# Delocalization in infinite disordered 2D lattices of different geometry

## TL;DR

This paper demonstrates the existence of extended electronic states in disordered 2D lattices of various geometries using a spectral approach, challenging traditional scaling theory predictions and supporting experimental findings.

## Contribution

It introduces a spectral method to show delocalization in 2D disordered lattices, providing new insights into transport behavior across different geometries.

## Key findings

- Extended states exist for nonzero disorder in 2D lattices.
- Triangular and honeycomb lattices show similar transport transitions.
- Honeycomb lattice transition is more abrupt due to fewer neighbors.

## Abstract

The spectral approach to infinite disordered crystals is applied to an Anderson-type Hamiltonian to demonstrate the existence of extended states for nonzero disorder in 2D lattices of different geometries. The numerical simulations shown prove that extended states exist for disordered honeycomb, triangular, and square crystals. This observation stands in contrast to the predictions of scaling theory, and aligns with experiments in photonic lattices and electron systems. The method used is the only theoretical approach aimed at showing delocalization. A comparison of the results for the three geometries indicates that the triangular and honeycomb lattices experience transition in the transport behavior for same amount of disorder, which is to be expected from planar duality. This provides justification for the use of artificially-prepared triangular lattices as analogues for honeycomb materials, such as graphene. The analysis also shows that the transition in the honeycomb case happens more abruptly as compared to the other two geometries, which can be attributed to the number of nearest neighbors. We outline the advantages of the spectral approach as a viable alternative to scaling theory and discuss its applicability to transport problems in both quantum and classical 2D systems.

---
Source: https://tomesphere.com/paper/1706.02800