  Acoustic phonon in a crystalline solid is a well-known and ubiquitous example
of elementary excitation with a triple degeneracy in the band structure.
Because of the Nambu-Goldstone theorem, this triple degeneracy is always
present in the phonon band structure. Here, we show that the triple degeneracy
of acoustic phonons can be characterized by a topological charge $\mathfrak{q}$
that is a property of three-band systems with $\mathcal{PT}$ symmetry, where
$\mathcal{P}$ and $\mathcal{T}$ are the inversion and the time-reversal
symmetries, respectively. We therefore call triple points with nontrivial
$\mathfrak{q}$ the topological acoustic triple point (TATP). The topological
charge $\mathfrak{q}$ can equivalently be characterized by the skyrmion number
of the longitudinal mode, or by the Euler number of the transverse modes, and
this strongly constrains the nodal structure around the TATP. The TATP can also
be symmetry-protected at high-symmetry momenta in the band structure of phonons
and spinless electrons by the $O_h$ and the $T_h$ groups. The nontrivial
wavefunction texture around the TATP can induce anomalous thermal transport in
phononic systems and orbital Hall effect in electronic systems. Our theory
demonstrates that the gapless points associated with the Nambu-Goldstone
theorem are an avenue for discovering new classes of degeneracy points with
distinct topological characteristics.
