Efficient design for nodal structures characterized by Euler class

Recently, the real triple point characterized by Euler class in multigap systems, has been proposed beyond the conventional topological phase classifications. Here, we apply the subgroup lattice to study Euler topology and establish an explicit mapping between various undiscovered nodal line configurations (NLCs) evolving from the nontrivial real triple point and specific symmetry-breaking ways. Considering all symmetry-breaking paths from the magnetic point group $m$\ensuremath{-}$3m1$\ensuremath{'} to \ensuremath{-}1\ensuremath{'} with preserved $PT$ symmetry where $P$ and $T$ are inversion and time-reversal symmetries, respectively, 13 types of topological admissible NLCs are identified, with each subgroup exhibiting a one-to-one or one-to-two correspondence to them, which indicates an accurate design for a target nodal structure. In addition, through the analysis on the interplay between point-group symmetry and underlying non-Abelian topology, we promote an understanding on the formation mechanism of NLCs. Based on first-principles calculations, we verify the NLC evolutions in ten candidate phononic materials such as ${\mathrm{RbCaH}}_{3}$ and CuCl driven by external strain. The provided strain tensors to reach each corresponding subgroup can be a symmetry guidance for experimentally designing nodal line semimetals via strain engineering. Furthermore, we show that the optical conductivity purely induced by the quantum metric can serve as a nontrivial signature of distinct NLCs, and also can be manipulated along symmetry breaking. Our findings reveal the generality of realizing nodal structures characterized by the Euler class from a group-theoretical perspective that are applicable to real triple point materials in other systems.