{
  "id": "1304.2134",
  "version": "v1",
  "published": "2013-04-08T08:37:50.000Z",
  "updated": "2013-04-08T08:37:50.000Z",
  "title": "Dynamical barriers of pure and random ferromagnetic Ising models on fractal lattices",
  "authors": [
    "Cecile Monthus",
    "Thomas Garel"
  ],
  "comment": "24 pages, 7 figures",
  "journal": "J. Stat. Mech. (2013) P06007",
  "doi": "10.1088/1742-5468/2013/06/P06007",
  "categories": [
    "cond-mat.dis-nn"
  ],
  "abstract": "We consider the stochastic dynamics of the pure and random ferromagnetic Ising model on the hierarchical diamond lattice of branching ratio $K$ with fractal dimension $d_f=(\\ln (2K))/\\ln 2$. We adapt the Real Space Renormalization procedure introduced in our previous work [C. Monthus and T. Garel, J. Stat. Mech. P02037 (2013)] to study the equilibrium time $t_{eq}(L)$ as a function of the system size $L$ near zero-temperature. For the pure Ising model, we obtain the behavior $t_{eq}(L) \\sim L^{\\alpha} e^{\\beta 2J L^{d_s}} $ where $d_s=d_f-1$ is the interface dimension, and we compute the prefactor exponent $\\alpha$. For the random ferromagnetic Ising model, we derive the renormalization rules for dynamical barriers $B_{eq}(L) \\equiv (\\ln t_{eq}/\\beta)$ near zero temperature. For the fractal dimension $d_f=2$, we obtain that the dynamical barrier scales as $ B_{eq}(L)= c L+L^{1/2} u$ where $u$ is a Gaussian random variable of non-zero-mean. While the non-random term scaling as $L$ corresponds to the energy-cost of the creation of a system-size domain-wall, the fluctuation part scaling as $L^{1/2}$ characterizes the barriers for the motion of the system-size domain-wall after its creation. This scaling corresponds to the dynamical exponent $\\psi=1/2$, in agreement with the conjecture $\\psi=d_s/2$ proposed in [C. Monthus and T. Garel, J. Phys. A 41, 115002 (2008)]. In particular, it is clearly different from the droplet exponent $\\theta \\simeq 0.299$ involved in the statics of the random ferromagnet on the same lattice.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2013-04-08T08:37:50.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "random ferromagnetic ising model",
      "dynamical barrier",
      "fractal lattices",
      "real space renormalization procedure",
      "fractal dimension"
    ],
    "tags": [
      "journal article"
    ],
    "publication": {
      "journal": "Journal of Statistical Mechanics: Theory and Experiment",
      "year": 2013,
      "month": "Jun",
      "volume": 2013,
      "number": 6,
      "pages": "06007"
    },
    "note": {
      "typesetting": "TeX",
      "pages": 24,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "inspire": 1239233,
      "adsabs": "2013JSMTE..06..007M"
    }
  }
}