{
  "id": "0910.5490",
  "version": "v1",
  "published": "2009-10-28T20:53:50.000Z",
  "updated": "2009-10-28T20:53:50.000Z",
  "title": "Almost Commuting Matrices, Localized Wannier Functions, and the Quantum Hall Effect",
  "authors": [
    "M. B. Hastings",
    "T. A. Loring"
  ],
  "comment": "35 pages, 2 figures",
  "journal": "J. Math. Phys. 51, 015214 (2010).",
  "categories": [
    "math-ph",
    "math.MP",
    "math.OA",
    "quant-ph"
  ],
  "abstract": "For models of non-interacting fermions moving within sites arranged on a surface in three dimensional space, there can be obstructions to finding localized Wannier functions. We show that such obstructions are $K$-theoretic obstructions to approximating almost commuting, complex-valued matrices by commuting matrices, and we demonstrate numerically the presence of this obstruction for a lattice model of the quantum Hall effect in a spherical geometry. The numerical calculation of the obstruction is straightforward, and does not require translational invariance or introducing a flux torus. We further show that there is a $Z_2$ index obstruction to approximating almost commuting self-dual matrices by exactly commuting self-dual matrices, and present additional conjectures regarding the approximation of almost commuting real and self-dual matrices by exactly commuting real and self-dual matrices. The motivation for considering this problem is the case of physical systems with additional antiunitary symmetries such as time reversal or particle-hole conjugation. Finally, in the case of the sphere--mathematically speaking three almost commuting Hermitians whose sum of square is near the identity--we give the first quantitative result showing this index is the only obstruction to finding commuting approximations. We review the known non-quantitative results for the torus.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2009-10-28T20:53:50.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "73.43.-f",
      "02.10.Yn",
      "05.30.Fk",
      "02.40.-k"
    ],
    "keywords": [
      "quantum hall effect",
      "localized wannier functions",
      "commuting matrices",
      "obstruction",
      "commuting self-dual matrices"
    ],
    "tags": [
      "journal article"
    ],
    "publication": {
      "publisher": "AIP",
      "journal": "Journal of Mathematical Physics",
      "doi": "10.1063/1.3274817",
      "year": 2010,
      "month": "Jan",
      "volume": 51,
      "number": 1,
      "pages": 5214
    },
    "note": {
      "typesetting": "TeX",
      "pages": 35,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2010JMP....51a5214H"
    }
  }
}