Conductance of 1D quantum wires with anomalous electron-wavefunction localization

TL;DR

This paper investigates how anomalous electron wavefunction localization with decay $| ext{psi}| \\sim \\exp(-\\lambda r^{\\alpha})$ affects conductance statistics in 1D disordered systems, revealing dependence on the average log conductance and the decay exponent.

Contribution

It introduces a theoretical framework for conductance statistics in systems with anomalous localization, extending beyond conventional Anderson localization, and verifies predictions numerically for various decay exponents.

Findings

Conductance statistics depend on the average \\ln g and the decay exponent \\alpha.

Numerical simulations confirm theoretical predictions for \\alpha=1/2.

The model applies to systems with chiral symmetry and anomalous localization.

Abstract

We study the statistics of the conductance $g$ through one-dimensional disordered systems where electron wavefunctions decay spatially as $|\psi| \sim \exp (-\lambda r^{\alpha})$ for $0 <\alpha <1$, $\lambda$ being a constant. In contrast to the conventional Anderson localization where $|\psi| \sim \exp (-\lambda r)$ and the conductance statistics is determined by a single parameter: the mean free path, here we show that when the wave function is anomalously localized ($\alpha <1$) the full statistics of the conductance is determined by the average $<\ln g>$ and the power $\alpha$. Our theoretical predictions are verified numerically by using a random hopping tight-binding model at zero energy, where due to the presence of chiral symmetry in the lattice there exists anomalous localization; this case corresponds to the particular value $\alpha =1/2$. To test our theory for other values of $\alpha$, we introduce a statistical model for the random hopping in the tight binding Hamiltonian.