{
  "id": "0711.2641",
  "version": "v1",
  "published": "2007-11-16T16:03:32.000Z",
  "updated": "2007-11-16T16:03:32.000Z",
  "title": "Extensive nonadditive entropy in quantum spin chains",
  "authors": [
    "Filippo Caruso",
    "Constantino Tsallis"
  ],
  "comment": "9 pages, 4 figures, Invited Paper presented at the international conference CTNEXT07, satellite of STATPHYS23, 1-5 July 2007, Catania, Italy",
  "journal": "AIP conference Proceedings 965 (2007) p. 51",
  "doi": "10.1063/1.2828759",
  "categories": [
    "cond-mat.stat-mech",
    "quant-ph"
  ],
  "abstract": "We present details on a physical realization, in a many-body Hamiltonian system, of the abstract probabilistic structure recently exhibited by Gell-Mann, Sato and one of us (C.T.), that the nonadditive entropy $S_q=k [1- Tr \\hat{\\rho}^q]/[q-1]$ ($\\hat{\\rho}\\equiv$ density matrix; $S_1=-k Tr \\hat{\\rho} \\ln \\hat{\\rho}$) can conform, for an anomalous value of q (i.e., q not equal to 1), to the classical thermodynamical requirement for the entropy to be extensive. Moreover, we find that the entropic index q provides a tool to characterize both universal and nonuniversal aspects in quantum phase transitions (e.g., for a L-sized block of the Ising ferromagnetic chain at its T=0 critical transverse field, we obtain $\\lim_{L\\to\\infty}S_{\\sqrt{37}-6}(L)/L=3.56 \\pm 0.03$). The present results suggest a new and powerful approach to measure entanglement in quantum many-body systems. At the light of these results, and similar ones for a d=2 Bosonic system discussed by us elsewhere, we conjecture that, for blocks of linear size L of a large class of Fermionic and Bosonic d-dimensional many-body Hamiltonians with short-range interaction at T=0, we have that the additive entropy $S_1(L) \\propto [L^{d-1}-1]/(d-1)$ (i.e., $ \\ln L$ for $d=1$, and $ L^{d-1}$ for d>1), hence it is not extensive, whereas, for anomalous values of the index q, we have that the nonadditive entropy $S_q(L)\\propto L^d$ ($\\forall d$), i.e., it is extensive. The present discussion neatly illustrates that entropic additivity and entropic extensivity are quite different properties, even if they essentially coincide in the presence of short-range correlations.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2007-11-16T16:03:32.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05.30.-d",
      "05.70.-a",
      "03.65.Ud"
    ],
    "keywords": [
      "quantum spin chains",
      "extensive nonadditive entropy",
      "bosonic d-dimensional many-body hamiltonians",
      "abstract probabilistic structure",
      "many-body hamiltonian system"
    ],
    "tags": [
      "conference paper",
      "journal article"
    ],
    "publication": {
      "publisher": "AIP",
      "journal": "AIP Conf. Proc."
    },
    "note": {
      "typesetting": "TeX",
      "pages": 9,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2007AIPC..965...51C"
    }
  }
}