  Three-, four-, and six-fold excitations have significantly extended the
subjects of condensed matter physics. There is an urgent need for a realistic
material that can have coexisting 3-, 4-, and 6-fold excitations. However,
these materials are uncommon because these excitations in electronic systems
are usually broken by spin-orbit coupling (SOC) and normally far from the Fermi
level. Unlike the case in electronic systems, the phonon systems with
negligible SOC effect, not constrained by the Pauli exclusion principle,
provide a feasible platform to realize these excitations in a wide frequency
range. Hence, in this work, we demonstrate by first-principle calculations and
symmetry analysis that perfect 3-, 4-, 6-fold excitations appear in the phonon
dispersion rather than the band structures of Zr3Ni3Sb4, which is a well-known
indirect-gap semiconductor with an Y3Au3Sb4-type structure. This material
features 3-fold quadratic contact triple-point phonon, 4-fold Dirac point
phonon, and 6-fold point phonon. Moreover, these nodal-point phonons are very
robust to uniform strain. Two obvious phonon surface arcs of the [001] plane
are extended in the whole Brillouin zone, which will facilitate their detection
in future experimental studies. The current work provides an ideal model to
investigate the rich excitations in a single material.
