1208.4234 (A. Alexandradinata et al.)

Wilson-Loop Characterization of Inversion-Symmetric Topological
Insulators    [PDF]

A. Alexandradinata, Xi Dai, B. Andrei Bernevig

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 the inversion-symmetric topological insulator. Through the Wilson loop, we formulate a criterion for nontriviality that indicates a Z classification of 1D inversion-symmetric insulators. If the ground-state wavefunctions at momenta 0 and pi transform under different representations of inversion, we find that a subset of the Wilson-loop eigenvalues are robustly quantized to -1; we identify the number of -1 eigenvalues as a topological index N \in Z. Physical interpretations of N are provided in holonomy and in the geometric-phase theory of polarization. In addition, we identify N with the number of protected boundary modes in the entanglement spectrum. In 2D, we identify a relative winding number W which provides a Z classification of 2D inversion-symmetric insulators. For insulators with nonzero W, their Wilson-loop eigenspectra exhibit spectral flow that is protected only by inversion symmetry. Hence, W is the inversion-analog of the first Chern class C (for charge-conserving insulators) and the Z_2 invariant Xi (for time-reversal invariant insulators). Finally, we establish relations between the topological invariants (W, C, Xi) and the Wilson-loop eigenvalues at symmetric lines in the 2D Brillouin zone.

View original: http://arxiv.org/abs/1208.4234