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

Using a realistic cesium fluogermanate (Cs2GeF6) 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 Cs2GeF6 via first-principles calculations. The nodal-box phonons in the Cs2GeF6 material are formed from the bowtie-shaped nodal lines in the [110], [101], [011], [1¯10], [01¯1], and [101¯] planes. The nodal-chain states in the phonon dispersion of Cs2GeF6 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.