Fractality of wave functions on a Cayley tree: Difference between a tree and a locally tree-like graph without boundary

TL;DR

This paper analyzes the fractal nature of wave functions on Cayley trees and highlights the fundamental differences in eigenfunction behavior between loopless trees and locally tree-like graphs without boundaries.

Contribution

It provides analytical and numerical evidence that wave functions on Cayley trees are fractal in the delocalized phase, contrasting with ergodic extended states on boundary-less graphs.

Findings

Wave function amplitudes are fractal on Cayley trees.

Fractal exponents relate to front propagation parameters.

Extended states on locally tree-like graphs are ergodic.

Abstract

We investigate analytically and numerically eigenfunction statistics in a disordered system on a finite Bethe lattice (Cayley tree). We show that the wave function amplitude at the root of a tree is distributed fractally in a large part of the delocalized phase. The fractal exponents are expressed in terms of the decay rate and the velocity in a problem of propagation of a front between unstable and stable phases. We demonstrate a crucial difference between a loopless Cayley tree and a locally tree-like structure without a boundary (random regular graph) where extended wavefunctions are ergodic.