Optimally Localized Wannier Functions for 2D Chern Insulators

The construction of optimally localized Wannier functions (and Wannier functions in general) for a Chern insulator has been considered to be impossible owing to the fact that the second moment of such functions is generally infinite. In this manuscript, we propose a solution to this problem in the case of a single band. We accomplish this by drawing an analogy between the minimization of the variance and the minimization of the electrostatic energy of a periodic array of point charges in a smooth neutralizing background. In doing so, we obtain a natural regularization of the diverging variance and this leads to an analytical solution to the minimization problem. We demonstrate our results numerically for a particular model system. Furthermore, we show how the optimally localized Wannier functions provide a natural way of evaluating the electric polarization for a Chern insulator.