The most salient feature of electronic topological states of matter is the existence of exotic electronic modes localized at the surface or interface of a sample. In this work, in an electronic topological system, we demonstrate the existence of localized phonon modes at the domain wall between topologically trivial and nontrivial regions, in addition to the localized interfacial electronic states. In particular, we consider a theoretical model for the Dirac semimetal with a gap opened by external strains and study the phonon dynamics, which couples to electronic degrees of freedom via strong electron-phonon interaction. By treating the phonon modes as pseudogauge fields, we find that the axion type of terms for phonon dynamics can emerge in gapped Dirac semimetal model and lead to interfacial phonon modes localized at the domain wall between trivial and nontrivial regimes that possess the axion parameters 0 and $\pi$, respectively. We also discuss the physical properties and possible experimental probe of such interfacial phonon modes.

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• Acoustic phonons
• Electron-phonon coupling
• Topological materials
• Topological phases of matter