Electron in a tangled chain: multifractality at the small-world critical point

TL;DR

This paper investigates electron behavior in a small-world network model of conducting polymers, revealing a critical point at zero shortcut density where wave functions become multifractal, indicating a unique type of localization transition.

Contribution

It introduces a simple small-world network model for electron propagation in polymers and demonstrates a critical point at zero shortcut density with multifractal wave functions.

Findings

Critical point at p=0 with finite-size scaling

Wave functions exhibit multifractality

Critical exponent aligns with small-world effect rather than Anderson transition

Abstract

We study a simple model of conducting polymers in which a single electron propagates through a randomly tangled chain. The model has the geometry of a small-world network, with a small density $p$ of crossings of the chain acting as shortcuts for the electron. We use numerical diagonalisation and simple analytical arguments to discuss the density of states, inverse participation ratios and wave functions. We suggest that there is a critical point at $p=0$ and demonstrate finite-size scaling of the energy and wave functions at the lower band edge. The wave functions are multifractal. The critical exponent of the correlation length is consistent with criticality due to the small-world effect, as distinct from the previously discussed, dimensionality-driven Anderson transition.