{
  "id": "1609.06983",
  "version": "v1",
  "published": "2016-09-22T13:53:22.000Z",
  "updated": "2016-09-22T13:53:22.000Z",
  "title": "On the local equivalence between the canonical and the microcanonical distributions for quantum spin systems",
  "authors": [
    "Hal Tasaki"
  ],
  "comment": "15 pages",
  "categories": [
    "cond-mat.stat-mech",
    "quant-ph"
  ],
  "abstract": "We study a quantum spin system on the $d$-dimensional hypercubic lattice $\\Lambda$ with $N$ sites with translation invariant short-ranged Hamiltonian under periodic boundary conditions. For this model, we consider both the canonical ensemble with inverse temperature $\\beta_0$ and the microcanonical ensemble with the corresponding energy $U_N(\\beta_0)$. Take an arbitrary self-adjoint operator $\\hat{A}$ whose support is contained in a hypercubic block $B$ inside $\\Lambda$. When $\\beta_0$ is sufficiently small, we prove that the expectation values (with respect to these two ensembles) of $\\hat{A}$ are close to each other provided that the number of sites in $B$ is $o(N^{1/2})$ and $N$ is large. This establishes the equivalence of ensembles on the level of local states in a large but finite system. The result is essentially that of Brandao and Cramer (with a more restrictive setting and a bigger block), but our proof is elementary.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-09-22T13:53:22.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "quantum spin system",
      "local equivalence",
      "microcanonical distributions",
      "arbitrary self-adjoint operator",
      "dimensional hypercubic lattice"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 15,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}