{
  "id": "2008.09318",
  "version": "v1",
  "published": "2020-08-21T05:40:58.000Z",
  "updated": "2020-08-21T05:40:58.000Z",
  "title": "Eigenstate thermalization hypothesis and eigenstate-to-eigenstate fluctuations",
  "authors": [
    "Jae Dong Noh"
  ],
  "comment": "8 pages with 7 figures",
  "categories": [
    "cond-mat.stat-mech"
  ],
  "abstract": "We investigate the extent to which the eigenstate thermalization hypothesis~(ETH) is valid or violated in the non-integrable and the integrable spin-$1/2$ XXZ chain. We perform the energy-resolved analysis of the statistical properties of matrix elements $\\{O_{\\gamma\\alpha}\\}$ of an observable $\\hat{O}$ in the energy eigenstate basis. The Hilbert space is coarse-grained into energy shells of width $\\Delta_E$, with which one can define a block submatrix $\\tilde{O}^{(b,a)}$ consisting of elements between eigenstates in the $a$th and $b$th shells. Each block submatrix is characterized by constant values of $E_{\\gamma\\alpha}=(E_\\gamma+E_\\alpha)/2 \\simeq \\tilde{E}$ and $\\omega_{\\gamma\\alpha}= (E_\\gamma-E_\\alpha) \\simeq \\omega$ up to $\\Delta_E$. We will show that all matrix elements within a block are statistically equivalent to each other in the non-integrable case. Their distribution is characterized by $\\bar{E}$ and $\\omega$, and follows the prediction of the ETH. In stark contrast, eigenstate-to-eigenstate fluctuations persist in the integrable case. Consequently, matrix elements $O_{\\gamma\\alpha}$ cannot be characterized by the energy parameters $E_{\\gamma\\alpha}$ and $\\omega_{\\gamma\\alpha}$ only. Our result explains the origin for the breakdown of the fluctuation dissipation theorem in the integrable system. The eigenstate-to-eigenstate fluctuations sheds a new light on the meaning of the ETH.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-08-21T05:40:58.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "eigenstate thermalization hypothesis",
      "matrix elements",
      "block submatrix",
      "eigenstate-to-eigenstate fluctuations persist",
      "fluctuation dissipation theorem"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 8,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}