Topological acoustic triple point

Acoustic phonon is a classic example of triple degeneracy point in band structure. This triple point always appears in phonon spectrum because of the Nambu-Goldstone theorem. Here, we show that this triple point can carry a topological charge [Formula: see text] that is a property of three-band systems with space-time-inversion symmetry. The charge [Formula: see text] can equivalently be characterized by the skyrmion number of the longitudinal mode, or by the Euler number of the transverse modes. We call triple points with nontrivial [Formula: see text] the topological acoustic triple point (TATP). TATP can also appear at high-symmetry momenta in phonon and spinless electron spectrums when O_h or T_h groups protect it. The charge [Formula: see text] constrains the nodal structure and wavefunction texture around TATP, and can induce anomalous thermal transport of phonons and orbital Hall effect of electrons. Gapless points protected by the Nambu-Goldstone theorem form a new platform to study the topology of band degeneracies.