{
  "id": "2310.12801",
  "version": "v1",
  "published": "2023-10-19T14:57:27.000Z",
  "updated": "2023-10-19T14:57:27.000Z",
  "title": "Roughness and critical force for depinning at 3-loop order",
  "authors": [
    "Mikhail N. Semeikin",
    "Kay Joerg Wiese"
  ],
  "comment": "24 pages, 17 figures",
  "categories": [
    "cond-mat.dis-nn",
    "hep-th"
  ],
  "abstract": "A $d$-dimensional elastic manifold at depinning is described by a renormalized field theory, based on the Functional Renormalization Group (FRG). Here we analyze this theory to 3-loop order, equivalent to third order in $\\epsilon=4-d$, where $d$ is the internal dimension. The critical exponent reads $\\zeta = \\frac \\epsilon3 + 0.04777 \\epsilon^2 -0.068354 \\epsilon^3 + {\\cal O}(\\epsilon^4)$. Using that $\\zeta(d=0)=2^-$, we estimate $\\zeta(d=1)=1.266(20)$, $\\zeta(d=2)=0.752(1)$ and $\\zeta(d=3)=0.357(1)$. For Gaussian disorder, the pinning force per site is estimated as $f_{\\rm c}= {\\cal B} m^{2}\\rho_m + f_{\\rm c}^0$, where $m^2$ is the strength of the confining potential, $\\cal B$ a universal amplitude, $\\rho_m$ the correlation length of the disorder, and $f_{\\rm c}^0$ a non-universal lattice dependent term. For charge-density waves, we find a mapping to the standard $\\phi^4$-theory with $O(n)$ symmetry in the limit of $n\\to -2$. This gives $f_{\\rm c} = \\tilde {\\cal A}(d) m^2 \\ln (m) + f_{\\rm c}^0 $, with $\\tilde {\\cal A}(d) = -\\partial_n \\big[\\nu(d,n)^{-1}+\\eta(d,n)\\big]_{n=-2}$, reminiscent of log-CFTs.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-10-19T14:57:27.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "critical force",
      "non-universal lattice dependent term",
      "dimensional elastic manifold",
      "functional renormalization group",
      "renormalized field theory"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 24,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}