Realistic cesium fluogermanate: An ideal platform to realize the topologically nodal-box and nodal-chain phonons

Using a realistic cesium fluogermanate $({\mathrm{Cs}}_{2}{\mathrm{GeF}}_{6})$ material as an example, we discuss the symmetry conditions for the co-occurrence of nodal-box and nodal-chain phonons via symmetry analysis and prove that the above-mentioned topological phonons can be realized in ${\mathrm{Cs}}_{2}{\mathrm{GeF}}_{6}$ via first-principles calculations. The nodal-box phonons in the ${\mathrm{Cs}}_{2}{\mathrm{GeF}}_{6}$ material are formed from the bowtie-shaped nodal lines in the [110], [101], [011], $[\overline{1}10]$, $[0\overline{1}1]$, and $[10\overline{1}]$ planes. The nodal-chain states in the phonon dispersion of ${\mathrm{Cs}}_{2}{\mathrm{GeF}}_{6}$ run along all three directions, forming a network structure. Moreover, the topologically nontrivial signatures of the nodal-box and nodal-chain phonons were determined. The phonon surface states related to the nodal-box and nodal-chain phonons are obvious, which is expected to benefit experimental detection. This work shows that the nodal-box phonons can exist in existing materials and reveals the possibility of the co-occurrence of nodal-box and nodal-chain phonons in one material.