Phononic higher-order topology in two-dimensional crystalline materials has attracted significant interest in recent years. Compared with their two-dimensional counterparts, three-dimensional crystalline materials can host more diverse higher-order topological phases. However, research on phononic higher-order topology in three-dimensional crystalline materials remains limited. In this work, we reveal multiple phononic higher-order topological states in the three-dimensional crystalline materials ${\mathrm{Al}}_4{\mathrm{H}}_{12}$ and ${\mathrm{Zr}}_2{\mathrm{Ti}}_4{\mathrm{H}}_8$, in which the phononic higher-order topology can be characterized by bulk polarization. In the one-dimensional nanorods of ${\mathrm{Al}}_4{\mathrm{H}}_{12}$ and ${\mathrm{Zr}}_2{\mathrm{Ti}}_4{\mathrm{H}}_8$, the phonon topological states are localized at the surfaces and hinges. In zero-dimensional nanoparallelepipeds of ${\mathrm{Al}}_4{\mathrm{H}}_{12}$ and ${\mathrm{Zr}}_2{\mathrm{Ti}}_4{\mathrm{H}}_8$, phonon topological surface states, second-order topological hinge states, and third-order topological corner states coexist. Our work extends the dimensional scope of phononic higher-order topology in crystalline materials and paves the way for future explorations of phonon topological states.

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• Phonons
• Symmetry protected topological states
• Topological phases of matter
• Topological materials
• Density functional theory
• First-principles calculations