Exact evaluation of the causal spectrum and localization properties of electronic states on a scale-free network

TL;DR

This paper analytically characterizes the energy spectrum and localization properties of electronic states on a scale-free tree network using a causal chain approach, providing explicit formulas for eigenvalues and localization measures.

Contribution

It introduces an exact analytical method to evaluate the causal spectrum and localization of electronic states on a scale-free network, linking eigenvalues to localization properties.

Findings

Full energy spectrum expressed via a causal chain.

Analytical expressions for eigenstate localization (IPR).

Scaling behavior of IPR related to eigenvalue chain tails.

Abstract

A nearest-neighbor tight-binding model on a tree structure is investigated. The full energy spectrum of the normalized Hamiltonian can be expressed in terms of successively increasing number of contributions at any finite step of construction of the tree, resulting in a causal chain. The degree of quantum localization of any eigenstate, measured by the inverse participation ratio (IPR), is also analytically expressed by means of terms in corresponding eigenvalue chain. The resulting IPR scaling behavior is expressed by the tails of eigenvalue chains as well.