{
  "id": "1708.04978",
  "version": "v1",
  "published": "2017-08-16T17:10:55.000Z",
  "updated": "2017-08-16T17:10:55.000Z",
  "title": "Multifractality of wave functions on a Cayley tree: From root to leaves",
  "authors": [
    "M. Sonner",
    "K. S. Tikhonov",
    "A. D. Mirlin"
  ],
  "categories": [
    "cond-mat.dis-nn"
  ],
  "abstract": "We explore the evolution of wave-function statistics on a finite Bethe lattice (Cayley tree) from the central site (\"root\") to the boundary (\"leaves\"). We show that the eigenfunction moments $P_q=N \\left<|\\psi|^{2q}(i)\\right>$ exhibit a multifractal scaling $P_q\\propto N^{-\\tau_q}$ with the volume (number of sites) $N$ at $N\\to\\infty$. The multifractality spectrum $\\tau_q$ depends on the strength of disorder and on the parameter $s$ characterizing the position of the observation point $i$ on the lattice. Specifically, $s= r/R$, where $r$ is the distance from the observation point to the root, and $R$ is the \"radius\" of the lattice. We demonstrate that the exponents $\\tau_q$ depend linearly on $s$ and determine the evolution of the spectrum with increasing disorder, from delocalized to the localized phase. Analytical results are obtained for the $n$-orbital model with $n \\gg 1$ that can be mapped onto a supersymmetric $\\sigma$ model. These results are supported by numerical simulations (exact diagonalization) of the conventional ($n=1$) Anderson tight-binding model.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-08-16T17:10:55.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "cayley tree",
      "wave functions",
      "multifractality",
      "observation point",
      "finite bethe lattice"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}