Fast Generation of Pipek-Mezey Wannier Functions via the Co-Iterative Augmented Hessian Method

We report a $k$-point extension of the second-order co-iterative augmented Hessian (CIAH) algorithm, termed $k$-CIAH, for Pipek-Mezey (PM) localization of Wannier functions (WFs). By exploiting an efficient evaluation of the Hessian-vector product, $k$-CIAH achieves $O(N_k^2 n^3)$ scaling in both CPU time and memory, matching that of previously reported first-order $k$-space approaches while improving upon the $O(N_k^3 n^3)$ scaling of $\Gamma$-point CIAH, where $N_k$ denotes the number of $k$-points sampling the first Brillouin zone and $n$ characterizes the unit-cell size. Benchmark calculations on a diverse set of solids -- including insulators, semiconductors, metals, and surfaces -- demonstrate the fast and robust convergence of $k$-CIAH-based PMWF optimization, which yields an overall computational efficiency approximately 2-3--fold higher than first-order $k$-space methods and orders of magnitude higher than $\Gamma$-point CIAH for localizing 1000-5000 orbitals. The quality of the resulting PMWFs is further validated by accurate electronic band structures obtained via PMWF-based Wannier interpolation.