  Topological phases associated with non-Abelian charges can exhibit a
distinguished bulk-edge correspondence compared to Abelian phases, although
elucidating this relationship remains challenging in traditional solid-state
systems. In this paper, we propose a theoretical framework for synthesizing
non-Abelian quaternion charges in degenerate Fermi gases. By designing
artificial spin-orbit coupling patterns, the topological edge modes demonstrate
a clear correspondence with the band topology determined by various quaternion
charges. This paves the way for observing the interface modes whose existence
is attributed to the non-conservation multiplication relation, which is
fundamental to non-Abelian charges. This scheme can be readily implemented
using current ultracold atom techniques, offering a promising approach to
explore the intriguing non-Abelian characteristics of the system.
