Numerical construction of Wannier functions through homotopy

Abstract

We provide a mathematically proven, simple and efficient algorithm to build localised Wannier functions, with the only requirement that the system has vanishing Chern numbers. Our algorithm is able to build localised Wannier for topological insulators such as the Kane-Mele model. It is based on an explicit and constructive proof of homotopies for the unitary group. We provide numerical tests validating the methods for several systems, including the Kane-Mele model.

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• Parseval frames of exponentially localized magnetic Wannier functions
• Symmetric Fermi projections and Kitaev's table: topological phases of matter in low dimensions
• Localised Wannier functions in metallic systems
• Topology vs localization in synthetic dimensions