Quantized Berry Phase and Surface States under Reflection Symmetry or Space-Time Inversion Symmetry

As reflection symmetry or space-time inversion symmetry is preserved, with a non-contractible integral loop respecting the symmetry in the Brilliouin zone, Berry phase is quantized in proper basis. Topological nodal lines can be enclosed in the integral loop and $\pi$-Berry phase topologically protects the nodal lines. In this work, we show that to have quantized Berry phase restricted by the symmetry in any crystal structure, we choose to use the cell-periodic convention and define the origin point in the real space at one of the reflection (inversion) centers. In addition, $\pi$-Berry phase is not the sufficient condition leading to the presence of the stable surface states. Their presence crucially depends on the location of the termination and the crystal structure in the unit cell. By using these new conditions we further reexamine if stable surface states exist in the known topological nodal line materials stemming from reflection symmetry or space-time inversion symmetry.