Lyapunov exponents in 1d disordered system with long-range memory

TL;DR

This paper investigates Lyapunov exponents in a one-dimensional disordered system with long-range correlated Gaussian potential, revealing exponential growth of eigenfunction moments and positive Lyapunov exponents indicating localization.

Contribution

It introduces analysis of Lyapunov exponents in 1D disordered systems with power-law correlated potentials, highlighting the impact of long-range memory.

Findings

Exponential growth of eigenfunction moments observed.

Positive Lyapunov exponents identified.

Correlation decay influences localization properties.

Abstract

The Lyapunov exponents for Anderson localization are studied in a one dimensional disordered system. A random Gaussian potential with the power law decay $\sim 1/|x|^q$ of the correlation function is considered. The exponential growth of the moments of the eigenfunctions and their derivative is obtained. Positive Lyapunov exponents, which determine the asymptotic growth rate are found.