Phononic higher-order topology induced by strain or twist in HfTe2

Higher-order phononic topology has garnered significant attention in recent years, with many novel phonon topological modes that have been unveiled. However, the exploration of efficient control of phononic higher-order topology in crystalline materials is rarely reported. In the present work, we demonstrate that strain and twist can effectively manipulate the phonon topological properties of materials. Specifically, by applying strain to the $1T\text{\ensuremath{-}}{\mathrm{HfTe}}_{2}$ monolayer and twisting the AAA-stacked (all three layers are perfectly aligned without relative displacement or rotation) ${\mathrm{HfTe}}_{2}$ trilayer, the phonon bands in both systems transition from topologically trivial to nontrivial. Phononic higher-order topology is characterized by the fractional quadrupole in the strained $1T\text{\ensuremath{-}}{\mathrm{HfTe}}_{2}$ monolayer, and by the fractional bulk dipole polarization in the ${\mathrm{HfTe}}_{2}$ twisted smallest moir\'e inversion lattice. Our work not only offers strategies for searching topological phonon modes, but also provides insights into controlling phononic higher-order topology in crystalline materials.