{
  "id": "2309.08059",
  "version": "v1",
  "published": "2023-09-14T23:03:14.000Z",
  "updated": "2023-09-14T23:03:14.000Z",
  "title": "Cellular automata in $d$ dimensions and ground states of spin models in $(d+1)$ dimensions",
  "authors": [
    "Konstantinos Sfairopoulos",
    "Luke Causer",
    "Jamie F. Mair",
    "Juan P. Garrahan"
  ],
  "comment": "22 pages, 21 figures",
  "categories": [
    "cond-mat.stat-mech",
    "quant-ph"
  ],
  "abstract": "We show how the trajectories of $d$-dimensional cellular automata (CA) can be used to determine the ground states of $(d+1)$-dimensional classical spin models, and we characterise their quantum phase transition, when in the presence of a transverse magnetic field. For each of the 256 one-dimensional elementary CA we explicitly construct the simplest local two-dimensional classical spin model associated to the given CA, and we also describe this method for $d>1$ through selected examples. We illustrate our general observations with detailed studies of: (i) the $d=1$ CA Rule 150 and its $d=2$ four-body plaquette spin model, (ii) the $d=2$ CA whose associated model is the $d=3$ square-pyramid plaquette model, and (iii) two counter-propagating $d=1$ Rule 60 CA that correspond to the two-dimensional Baxter-Wu spin model. For the quantum spin models, we show that the connection to CAs implies a sensitivity on the approach to the thermodynamic limit via finite size scaling for their quantum phase transitions.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-09-14T23:03:14.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "cellular automata",
      "ground states",
      "dimensions",
      "quantum phase transition",
      "two-dimensional baxter-wu spin model"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 22,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}