Realization of Hopf-link structure in phonon spectra: Symmetry guidance and High-throughput investigation

The realization of Hopf-link structure in the Brillouin zone is rather rare hindering the comprehensive exploration and understanding of such exotic nodal loop geometry. Here we first tabulate 141 space groups hosting Hopf-link structure and then investigate Phonon Database at Kyoto University consisting of 10034 materials to search for phonon realization of the Hopf-link nodal structure. It is found that almost all the investigated materials own nodal loops or nodal chains while only 113 materials can host Hopf-link structure in phonon spectra, among which 8 representative materials are manually selected to showcase relatively clean Hopf-link structure including LiGaS\(_2\), LiInSe\(_2\), Ca\(_2\)Al\(_2\)Si(HO\(_4\))\(_2\), Ca\(_7\)GeN\(_6\), Al(HO)\(_3\), NaNd(GaS\(_2\))\(_4\), Ga\(_5\)(PS)\(_3\) and RbTh\(_3\)F\(_{13}\). The visible phonon drumhead surface states corresponding to the nodal loops in the Hopf-link structure are further demonstrated using Ga\(_5\)(PS)\(_3\) as an example.The listed 113 crystalline materials provide a good platform for experimentalists to further explore the interesting properties related to Hopf-link structure.