Quantized Berry Phase and Surface States in the Presence of Reflection or Space-Time Inversion Symmetry

Reflection symmetry or space-time inversion symmetry quantizes the Berry phase to 0 or \pi. This in turn can lead to the protection of nodal lines in systems with negligible spin-orbit coupling. In this work, we show that to prove the Berry phase quantization, it is most convenient to use the cell-periodic convention and to choose the coordinates such that the origin is at the reflectioin (inversion) center. In addition, we show that a \pi Berry phase is not a sufficient condition for the presence of stable surface states. Their presence crucially depends on the surface termination and the atom distribution in the unit cell. By using these new conditions we examine if stable surface states exist in known topological nodal line materials.