  Recent theoretical works on effective, four-band models of three-dimensional,
Dirac semimetals suggest the generic planes in momentum space, orthogonal to
the direction of nodal separation, and lying between two Dirac points are
higher-order topological insulators, supporting gapped, edge-states.
Furthermore, the second homotopy classification of four-band models shows the
higher-order topological insulators support quantized, non-Abelian Berry's flux
and the Dirac points are monopoles of $SO(5)$ Berry's connections. Due to the
lack of suitable computational scheme, such bulk topological properties are yet
to be determined from the \emph{ab initio} band structures of Dirac materials.
In this work, we report first, comprehensive topological classification of
\emph{ab initio} band structures of Na$_3$Bi, by computing Wilson loops of
non-Abelian, Berry's connections for several, Kramers-degenerate bands. Our
work shows the quantized, non-Abelian, Berry's flux can be used as a stable,
bulk invariant for describing higher-order topology and topological phase
transitions.
