Anderson localisation in tight-binding models with flat bands

TL;DR

This paper investigates how weak disorder affects eigenstates in flat-band tight-binding models, revealing that these states are typically critical and neither localized nor extended, with insights supported by numerical simulations.

Contribution

It demonstrates that flat-band eigenstates become critical under weak disorder and establishes a link to resonance phenomena in random impedance networks.

Findings

Flat-band states are critical with weak disorder.

Mapping to resonance problems in impedance networks.

Numerical evidence from 2D lattice models.

Abstract

We consider the effect of weak disorder on eigenstates in a special class of tight-binding models. Models in this class have short-range hopping on periodic lattices; their defining feature is that the clean systems have some energy bands that are dispersionless throughout the Brillouin zone. We show that states derived from these flat bands are generically critical in the presence of weak disorder, being neither Anderson localised nor spatially extended. Further, we establish a mapping between this localisation problem and the one of resonances in random impedance networks, which previous work has suggested are also critical. Our conclusions are illustrated using numerical results for a two-dimensional lattice, known as the square lattice with crossings or the planar pyrochlore lattice.