{"paper":{"title":"Fast Generation of Pipek-Mezey Wannier Functions via the Co-Iterative Augmented Hessian Method","license":"http://creativecommons.org/licenses/by/4.0/","headline":"k-CIAH extends second-order CIAH optimization to k-point Pipek-Mezey Wannier functions while achieving O(N_k² n³) scaling.","cross_cats":["cond-mat.mtrl-sci"],"primary_cat":"physics.chem-ph","authors_text":"Gengzhi Yang, Hong-Zhou Ye","submitted_at":"2026-02-12T20:24:02Z","abstract_excerpt":"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 calculat"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"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 Γ-point CIAH... yields an overall computational efficiency approximately 2-3-fold higher than first-order k-space methods and orders of magnitude higher than Γ-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.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"That the efficient Hessian-vector product evaluation in the k-CIAH extension maintains numerical stability and robust convergence for all tested systems (insulators, semiconductors, metals, surfaces) without hidden costs or post-hoc adjustments that affect the claimed scaling.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"k-CIAH enables efficient second-order optimization of Pipek-Mezey Wannier functions with O(N_k² n³) scaling, yielding 2-3x speedup over first-order k-space methods.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"k-CIAH extends second-order CIAH optimization to k-point Pipek-Mezey Wannier functions while achieving O(N_k² n³) scaling.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"35aee3e633c0a753bdce3cff54283019ccb98c3e0c43ccd7b18885aea70a7975"},"source":{"id":"2602.12382","kind":"arxiv","version":2},"verdict":{"id":"9181938b-e0af-4e8b-a1d4-d01dc8c4b60a","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-16T05:11:33.636827Z","strongest_claim":"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 Γ-point CIAH... yields an overall computational efficiency approximately 2-3-fold higher than first-order k-space methods and orders of magnitude higher than Γ-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.","one_line_summary":"k-CIAH enables efficient second-order optimization of Pipek-Mezey Wannier functions with O(N_k² n³) scaling, yielding 2-3x speedup over first-order k-space methods.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"That the efficient Hessian-vector product evaluation in the k-CIAH extension maintains numerical stability and robust convergence for all tested systems (insulators, semiconductors, metals, surfaces) without hidden costs or post-hoc adjustments that affect the claimed scaling.","pith_extraction_headline":"k-CIAH extends second-order CIAH optimization to k-point Pipek-Mezey Wannier functions while achieving O(N_k² n³) scaling."},"references":{"count":34,"sample":[{"doi":"","year":null,"title":"The special case in which this real-space unitary is constrained to be real-valued is discussed in sec- tion II G","work_id":"1d476f56-8fee-4eea-a0c8-0124e4b4c97d","ref_index":1,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":null,"title":"using the second-order co-iterative augmented Hes- sian (CIAH) algorithm. 43 CIAH is a modified trust-region Newton method46 that has been successfully applied to orbital localization in molecules 43,4","work_id":"e2c9356b-dabb-4659-a7ad-ecb4f50e5216","ref_index":2,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":null,"title":"(The Hessian diagonals are used to precondition the Davidson update","work_id":"911c0253-c00b-4259-87da-7ff04a1edb73","ref_index":3,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":null,"title":"We summarize the working equations be- low. We first define two types of matrix elements of the atomic projection operators, (PTTT A,kkkkkk′)i j = 1 Nk ⟨φ kkki|ˆPTTT A|φ kkk′j⟩ = 1 Nk ∑ µ ∈ A O∗ TTT µ ,","work_id":"35f64ee8-e8a5-41de-8734-c43529694bb3","ref_index":4,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":null,"title":"35,38 In section IV, we compare the performance and computational efficiency of k-CIAH- and k-BFGS-based PMWF optimization","work_id":"a1bb785e-2186-4c3d-a10c-17bbddbbad04","ref_index":5,"cited_arxiv_id":"","is_internal_anchor":false}],"resolved_work":34,"snapshot_sha256":"94ea7ea2469ddbd4dfa5b3db9f5bebf0d831007c8cd930daaf4fe2b1af8f7efa","internal_anchors":0},"formal_canon":{"evidence_count":2,"snapshot_sha256":"94c7d7f621b035fbbc709b268b9900a6ed897eb797e49c0e5d3d94f33ec9ab47"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}