Localized interfacial phonon modes at the electronic axion domain wall

One of the most salient features of the topological state is the existence of exotic electronic modes localized at the surface or interface of a sample. In this work, we show that along with the electronic state, the phonon mode can also be localized at the domain wall between topologically trivial and non-trivial regions. We consider the electron-phonon interaction in a gapped Dirac semimetal and by treating the phonon degrees of freedom as a pseudo-gauge field, we find that the valley axion term in a gapped Dirac semimetal can influence the phonon dynamics which leads to localized interfacial phonon modes at the domain wall between trivial and non-trivial regimes (determined by the valley axion parameter). We also discuss the physical properties of this interfacial phonon mode and a possible experimental probe.