  The conventional k.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 new and 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 eleven inequivalent nodal
line Fermi surfaces together with their related constraints. By means of
first-principles calculations, we first propose a family of real materials,
beta phase of ternary nitrides X2GeN2 (X = Ca; Sr; Ba), that support one kind
of these novel Fermi surfaces. Therefore, our work deepens the understanding of
the nodal line structures and promotes the experimental progress of topological
nodal line semimetals.
