We consider the problem of constructing Wannier functions for ${\mathbb{Z}}_2$ topological insulators in two dimensions. It is well known that there is a topological obstruction to the construction of Wannier functions for Chern insulators, but it has been unclear whether this is also true for the ${\mathbb{Z}}_2$case. We consider the Kane-Mele tight-binding model, which exhibits both normal (${\mathbb{Z}}_2$-even) and topological (${\mathbb{Z}}_2$-odd) phases as a function of the model parameters. In the ${\mathbb{Z}}_2$-even phase, the usual projection-based scheme can be used to build the Wannier representation. In the ${\mathbb{Z}}_2$-odd phase, we do find a topological obstruction, but only if one insists on choosing a gauge that respects the time-reversal symmetry, corresponding to Wannier functions that come in time-reversal pairs. If, instead, we are willing to violate this gauge condition, a Wannier representation becomes possible. We present an explicit construction of Wannier functions for the ${\mathbb{Z}}_2$-odd phase of the Kane-Mele model via a modified projection scheme, followed by maximal localization, and confirm that these Wannier functions correctly represent the electric polarization and other electronic properties of the insulator.

• Received 7 September 2010

DOI:https://doi.org/10.1103/PhysRevB.83.035108