{
  "id": "1908.06700",
  "version": "v1",
  "published": "2019-08-19T11:17:58.000Z",
  "updated": "2019-08-19T11:17:58.000Z",
  "title": "The Zak phase and Winding number",
  "authors": [
    "Han-Ting Chen",
    "Chia-Hsun Chang",
    "Hsien-chung Kao"
  ],
  "comment": "13 pages, 39 figures",
  "categories": [
    "cond-mat.mes-hall"
  ],
  "abstract": "Bulk-edge correspondence is one of the most distinct properties of topological insulators. In particular, the 1-d winding number $\\n$ has a one-to-one correspondence to the number of edge states in a chain of topological insulators with boundaries. By properly choosing the unit cells, we show explicitly in the so-called extended SSH model that the winding numbers corresponding to the left and right unit cells may be used to predict the numbers of edge states on the two boundaries in a finite chain. Moreover, by modifying the definition of the Zak phase $\\g$ to be summing over all the bands of the system, we show for a general two-band model that the modified Zak phase obeys $\\g = 2\\pi\\n$. It is thus always quantized even if there is no chiral symmetry in the system so that it is classified as trivial in the so-called periodic table of topological materials. We also carry out numerical calculation to demonstrate explicitly that the bulk-edge correspondence may indeed be generalized to this kind of systems.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-08-19T11:17:58.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "winding number",
      "bulk-edge correspondence",
      "edge states",
      "topological insulators",
      "right unit cells"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 13,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}