{
  "id": "2102.07287",
  "version": "v1",
  "published": "2021-02-15T00:51:14.000Z",
  "updated": "2021-02-15T00:51:14.000Z",
  "title": "On the stability of the area law for the entanglement entropy of the Landau Hamiltonian",
  "authors": [
    "Paul Pfeiffer"
  ],
  "comment": "32 pages",
  "categories": [
    "math-ph",
    "cond-mat.quant-gas",
    "math.MP"
  ],
  "abstract": "We consider the two-dimensional ideal Fermi gas subject to a magnetic field which is perpendicular to the Euclidean plane $\\mathbb R^2$ and whose strength $B(x)$ at $x\\in\\mathbb R^2$ converges to some $B_0>0$ as $\\|x\\|\\to\\infty$. Furthermore, we allow for an electric potential $V_\\varepsilon$ which vanishes at infinity. They define the single-particle Landau Hamiltonian of our Fermi gas (up to gauge fixing). Starting from the ground state of this Fermi gas with chemical potential $\\mu\\ge B_0$ we study the asymptotic growth of its bipartite entanglement entropy associated to $L\\Lambda$ as $L\\to\\infty$ for some fixed bounded region $\\Lambda\\subset\\mathbb R^2$. We show that its leading order in $L$ does not depend on the perturbations $B_\\varepsilon := B_0 - B$ and $V_\\varepsilon$ if they satisfy some mild decay assumptions. Our result holds for all $\\alpha$-R\\' enyi entropies $\\alpha>1/3$; for $\\alpha\\le 1/3$, we have to assume in addition some differentiability of the perturbations $B_\\varepsilon$ and $V_\\varepsilon$. The case of a constant magnetic field $B_\\varepsilon = 0$ and with $V_\\varepsilon= 0$ was treated recently for general $\\mu$ by Leschke, Sobolev and Spitzer. Our result thus proves the stability of that area law under the same regularity assumptions on the boundary $\\partial \\Lambda$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-02-15T00:51:14.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "area law",
      "two-dimensional ideal fermi gas subject",
      "magnetic field",
      "single-particle landau hamiltonian",
      "bipartite entanglement entropy"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 32,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}