{
  "id": "1506.07128",
  "version": "v1",
  "published": "2015-06-23T18:38:18.000Z",
  "updated": "2015-06-23T18:38:18.000Z",
  "title": "Introduction to Mean-Field Theory of Spin Glass Models",
  "authors": [
    "Václav Janiš"
  ],
  "comment": "28 pages, Lecture Notes for the Autumn School on Correlated Electrons, J\\\"ulich, 2015 (http://www.cond-mat.de/events/correl15). arXiv admin note: text overlap with arXiv:1501.01653",
  "categories": [
    "cond-mat.dis-nn"
  ],
  "abstract": "We discuss the mean-field theory of spin-glass models with frustrated long-range random spin exchange. We analyze the reasons for breakdown of the simple mean-field theory of Sherrington and Kirkpatrick. We relate the replica-symmetry breaking to ergodicity breaking and use the concept of real replicas to restore thermodynamic homogeneity of the equilibrium free energy in a replicated phase space. Embedded replications of the spin variables result in a set of hierarchical free energies and overlap susceptibilities between replica hierarchies as order parameters. The limit to infinite number of replica hierarchies leads to the Parisi solution with a continuous replica-symmetry breaking. We present a closed-form representation of the Parisi mean-field theory that is independent from stability of solutions with finite-many replica hierarchies. Hence, solutions with continuous and discrete replica-symmetry breaking can coexist. We demonstrate the construction of the spin-glass mean-field solutions via real replicas on three spin-glass models: Ising, Potts and p-spin. An asymptotic expansion in each model is used to demonstrate various types of the transition from the paramagnetic to the glassy phase.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-06-23T18:38:18.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "mean-field theory",
      "spin glass models",
      "replica hierarchies",
      "introduction",
      "spin-glass models"
    ],
    "tags": [
      "lecture notes"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 28,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015arXiv150607128J"
    }
  }
}