LAUSR

The conventional $\mathbit{k}\ifmmode\cdot\else\textperiodcentered\fi{}\mathbit{p}$ method fails to capture the full and essential physics of many symmetry enriched multiple nodal line structures in the three-dimensional Brillouin zone. Here, we present a systematical method to construct the effective lattice model of mirror symmetry protected three-dimensional multiple nodal line semimetals, when the spin-orbit interaction is ignored. For systems with a given pair of perpendicular nodal rings, we obtain all the effective lattice models and 11 inequivalent nodal line Fermi surfaces together with their related constraints. By means of first-principles calculations, we first propose a family of real materials, the $\ensuremath{\beta}$ phase of ternary nitrides ${X}_{2}{\mathrm{GeN}}_{2}$ ($X=\mathrm{Ca},\mathrm{Sr},\mathrm{Ba}$), that support one kind of these Fermi surfaces. Therefore, our work deepens the understanding of the nodal line structures and promotes the experimental progress of topological nodal line semimetals.