Most wave manipulation in metamaterials relies on abelian geometric phases -- Berry phases -- where the order of operations does not matter. Kumar, Kaufmann, and Semperlotti demonstrate elastic metamaterials that exhibit genuinely nonabelian behavior: sequentially coupling two waveguide segments produces an output that depends on the order in which they are traversed. Reversing the sequence changes the result.
The design uses multiple cylindrical waveguides connected through spatially varying coupling elements. These elements trace paths in parameter space that cross degeneracies rather than merely circling or avoiding them. This crossing is the essential departure from standard approaches. Abelian Berry phases arise from adiabatic transport around a degeneracy; Wilczek-Zee phases arise from transport within a degenerate subspace. The nonabelian holonomies in this system arise from a different mechanism -- the parameter path passes through the degeneracy point, and the resulting transformation depends on the direction and order of crossing. This produces a holonomy group that is genuinely noncommutative: the composition of two paths yields different results depending on sequence.
The practical consequence is that a single metamaterial platform can route longitudinal excitation between rods in a manner that depends on the history of the signal's path through the device. This is stable across broad frequency ranges and robust to geometric perturbations. The implications extend beyond wave routing: nonabelian operations on wave states are the physical substrate for on-material computation, where the waveguide network performs logic through the geometry of its coupling rather than through external electronic control.
(arXiv:2603.13979)