Phonon realization of real triple points

Recently, spinless $PT$ (product of spatial and temporal inversion operations) symmetric systems have been shown to display multigap band topology for the real-valued Hamiltonians that goes beyond conventional topological classification, e.g., that by Chern number for Weyl points. The $3\ifmmode\times\else\texttimes\fi{}3$ low-energy models around triply degenerate points [denoted as real triple points (RTPs) in this work] can be made real-valued by choosing a suitable basis in such systems with a minimal number of bands for multigap topology. Interestingly, RTPs have been shown to occur ubiquitously as the Nambu-Goldstone (NG) mode for acoustic phonons, and be characterized by Euler number as a topological invariant even though the Chern number vanishes due to the $PT$ symmetry. Here, combining symmetry analysis, $k\ifmmode\cdot\else\textperiodcentered\fi{}p$ models, and first-principles calculations, we perform a comprehensive study of RTPs realizable in phonon spectra. We provide $k\ifmmode\cdot\else\textperiodcentered\fi{}p$ models around RTPs by NG mode and symmetry-protected RTPs pinned at high-symmetry points (HSPs) in the 1651 magnetic space groups containing $PT$ symmetry. On the other hand, we list the numbers of occurrences of RTPs and other coexisting topological phonons at HSPs in the 15478 materials from the Inorganic Crystal Structure Database. In addition, based on the computed force constants from the Phonon Database at Kyoto University, we catalog 386 materials with at least one RTP owning a nonvanishing Euler number. We take ${\mathrm{KMgF}}_{3}$ in space group 221 to demonstrate the resulting link nodal structure by external strain. The complete lists of RTPs, effective models, coexisting different types of topological phonons, and predicted materials are believed to promote future studies on multigap band topology and related thermal transport of phonons.