Fractional Excitations in Non-Euclidean Elastic Plates

We show that minimal-surface non-Euclidean elastic plates share the same low-energy effective theory as Haldane’s dimerized quantum spin chain. As a result, such elastic plates support fractional excitations, which take the form of charge-$1/2$ solitons between degenerate states of the plate, in strong analogy to their quantum counterpart. These fractional excitations exhibit properties similar to fractional excitations in quantum fractional topological states and in Haldane’s dimerized quantum spin chain, including deconfinement and braiding, as well as unique new features such as holographic properties and diodelike nonlinear response, demonstrating great potentials for applications as mechanical metamaterials.

Figure 1

(a) The helicoid-catenoid associate family. (b),(c) fractional excitation configurations from FEA of ribbons with helicoids (b) and catenoids (c) ground states.

Figure 2

Analogous fractional excitations in (a)–(d) a dimerized spin chain and (e)–(h) a helicoid ribbon. The spin chain has two degenerate ground states (a) and (b). A spin-1 excitation can be created via local perturbations (c), which splits into two deconfined spin-$1/2$ excitations (d). (e)–(f) The two degenerate ground states with opposite chirality of a helicoid ribbon. (g) A charge-1 soliton can be created locally, and split into two charge-$1/2$ solitons (h).

Figure 3

Fractional excitations and topological degeneracy. (a)–(d) A quantum $Z_2$ spin liquid on an annulus, with two degenerate ground states (a),(b) due to topological degeneracy. (c) A spin-1 excitation is introduced via local perturbations and split into two spin-$1/2$ fractional excitations. (d) If these two fractional excitations move around the annulus and annihilated with each other, the system turns from ground state (a) to (b). (e)–(h) A catenoid with the same geometry shows the same property. The two degenerate ground states correspond to swap the two sides of the 2D manifold (e),(f). One can create two charge-$1/2$ solitons (g) and move them around the catenoid (h), flipping the catenoid to ground state (f).

Figure 4

Diodelike torque-rotation response. Three representative configurations are shown for the ground state (middle), a state with counterclockwise rotation (orange arrow) where the helicoid is tightened (right), and a state with clockwise rotation where a soliton (red arrow) is generated (left).