Multifractality of wave functions on a Cayley tree: From root to leaves

M. Sonner, K. S. Tikhonov, A. D. Mirlin

Research output: Contribution to journalArticlepeer-review

33 Citations (Scopus)


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 Pq=N(|ψ|2q(i)) exhibit a multifractal scaling Pq∞N-τq with the volume (number of sites) N at N→∞. The multifractality spectrum τq 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 τq 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≫1 that can be mapped onto a supersymmetric σ model. These results are supported by numerical simulations (exact diagonalization) of the conventional (n=1) Anderson tight-binding model.

Original languageEnglish
Article number214204
JournalPhysical Review B
Issue number21
Publication statusPublished - 20 Dec 2017
Externally publishedYes


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