{
  "abstract": "Research on topological phases of matter is a core field in modern condensed matter physics. Free fermion systems, such as topological insulators and superconductors, have been studied using the \"Tenfold Way\" and K-theory. Building on Kitaev's idea of $Ω$-spectrum and classifying space, as well as Freed-Moore's K-theory, this work demonstrates that free fermionic systems form a genuine $G$-$Ω$-spectrum and clarifies its connection to several distinct classification schemes appearing in the physical literature. By introducing the $\\mathbb{Z}_2$-graded algebra $A_{\\mathrm{sym}}^V$, the classification problem for systems with general symmetries, including antilinear symmetries, antisymmetries, projective representations, and point group symmetries, is turned into an extension problem in representation theory. To solve this, a computational method for the $\\mathbb{Z}_2$-graded Wedderburn-Artin decomposition of $A_{\\mathrm{sym}}^V$ is developed. This decomposition not only yields a classification but also enables the explicit construction of the corresponding Dirac Hamiltonian. Furthermore, a GAP programming package has been developed to automate these calculations.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": "Journal reference: Phys. Rev. B 114, 165101 (2026)",
  "arxiv": "2605.11655",
  "author_total": 2,
  "authors": [
    {
      "name": "Tian Yuan"
    },
    {
      "name": "Yang Qi"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/6rme4bbvahxa5.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2605.11655",
  "fulltext_url": null,
  "id": "6rme4bbvahxa5",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2605.11655",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "cond-mat.mes-hall"
    ],
    [
      "arxiv",
      "math-ph"
    ],
    [
      "arxiv",
      "math.MP"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2605.11655",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-03",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/6rme4bbvahxa5.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:18:31Z",
      "native_id": "2605.11655",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2605.11655"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math-ph",
      "subfield": "mathematical-physics"
    },
    {
      "field": "physics",
      "origin": "arxiv:cond-mat.mes-hall",
      "subfield": "condensed-matter"
    }
  ],
  "title": "Algebra of free fermions: Classifying spaces, Hamiltonians, and computation",
  "title_text": "Algebra of free fermions: Classifying spaces, Hamiltonians, and computation",
  "type": "preprint",
  "updated_at": "2026-10-30T12:18:36Z",
  "url": "https://research.draconitic.com/paper/6rme4bbvahxa5",
  "venue": "arXiv",
  "versions": [
    {
      "doi": "10.1103/llyv-nn5q",
      "pid": null,
      "relation": "published-as",
      "title": null,
      "venue": null
    }
  ]
}