{
  "id": "1704.00932",
  "version": "v1",
  "published": "2017-04-04T09:36:36.000Z",
  "updated": "2017-04-04T09:36:36.000Z",
  "title": "Parseval frames of localized Wannier functions",
  "authors": [
    "Horia D. Cornean",
    "Domenico Monaco"
  ],
  "comment": "18 pages",
  "categories": [
    "math-ph",
    "cond-mat.mes-hall",
    "math.MP"
  ],
  "abstract": "Let $d \\le 3$ and consider a real analytic and $\\mathbb{Z}^d$-periodic family $\\{P(\\mathbf{k})\\}_{\\mathbf{k} \\in \\mathbb{R}^d}$ of orthogonal projections of rank $m$. A moving orthonormal basis of $\\mathrm{Ran}\\, P(\\mathbf{k})$ consisting of real analytic and $\\mathbb{Z}^d$-periodic Bloch vectors can be constructed if and only if the first Chern number(s) of $P$ vanish(es). Here we are mainly interested in the topologically obstructed case. First, by dropping the generating condition, we can construct a collection of $m-1$ orthonormal, real analytic, and $\\mathbb{Z}^d$-periodic Bloch vectors. Second, by dropping the orthonormality condition, we can construct a Parseval frame of $m+d-1$ real analytic and $\\mathbb{Z}^d$-periodic Bloch vectors which generate $\\mathrm{Ran}\\, P(\\mathbf{k})$. Both constructions are based on a two-step logarithm method which produces a moving orthonormal basis in the topologically trivial case. In applications to condensed matter systems, a moving Parseval frame of analytic, $\\mathbb{Z}^d$-periodic Bloch vectors generates a Parseval frame of exponentially localized composite Wannier functions for the occupied states of a gapped periodic Hamiltonian.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-04-04T09:36:36.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "81Q30",
      "81Q70"
    ],
    "keywords": [
      "parseval frame",
      "localized wannier functions",
      "real analytic",
      "moving orthonormal basis",
      "periodic bloch vectors generates"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 18,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}