies throughout their cycle. The resulting holonomy transfers an input longitudinal excitation in one rod to an output response in another rod. When two such waveguides are concatenated, their ordering dictates the output response, thereby revealing the emergence of nonabelian dynamics. The nonabelian behavior persists across a broad range of frequencies and under perturbations to the geometry of coupling elements or cylinder diameters. These results establish a robust, effective, and practical route to leverage nonabelian physics in elastic metamaterials.

Figures (4)

• Figure 1: (a) Schematic illustration of parameter variations (red loop) that result in a topological geometric phase. The parameter space consists of three quantities $[p_1,p_2,p_3]$. The black solid line denotes parameter values supporting degeneracies in the system. The black dot denotes the start and end point of the loop. (b) Schematic illustration of parameter variations (red loop) that result in a Wilczek-Zee phase. The parameter space consists of three quantities $[p_1,p_2,p_3]$. The gray surface denotes parameter values supporting degeneracies in the system. The black dot denotes the start and end point of the loop. (c) Schematic illustration of parameter variations (red loop $L$) that cross a degeneracy and result in a holonomy. The 3D parameter space $[h_{11},h_{22},h_{12}]$ denotes the thickness of the coupling plates in a system of two coupled rods shown in (d). Each point represents a set of three values of the thickness of the coupling plates at a cross section. Loop $L$ parametrized by $z$ corresponds to a waveguide of length $Z$ with a cyclic variation in the cross section. It comprises line $C_1$, curve $C_2$, and line $C_3$. The points on loop $L$ at $z=0,Z/8,$ and $7Z/8$ also report the associated coupling matrix as $\mathcal{K}(z)$. Perturbing $L$ to $L'$ leaves the holonomy unchanged. The yellow plane at $h_{12}=\mathrm{const}=0$ denotes uncoupled rods. The black solid line marks twofold degeneracies, for which $h_{11}=h_{22}$ and $h_{12}=0$. (d) Cross section of the composite elastic waveguide made of $N=2$ cylindrical rods.
• Figure 2: (a) A composite elastic waveguide consisting of two coupled rods. The waveguide is designed to exhibit the holonomy $\mathcal{H}_\mathrm{1,2D}$. The three stages of variations in cross-section are marked $C_1$, $C_2$, and $C_3$. Rod 1 is excited at $z=0$ (green arrow) according to its longitudinally polarized mode, and the ensuing steady-state longitudinal displacement response is shown by the colormap. The output response at $z=Z$ is localized on rod 2 (orange arrow). The displacements are normalized against the input excitation. For clarity in the visualization, the z-axis is compressed by a factor of 40. (b-f) Cross section of the waveguide at $z=0$, $z=Z/8$, $z=Z/2$, $z=7Z/8$, and $z=Z$, respectively. The colormap indicates the normalized longitudinal displacement after factoring out the dynamical phase. (g) Longitudinal displacements of rod 1 (green) and rod 2 (orange) measured on a line parallel to the axis of the corresponding cylinder and lying on its surface. The thick lines denote the components of the displacement after factoring out the dynamical phase.
• Figure 3: (a) The composite elastic waveguide WG2 comprising three coupled rods. The waveguide is designed to exhibit the holonomy $\mathcal{H}_2$. The three stages of variation of the cross-section are marked $C_1$, $C_2$, and $C_3$. Rod 1 is excited at $z=0$ (green arrow) according to its longitudinally polarized mode, and the ensuing steady-state longitudinal displacement response is shown by the colormap. The output response at $z=Z$ is localized on rod 3 (purple arrow). The displacements are normalized against the input excitation. For clarity in the visualization, the z-axis is compressed by a factor of 70. (b-d) Cross section of the waveguide at $z=0$, $z=Z/2$, and $z=Z$, respectively. The colormap indicates the normalized longitudinal displacement after factoring out the dynamical phase. (e) Longitudinal displacements of rod 1 (green), rod 2 (orange), and rod 3 (purple) measured on a line parallel to the axis of the corresponding cylinder and lying on its surface. The thick lines denote the components of the displacement after factoring out the dynamical phase.
• Figure 4: (a) The elastic waveguide WG12 consisting of waveguide WG1 followed by waveguide WG2. Rod 1 is excited at $z=0$ (green arrow, left end) according to its longitudinally polarized mode, and the ensuing steady-state longitudinal displacement response is shown by the colormap. The output response is localized on rod 1 (green arrow, right end). The displacements are normalized against the input excitation. For clarity in the visualization, the z-axis is compressed by a factor of 55. (b-d) Cross section of the waveguide at the start of WG1, at the end of WG1, and at the end of WG2, respectively. The colormap indicates the normalized longitudinal displacement after factoring out the dynamical phase. (e) Longitudinal displacements of rod 1 (green), rod 2 (orange), and rod 3 (purple) measured on a line parallel to the axis of the corresponding cylinder and lying on its surface. The thick lines denote the components of the displacement after factoring out the dynamical phase. (f) The elastic waveguide WG21 and its longitudinal displacement response when rod 1 is excited at $z=0$. For clarity in the visualization, the z-axis is compressed by a factor of 55. (g-i) Cross section of the waveguide at the start of WG2, at the end of WG2, and at the end of WG1, respectively. (j) Longitudinal displacements of the rod 1 (green), rod 2 (orange), and rod 3 (purple) measured on a line parallel to the axis of the corresponding cylinder and lying on its surface.