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As reflection symmetry or space-time inversion symmetry is preserved, with anon-contractible integral loop respecting the symmetry in the Brilliouin zone,Berry phase is quantized in proper basis. Topological nodal lines can beenclosed in the integral loop and $\pi$-Berry phase topologically protects thenodal lines. In this work, we show that to have quantized Berry phaserestricted by the symmetry in any crystal structure, we choose to use thecell-periodic convention and define the origin point in the real space at oneof the reflection (inversion) centers. In addition, $\pi$-Berry phase is notthe sufficient condition leading to the presence of the stable surface states.Their presence crucially depends on the location of the termination and thecrystal structure in the unit cell. By using these new conditions we furtherreexamine if stable surface states exist in the known topological nodal linematerials stemming from reflection symmetry or space-time inversion symmetry.

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