  Materials with triply-degenerate nodal points in their low-energy electronic
spectrum produce crystalline-symmetry-enforced three-fold fermions, which
conceptually lie between the two-fold Weyl and four-fold Dirac fermions. Here
we show how a silver-based Dirac semimetal BaAgAs realizes three-fold fermions
through our first-principles calculations combined with a low-energy effective
$\mathbf{k.p}$ model Hamiltonian analysis. BaAgAs is shown to harbor
triply-degenerate nodal points, which lie on its $C_{3}$ rotation axis, and are
protected by the $C_{6v}$($C_2\otimes C_{3v}$) point-group symmetry in the
absence of spin-orbit coupling (SOC) effects. When the SOC is turned on, BaAgAs
transitions into a nearly-ideal Dirac semimetal state with a pair of Dirac
nodes lying on the $C_{3}$ rotation axis. We show that breaking inversion
symmetry in the BaAgAs$_{1-x}$P$_x$ alloy yields a clean and tunable three-fold
fermion semimetal. Systematic relaxation of other symmetries in BaAgAs
generates a series of other topological phases. BaAgAs materials thus provide
an ideal platform for exploring tunable topological properties associated with
a variety of different fermionic excitations.
