Topological acoustic triple point

Abstract

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 q that is a property of three-band systems with PT symmetry, where P and T are the inversion and the time-reversal symmetries, respectively. We therefore call triple points with nontrivial q the topological acoustic triple point (TATP). The topological charge 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 symmetryprotected 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.

Topological continuum charges of acoustic phonons in 2D

“…We note that a similar analysis was recently carried out in 3D in Ref. [63], and we comment on the connection of their results to ours throughout.…”

“…The quadratic band corresponds to the out-of plane flexural mode, and it is well-known [65,68,69] in Ref. [63]. We note that the flexural band is completely decoupled from the in-plane modes.…”

“…We begin by introducing a continuum model for acoustic phonons based on classical elasticity theory in 2D [64]. The analogous model in 3D was studied in [63], and we include it for completeness in Appendix B 1 where we also discuss the model in 1D and show that it is trivial.…”

Nodal lines in momentum space: topological invariants and recent realizations in photonic and other systems

“…Each ring of the nodal chain/link can be formed by the same or a different pair of bands. Especially, if the two rings of the nodal chain originate from different set of bands in a three-band system, the three bands meet at a single point where the nodal rings touch [17,[106][107][108]. This is called a triple point.…”

“…First, the mixed shape of the nodal rings appear as earring nodal links [17,29], multiple Hopf links [25,91,95,110], mixed nodal links [29], and the linked nodal ring and a chain (Figure 2d) [31,111]. Second, the nodal lines and a nodal ring/chain can be linked to show the non-touching between nodal lines and rings [29,31,107]. Inversely, the nodal lines and nodal ring/link can be chained to show the touching between nodal lines and ring [32,94,111,112] or the touching between nodal lines and link [17,111], as shown in Figure 2e.…”