{
  "id": "0911.4706",
  "version": "v1",
  "published": "2009-11-24T20:23:30.000Z",
  "updated": "2009-11-24T20:23:30.000Z",
  "title": "Quantization of Hall Conductance For Interacting Electrons Without Averaging Assumptions",
  "authors": [
    "Matthew B. Hastings",
    "Spyridon Michalakis"
  ],
  "comment": "36 pages, 4 figures",
  "categories": [
    "math-ph",
    "math.MP",
    "quant-ph"
  ],
  "abstract": "We consider two-dimensional Hamiltonians on a torus with finite range, finite strength interactions and a unique ground state with a non-vanishing spectral gap, and a conserved local charge, as defined precisely in the text. Using the local charge operators, we introduce a boundary magnetic flux in the horizontal and vertical direction and evolve the ground state quasi-adiabatically around a square of size one magnetic flux, in flux space. At the end of the evolution we obtain a trivial Berry phase, which we compare, via a method reminiscent of Stokes' Theorem, to the Berry phase obtained from an evolution around a small loop near the origin. As a result, we prove, without any averaging assumption, that the Hall conductance for interacting electron systems is quantized in integer multiples of e^2/h up to small corrections bounded by a function that decays as a stretched exponential in the linear size L. Finally, we discuss extensions to the fractional case under an additional topological order assumption to describe the multiple degenerate ground states.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2009-11-24T20:23:30.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "81V70",
      "82B10",
      "82B20"
    ],
    "keywords": [
      "hall conductance",
      "interacting electron",
      "averaging assumption",
      "quantization",
      "multiple degenerate ground states"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 36,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2009arXiv0911.4706H"
    }
  }
}