{
  "id": "2307.12773",
  "version": "v1",
  "published": "2023-07-24T13:22:54.000Z",
  "updated": "2023-07-24T13:22:54.000Z",
  "title": "Violation of Ferromagnetic Ordering of Energy Levels in Spin Rings by Weak Paramagnetism of the Singlet",
  "authors": [
    "David Heson",
    "Shannon Starr",
    "Jacob Thornton"
  ],
  "comment": "15 pages, 6 figures",
  "categories": [
    "math-ph",
    "cond-mat.stat-mech",
    "math.MP"
  ],
  "abstract": "For the quantum Heisenberg antiferromagnet with spin-$j$ on a bipartite, balanced graph, the Lieb-Mattis theorem, ``Ordering of energy levels,'' guarantees that the ground state is a spin singlet, and moreover, defining $E^{\\textrm{AF}}_{\\min}(S)$ to be the minimum eigenvalue of the Hamiltonian in the invariant subspace consisting of all spin $S$ vectors, $\\boldsymbol{S}_{\\mathrm{tot}}^2 \\psi = S(S+1)\\psi$, the function $E^{\\textrm{AF}}_{\\min}(S)$ is monotonically increasing for $0\\leq S\\leq j|\\mathcal{V}|$. For the ferromagnet, the absolute ground state is $E_{\\min}^{\\textrm{FM}}(j|\\mathcal{V}|)$. We say that the graph satisfies ``ferromagnetic ordering of energy levels'' at order $n$, or FOEL-$n$, if two properties hold: (1) $E_{\\min}^{\\textrm{FM}}(j|\\mathcal{V}|)\\leq \\dots \\leq E_{\\min}^{\\mathrm{FM}}(j|\\mathcal{V}|-n)$, and (2) $E_{\\min}^{\\mathrm{FM}}(j|\\mathcal{V}|-n)\\leq E_{\\min}^{\\mathrm{FM}}(j|\\mathcal{V}|-m)$ for all $m\\geq n$. Caputo, Liggett and Richthammer proved a theorem which generally implies FOEL-$1$ is true. Apparently $E_0^{\\mathrm{FM}}(0) <E_0^{\\mathrm{FM}}(1)$ for sufficiently long spin rings, $\\mathbb{Z}/L\\mathbb{Z}$ with even length $L$. So FOEL-$n$ does not hold for $n=jL-1$. We consider $E_0^{\\mathrm{FM}}(1)-E_0^{\\mathrm{FM}}(0)$ using linear spin-wave analysis and numerical computation. Using the Bethe ansatz, Sutherland already considered the spin ring with $j=1/2$ and notably proved weak paramagnetism. But we also present evidence for $j>1/2$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-07-24T13:22:54.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "82B10",
      "81R05",
      "81R50"
    ],
    "keywords": [
      "energy levels",
      "weak paramagnetism",
      "ferromagnetic ordering",
      "absolute ground state",
      "quantum heisenberg antiferromagnet"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 15,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}