  Topological phases of matter lie at the heart of physics, connecting elegant
mathematical principles to real materials that are believed to shape future
electronic and quantum computing technologies. To date, studies in this
discipline have almost exclusively been restricted to single-gap band topology
because of the Fermi-Dirac filling effect. Here, we theoretically analyze and
experimentally confirm a novel class of multi-gap topological phases, which we
will refer to as non-Abelian topological semimetals, on kagome geometries.
These unprecedented forms of matter depend on the notion of Euler class and
frame charges which arise due to non-Abelian charge conversion processes when
band nodes of different gaps are braided along each other in momentum space. We
identify such exotic phenomena in acoustic metamaterials and uncover a rich
topological phase diagram induced by the creation, braiding and recombination
of band nodes. Using pump-probe measurements, we verify the non-Abelian charge
conversion processes where topological charges of nodes are transferred from
one gap to another. Moreover, in such processes, we discover symmetry-enforced
intermediate phases featuring triply-degenerate band nodes with unique
dispersions that are directly linked to the multi-gap topological invariants.
Furthermore, we confirm that edge states can faithfully characterize the
multi-gap topological phase diagram. Our study unveils a new regime of
topological phases where multi-gap topology and non-Abelian charges of band
nodes play a crucial role in understanding semimetals with inter-connected
multiple bands.
