{
  "id": "1908.07502",
  "version": "v1",
  "published": "2019-08-20T17:27:12.000Z",
  "updated": "2019-08-20T17:27:12.000Z",
  "title": "Fractal dimension of critical curves in the $O(n)$-symmetric $φ^4$-model and crossover exponent at 6-loop order: Loop-erased random walks, self-avoiding walks, Ising, XY and Heisenberg models",
  "authors": [
    "Mikhail Kompaniets",
    "Kay Joerg Wiese"
  ],
  "comment": "16 pages, 18 figures, 7 tables",
  "categories": [
    "cond-mat.stat-mech",
    "hep-th"
  ],
  "abstract": "We calculate the fractal dimension $d_{\\rm f}$ of critical curves in the $O(n)$ symmetric $(\\vec \\phi^2)^2$-theory in $d=4-\\varepsilon$ dimensions at 6-loop order. This gives the fractal dimension of loop-erased random walks at $n=-2$, self-avoiding walks ($n=0$), Ising lines $(n=1)$, and XY lines ($n=2$), in agreement with numerical simulations. It can be compared to the fractal dimension $d_{\\rm f}^{\\rm tot}$ of all lines, i.e. backbone plus the surrounding loops, identical to $d_{\\rm f}^{\\rm tot} = 1/\\nu$. The combination $\\phi_{\\rm c}= d_{\\rm f}/d_{\\rm f}^{\\rm tot} = \\nu d_{\\rm f}$ is the crossover exponent, describing a system with mass anisotropy. Introducing a novel self-consistent resummation procedure, and combining it with analytic results in $d=2$ allows us to give improved estimates in $d=3$ for all relevant exponents at 6-loop order.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-08-20T17:27:12.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "fractal dimension",
      "loop-erased random walks",
      "crossover exponent",
      "self-avoiding walks",
      "critical curves"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 16,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}