{
  "id": "2209.01214",
  "version": "v1",
  "published": "2022-09-03T02:15:49.000Z",
  "updated": "2022-09-03T02:15:49.000Z",
  "title": "General theory of emergent elasticity and topological stability",
  "authors": [
    "Yangfan Hu"
  ],
  "categories": [
    "cond-mat.mes-hall",
    "physics.class-ph"
  ],
  "abstract": "The raise of the symmetry breaking mechanism by Landau[1] is a landmark in the studies of phase transitions. The Kosterlitz-Thouless phase transition[2-3] and the Fractional quantum Hall effect[4], however, are believed to be induced by another mechanism: topology change. Despite rapid development of the theory of topological orders[5-7], a unified mathematical language describing this new paradigm of phase transition and its relation to the Landau paradigm, is not seen. Here, we show that the critical condition for any second-order phase transitions including topological phase transitions is loss of positive-definiteness of a second-order variation of the free energy. For topological phase transitions, the variation is performed with respect to the emergent displacements of the state variables, which is introduced by constructing an emergent elasticity problem for any stable field solution. The Landau paradigm of phase transitions studies global property changes induced instability, while topology phase transitions study local property changes induced topological instability. By applying the theory to study several examples of the band structure of crystals, we show that it can also be used to study changes of global topological properties. Every field is emergent elastic, with a spatially dependent stiffness, and topological instability initiates at the softened points.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-09-03T02:15:49.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "emergent elasticity",
      "topological",
      "general theory",
      "transitions study local property",
      "studies global property changes"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}