Topological insulators and higher-order topological insulators from gauge-invariant one-dimensional lines

In this paper, we study the interplay between symmetry and topology with a focus on the Z(2) topological index of two-dimensional and three-dimensional (2D/3D) topological insulators and high-order topological insulators. We show that in the presence of either a two-fold-rotational symmetry or a mirror symmetry, a gauge-invariant quantity can be defined for arbitrary one-dimensional (1D) lines in the Brillouin zone. Such 1D quantities provide a new pathway to compute the Z(2) index of topological insulators. In contrast to the generic setup, where the Z(2) index generally involves 2D planes in the Brillouin zone with a globally defined smooth gauge, this 1D approach only involves some 1D lines in the Brillouin zone without requiring a global gauge. Such a simplified approach can be used in any time-reversal invariant insulators with a two-fold crystalline symmetry, which can be found in 30 of the 32 point groups. In addition, this 1D quantity can be further generalized to higher-order topological insulators to compute the magnetoelectric polarization P-3.