Quantum Transports in Two-Dimensions with Long Range Hopping: Shielding, Localization and the Extended Isolated State

TL;DR

This paper explores how long-range hopping and disorder influence quantum transport in two-dimensional systems, revealing a hybrid localization-delocalization state, shielding effects, and a robust isolated extended state.

Contribution

It demonstrates the emergence of a marginal phase with hybrid localization features and identifies a robust isolated extended state due to all-to-all coupling.

Findings

Perfect conducting channels in weak disorder due to shielding

Wavefunctions exhibit hybrid localization with fractal dimensions

Existence of a robust isolated extended state far out of the band

Abstract

We investigate the effects of disorder and shielding on quantum transports in a two dimensional system with all-to-all long range hopping. In the weak disorder, cooperative shielding manifests itself as perfect conducting channels identical to those of the short range model, as if the long range hopping does not exist. With increasing disorder, the average and fluctuation of conductance are larger than those in the short range model, since the shielding is effectively broken and therefore long range hopping starts to take effect. Over several orders of disorder strength (until $\sim 10^4$ times of nearest hopping), although the wavefunctions are not fully extended, they are also robustly prevented from being completely localized into a single site. Each wavefunction has several localization centers around the whole sample, thus leading to a fractal dimension remarkably smaller than 2 and also remarkably larger than 0, exhibiting a hybrid feature of localization and delocalization. The size scaling shows that for sufficiently large size and disorder strength, the conductance tends to saturate to a fixed value with the scaling function $\beta\sim 0$, which is also a marginal phase between the typical metal ($\beta>0$) and insulating phase ($\beta<0$). The all-to-all coupling expels one isolated but extended state far out of the band, whose transport is extremely robust against disorder due to absence of backscattering. The bond current picture of this isolated state shows a quantum version of short circuit through long hopping.