Topological Nodal Lines from Crystalline Symmetry

Topological nodal lines protected by the time-reversal symmetry T and space-inversion symmetry P can be classified into two categories, depending on the value of the Z₂ monopole topological index. Nodal lines with trivial (non-trivial) monopole charge can (cannot) be gapped out via shrinking the nodal line into a point. In this study, we show that for four-band models with T ²=+1, there are two special points in phase space where the system has a Z₂ symmetry, and any Hamiltonian can be adiabatically connected to one of these points. Utilizing this result, we provide a general principle for finding topological nodal lines with nontrivial monopole charge based on the interplay between lattice crystalline symmetries and the time-reversal symmetry. In addition, this study also reveals a new family of topological nodal lines, protected by the time-reversal symmetry and point group symmetries.