{
  "id": "1511.04230",
  "version": "v1",
  "published": "2015-11-13T11:00:32.000Z",
  "updated": "2015-11-13T11:00:32.000Z",
  "title": "Relation between two-phase quantum walks and the topological invariant",
  "authors": [
    "Takako Endo",
    "Norio Konno",
    "Hideaki Obuse"
  ],
  "comment": "49 pages, 13 figures",
  "categories": [
    "math-ph",
    "cond-mat.mes-hall",
    "math.MP",
    "quant-ph"
  ],
  "abstract": "We study a position-dependent discrete-time quantum walk (QW) in one dimension, whose time-evolution operator is built up from two coin operators which are distinguished by phase factors from $x\\geq0$ and $x\\leq-1$. We call the QW the ${\\it complete\\;two}$-${\\it phase\\;QW}$ to discern from the two-phase QW with one defect[13,14]. Because of its localization properties, the two-phase QWs can be considered as an ideal mathematical model of topological insulators which are novel quantum states of matter characterized by topological invariants. Employing the complete two-phase QW, we present the stationary measure, and two kinds of limit theorems concerning ${\\it localization}$ and the ${\\it ballistic\\;spreading}$, which are the characteristic behaviors in the long-time limit of discrete-time QWs in one dimension. As a consequence, we obtain the mathematical expression of the whole picture of the asymptotic behavior of the walker in the long-time limit. We also clarify relevant symmetries, which are essential for topological insulators, of the complete two-phase QW, and then derive the topological invariant. Having established both mathematical rigorous results and the topological invariant of the complete two-phase QW, we provide solid arguments to understand localization of QWs in term of topological invariant. Furthermore, by applying a concept of ${\\it\\;topological\\;protections}$, we clarify that localization of the two-phase QW with one defect, studied in the previous work[13], can be related to localization of the complete two-phase QW under symmetry preserving perturbations.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-11-13T11:00:32.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "topological invariant",
      "complete two-phase qw",
      "two-phase quantum walks",
      "position-dependent discrete-time quantum walk",
      "long-time limit"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 49,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015arXiv151104230E"
    }
  }
}