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

Abstract

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.

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