Wilson-loop Classification of Inversion-Symmetric Topological Insulators and the Z$_2$ Crystalline Topological Insulator

In the context of translationally-invariant insulators, Wilson loops describe the adiabatic evolution of the ground state around a closed circuit in the Brillouin zone. We propose that the Wilson-loop eigenspectrum provides a complete characterization of (a) the inversion-symmetric topological insulator, and (b) the \textbf{Z}$_2$ crystalline topological insulator: time-reversal symmetric insulators with either C$_4$ or C$_6$ rotational symmetry, but with no spin-orbit coupling. For the 1D inversion-symmetric insulator, we formulate a \textbf{Z} Wilson-loop index, which is identifiable with the number of protected boundary modes in the entanglement spectrum. For the 2D inversion-symmetric insulator, we identify a \textbf{Z} relative-winding number, which is the inversion-analog of the first Chern class (for charge-conserving insulators). For the \textbf{Z}$_2$ crystalline topological insulator, we show how the \textbf{Z}$_2$ invariant can be extracted from the Wilson-loop eigenspectrum; this aids the identification of materials that realize this phase.