{
  "id": "1404.5804",
  "version": "v2",
  "published": "2014-04-23T12:30:54.000Z",
  "updated": "2015-06-05T19:48:45.000Z",
  "title": "Classification of \"Quaternionic\" Bloch-bundles: Topological Quantum Systems of type AII",
  "authors": [
    "Giuseppe De Nittis",
    "Kiyonori Gomi"
  ],
  "comment": "45 pages, 5 tables. key words: Topological insulators, Bloch-bundle, \"Quaternionic\" vector bundle, FKMM-invariant, Fu-Kane-Mele index. v2 Title, abstract and introduction have been changed. Several typos have been fixed. Journal of reference Commun. Math. Phys",
  "categories": [
    "math-ph",
    "cond-mat.mes-hall",
    "math.MP"
  ],
  "abstract": "We provide a classification of type AII topological quantum systems in dimension d=1,2,3,4. Our analysis is based on the construction of a topological invariant, the FKMM-invariant, which completely classifies \"Quaternionic\" vector bundles (a.k.a. \"symplectic\" vector bundles) in dimension d<4. This invariant takes value in a proper equivariant cohomology theory and, in the case of examples of physical interest, it reproduces the familiar Fu-Kane-Mele index. In the case d=4 the classification requires a combined use of the FKMM-invariant and the second Chern class. Among the other things, we prove that the FKMM-invariant is a bona fide characteristic class for the category of \"Quaternionic\" vector bundles in the sense that it can be realized as the pullback of a universal topological invariant.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-04-23T12:30:54.000Z",
      "title": "Classification of \"Quaternionic\" Bloch-bundles: Topological Insulators of type AII",
      "abstract": "We provide a classification of type AII topological insulators in dimension d=1,2,3,4. Our analysis is based on the construction of a topological invariant, the FKMM-invariant, which completely classifies \"Quaternionic\" vector bundles (a.k.a. \"symplectic\" vector bundles) in dimension d<4. This invariant takes value in a proper equivariant cohomology theory and, in the case of examples of physical interest, it reproduces the familiar Fu-Kane-Mele index. In the case d=4 the classification requires a combined use of the FKMM-invariant and the second Chern class. Among the other things, we prove that the FKMM-invariant is a bona fide characteristic class for the category of \"Quaternionic\" vector bundles in the sense that it can be realized as the pullback of a universal topological invariant.",
      "comment": "45 pages, 5 tables. key words: Topological insulators, Bloch-bundle, \"Quaternionic\" vector bundle, FKMM-invariant, Fu-Kane-Mele index",
      "journal": null,
      "doi": null
    },
    {
      "version": "v2",
      "updated": "2015-06-05T19:48:45.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "57R22",
      "55N25",
      "53C80",
      "19L64"
    ],
    "keywords": [
      "vector bundles",
      "classification",
      "quaternionic",
      "bloch-bundles",
      "bona fide characteristic class"
    ],
    "publication": {
      "doi": "10.1007/s00220-015-2390-0",
      "journal": "Communications in Mathematical Physics",
      "year": 2015,
      "month": "Oct",
      "volume": 339,
      "number": 1,
      "pages": 1
    },
    "note": {
      "typesetting": "TeX",
      "pages": 45,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015CMaPh.339....1D"
    }
  }
}