# Numerical construction of Wannier functions through homotopy

## TL;DR

This paper introduces a mathematically rigorous and efficient algorithm for constructing localized Wannier functions in systems with zero Chern numbers, applicable to topological insulators like the Kane-Mele model.

## Contribution

It presents a new constructive homotopy-based algorithm for Wannier functions, validated through numerical tests on topological insulators.

## Key findings

- Successfully constructs Wannier functions for the Kane-Mele model
- Proves the homotopy existence for the unitary group in this context
- Demonstrates the algorithm's efficiency and accuracy

## 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.