Two-dimensional quantum percolation on anisotropic lattices

TL;DR

This study investigates quantum percolation on anisotropic 2D lattices, revealing evidence of a delocalized phase at low disorder and clarifying discrepancies in previous research regarding localization transitions.

Contribution

It provides a detailed analysis of anisotropic lattice geometries, demonstrating the existence of delocalized states and phase transitions using transmission coefficients and participation ratios.

Findings

Localization length does not converge at low dilution for anisotropic strips.

Evidence of a delocalized phase at small disorder levels.

Phase transition from delocalized to localized near specific dilutions.

Abstract

In a previous work [Dillon and Nakanishi, Eur. Phys.J B {\bf 87}, 286 (2014)], we calculated the transmission coefficient of the two-dimensional quantum percolation model and found there to be three regimes, namely, exponentially localized, power-law localized, and delocalized. However, the existence of these phase transitions remains controversial, with many other works divided between those which claim that quantum percolation in 2D is always localized, and those which assert there is a transition to a less localized or delocalized state. It stood out that many works based on highly anisotropic two-dimensional strips fall in the first group, whereas our previous work and most others in the second group were based on an isotropic square geometry. To better understand the difference in our results and those based on strip geometry, we apply our direct calculation of the transmission coefficient to highly anisotropic strips of varying widths at three energies and a wide range of dilutions. We find that the localization length of the strips does not converge at low dilution as the strip width increases toward the isotropic limit, indicating the presence of a delocalized state for small disorder. We additionally calculate the inverse participation ratio of the lattices and find that it too signals a phase transition from delocalized to localized near the same dilutions.