Study of the Localization Transition on a Cayley-tree via Spectral Statistics

TL;DR

This paper investigates the spectral statistics of a Cayley-tree, revealing a localization transition influenced by boundary conditions and identifying a critical disorder point with associated critical behavior.

Contribution

It demonstrates how boundary connections restore universal spectral statistics and characterizes the localization transition with precise critical parameters.

Findings

Spectral statistics are non-universal due to boundary effects.

Connecting boundary sites restores universal spectral statistics.

A clear localization transition occurs at a critical disorder W_c=11.44.

Abstract

The spectral statistics of a Cayley-tree is numerically studied. The statistics are non-universal due to the high ratio of boundary sites. Once the boundary sites are connected to each other in a way that preserves the local structure of the tree the universal statistics of the spectra is recovered. A clear localization transition is observed as function of on-site disorder strength, with a critical disorder $W_c=11.44^{+0.08}_{-0.04}$ and critical index $\nu=0.51^{+0.05}_{-0.04}$. The value of $\nu$ fits nicely to its mean field value, while the value of $W_c$ is puzzling.