{"paper":{"title":"Atiyah-Hirzebruch spectral sequence for topological insulators and superconductors: $E_2$ pages for 1651 magnetic space groups","license":"http://creativecommons.org/licenses/by/4.0/","headline":"The Atiyah-Hirzebruch spectral sequence determines the topological K-groups for about 59 percent of three-dimensional magnetic space group settings.","cross_cats":["cond-mat.supr-con"],"primary_cat":"cond-mat.mes-hall","authors_text":"Ken Shiozaki, Seishiro Ono","submitted_at":"2023-04-04T14:31:20Z","abstract_excerpt":"We compute the $E_2$ pages of the momentum-space and real-space Atiyah-Hirzebruch spectral sequence (AHSS) for topological crystalline insulators and superconductors up to three spatial dimensions, considering the cell decomposition in which if a group action fixes a cell setwise then its group action fixes the same cell pointwise. We provide a detailed description of the implementation for computing the $E_2$ pages of AHSS. Under a physically reasonable assumption, we enumerate all possible $K$-groups that are compatible with the $E_2$ pages for both momentum and real-space AHSS. As a result,"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"We determine the K-groups for approximately 59% of symmetry settings in three spatial dimensions.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"Under a physically reasonable assumption, we enumerate all possible K-groups that are compatible with the E2 pages for both momentum and real-space AHSS.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Computes E2 pages of momentum- and real-space AHSS for 1651 magnetic space groups and determines compatible K-groups for ~59% of 3D symmetry settings under a physical assumption.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"The Atiyah-Hirzebruch spectral sequence determines the topological K-groups for about 59 percent of three-dimensional magnetic space group settings.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"0e3c21b941e529aa674bcec2d4c8aab7f6a3f82a8673878849cd86864bb49a37"},"source":{"id":"2304.01827","kind":"arxiv","version":2},"verdict":{"id":"e5579b0a-9ff6-4fc9-a07c-28199c93c0e1","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-24T09:09:36.754528Z","strongest_claim":"We determine the K-groups for approximately 59% of symmetry settings in three spatial dimensions.","one_line_summary":"Computes E2 pages of momentum- and real-space AHSS for 1651 magnetic space groups and determines compatible K-groups for ~59% of 3D symmetry settings under a physical assumption.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"Under a physically reasonable assumption, we enumerate all possible K-groups that are compatible with the E2 pages for both momentum and real-space AHSS.","pith_extraction_headline":"The Atiyah-Hirzebruch spectral sequence determines the topological K-groups for about 59 percent of three-dimensional magnetic space group settings."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2304.01827/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":74,"sample":[{"doi":"","year":null,"title":"ch shφh φhch sh ch shφh φhch sh TABLE I","work_id":"f59c3e89-25b9-41e3-97a7-f410fd7361b1","ref_index":1,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":null,"title":"(75) We note that since the p-cells inCp are oriented sym- metrically, there is no sign change due to the mismatch of orientations","work_id":"07442882-4e45-4a9c-9b61-708af6dc1382","ref_index":2,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":null,"title":"crys- talline topological liquid","work_id":"0a75f787-4f1c-4452-a64f-830229f6422b","ref_index":3,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":null,"title":"Pick a representative p-cell Dp i of an or- bit of equivalent p-cells inCp","work_id":"693eaac2-b2d7-4ab8-848e-623eb2a7f229","ref_index":4,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":null,"title":"Z2 classification For an irrep αr(xp i ), if the classification of degree −n is Z2, set bp,−n ir = dim (H), the matrix dimension listed in Table II. 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