Optimally Localized Wannier Functions for 2D Chern Insulators

Abstract

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.

16 pages, 4 figures

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• Wannier representation of Z_2 topological insulators
• Exponential localization of Wannier functions in insulators
• The insulator/Chern-insulator transition in the Haldane model
• Electric polarization in a Chern insulator
• Vortex lattice transitions in cyclic spinor condensates
• Quantized charge polarization as a many-body invariant in (2+1)D crystalline topological states and Hofstadter butterflies
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• Reduced Wannier representation for topological bands
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