{
  "abstract": "We define a relative $\\mathbb{Z}_2$-index $\\mathcal{N}(ω_1,ω_2)$ for locally-comparable, parity-invariant pure states on a unital $C^*$-algebra. We prove that the index is well-defined, multiplicative, invariant under parity-preserving automorphisms, and locally constant in the norm topology. For the one-dimensional self-dual CAR algebra, we apply this construction to the half-chain automorphism $σ$ and define the many-body Majorana number $\\mathcal{N}σ(ω):=\\mathcal{N}(ω,ω\\circσ)$ for parity-invariant $σ$-local pure states. In the quasi-free Hilbert--Schmidt regime, this index agrees with the usual single-body Majorana number. We then introduce symmetric local automorphism paths and prove that $\\mathcal{N}_σ$ completely classifies the resulting automorphic-path-components. This automorphic-path equivalence retains the bulk Majorana number but may forget a relative zero-dimensional parity obstruction.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.28796",
  "author_total": 2,
  "authors": [
    {
      "name": "Anna Mazhar"
    },
    {
      "name": "Jacob Shapiro"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/10vbwj9wtjqbe.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.28796",
  "fulltext_url": null,
  "id": "10vbwj9wtjqbe",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.28796",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math-ph"
    ],
    [
      "arxiv",
      "math.MP"
    ],
    [
      "arxiv",
      "quant-ph"
    ],
    [
      "openalex",
      "3107|Topological Materials and Phenomena"
    ]
  ],
  "openalex": "W7204957308",
  "pdf_url": "https://arxiv.org/pdf/2608.28796",
  "pmcid": null,
  "pmid": null,
  "published": "2026-08-28",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/10vbwj9wtjqbe.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:17:41Z",
      "native_id": "2608.28796",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.28796"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math-ph",
      "subfield": "mathematical-physics"
    },
    {
      "field": "physics",
      "origin": "arxiv:quant-ph",
      "subfield": "quantum"
    }
  ],
  "title": "The $\\mathbb{Z}_2$-Index of a Pair of Pure States and the Topology of Interacting 1D Superconductors",
  "title_text": "The ℤ_2-Index of a Pair of Pure States and the Topology of Interacting 1D Superconductors",
  "type": "preprint",
  "updated_at": "2026-10-30T12:17:48Z",
  "url": "https://research.draconitic.com/paper/10vbwj9wtjqbe",
  "venue": "arXiv",
  "versions": []
}