  Non-Abelian states of matter, in which the final state depends on the order
of the interchanges of two quasiparticles, can encode information immune from
environmental noise with the potential to provide a robust platform for
topological quantum computation. We demonstrate that phonons can carry
non-Abelian frame charges at the band crossing points of their frequency
spectrum, and that external stimuli can drive their braiding. We present a
general framework to understand the topological configurations of phonons from
first principles calculations using a topological invariant called Euler class,
and provide a complete analysis of phonon braiding by combining different
topological configurations. Taking a well-known dielectric material,
Al$_2$O$_3$, as a representative example, we demonstrate that electrostatic
doping gives rise to phonon band inversions that can induce redistribution of
the frame charges, leading to non-Abelian braiding of phonons. Our work
provides a new quasiparticle platform for realizable non-Abelian braiding in
reciprocal space, and expands the toolset for studying braiding processes.
