{
  "id": "2501.18569",
  "version": "v1",
  "published": "2025-01-30T18:39:44.000Z",
  "updated": "2025-01-30T18:39:44.000Z",
  "title": "Derivation of the free energy, entropy and specific heat for planar Ising models: Application to Archimedean lattices and their duals",
  "authors": [
    "Laurent Pierre",
    "Bernard Bernu",
    "Laura Messio"
  ],
  "comment": "22 pages, 48 with the appendices, 10 figures, 5 tables",
  "categories": [
    "cond-mat.stat-mech"
  ],
  "abstract": "The 2d ferromagnetic Ising model was solved by Onsager on the square lattice in 1944, and an explicit expression of the free energy density $f$ is presently available for some other planar lattices. An exact derivation of the critical temperature $T_c$ only requires a partial derivation of $f$ and has been performed on many lattices, including the 11 Archimedean lattices. We give general expressions of the free energy, energy, entropy and specific heat for planar lattices with a single type of non-crossing links. The specific heat exhibits a logarithmic singularity at $T_c$: $c_V(T)\\sim -A\\ln|1-T_c/T|$, in all the ferromagnetic and some antiferromagnetic cases. While the non-universal weight $A$ of the leading term has often been evaluated, this is not the case for the sub-leading order term $B$ such that $c_V(T)+A\\ln|1-T_c/T|\\sim B$, despite its strong impact on $c_V(T)$ values in the vicinity of $T_c$, particularly important in experimental measurements. Explicit values of these thermodynamic quantities and of $A$ and $B$ are given for the Archimedean lattices and their dual for both ferromagnetic and antiferromagnetic interactions.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2025-01-30T18:39:44.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "specific heat",
      "archimedean lattices",
      "planar ising models",
      "derivation",
      "application"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 22,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}