Decoherence-induced conductivity in the discrete 1D Anderson model: A novel approach to even-order generalized Lyapunov exponents

TL;DR

This paper investigates how decoherence affects electron transport in the 1D Anderson model, deriving exact Lyapunov exponents and analyzing the transition from localized to ohmic behavior based on coherence length.

Contribution

It provides exact calculations of generalized Lyapunov exponents for the Anderson model and introduces a new approximation for the localization length under decoherence effects.

Findings

Exact second-order Lyapunov exponent for infinite system

Calculation of higher even-order Lyapunov exponents

Effective approximation for localization length in weak disorder

Abstract

A recently proposed statistical model for the effects of decoherence on electron transport manifests a decoherence-driven transition from quantum-coherent localized to ohmic behavior when applied to the one-dimensional Anderson model. Here we derive the resistivity in the ohmic case and show that the transition to localized behavior occurs when the coherence length surpasses a value which only depends on the second-order generalized Lyapunov exponent $\xi^{-1}$. We determine the exact value of $\xi^{-1}$ of an infinite system for arbitrary uncorrelated disorder and electron energy. Likewise all higher even-order generalized Lyapunov exponents can be calculated, as exemplified for fourth order. An approximation for the localization length (inverse standard Lyapunov exponent) is presented, by assuming a log-normal limiting distribution for the dimensionless conductance $T$. This approximation works well in the limit of weak disorder, with the exception of the band edges and the band center.