{
  "id": "2203.03786",
  "version": "v1",
  "published": "2022-03-08T00:48:23.000Z",
  "updated": "2022-03-08T00:48:23.000Z",
  "title": "Locobatic theorem for disordered media and validity of linear response",
  "authors": [
    "Wojciech De Roeck",
    "Alexander Elgart",
    "Martin Fraas"
  ],
  "comment": "56 pages, 1 figure",
  "categories": [
    "math-ph",
    "cond-mat.dis-nn",
    "math.MP"
  ],
  "abstract": "Spectral localization is intrinsically unstable under perturbation. As a result, the adiabatic theorem of quantum mechanics cannot generally hold for localized eigenstates. However, it turns out that a remnant of the adiabatic theorem, which we name the \"locobatic theorem\", survives: The physical evolution of a typical eigenstate $\\psi$ for a random system remains close, with high probability, to the spectral flow for $\\psi$ associated with a restriction of the full Hamiltonian to a region where $\\psi$ is supported. We make the above statement precise for a class of Hamiltonians describing a particle in a disordered background. Our argument relies on finding a local structure that remains stable under the small perturbation of a random system. An application of this work is the justification of the linear response formula for the Hall conductivity of a two-dimensional system with the Fermi energy lying in a mobility gap. Additional results are concerned with eigenvector hybridization in a one-dimensional Anderson model and the construction of a Wannier basis for underlying spectral projections.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-03-08T00:48:23.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "locobatic theorem",
      "disordered media",
      "adiabatic theorem",
      "random system remains close",
      "one-dimensional anderson model"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 56,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}