Localization of the electronic states in a nonstationary chaotic field with long-range correlation

Abstract

The localization problem in nonstationary chaotic field with a long-range correlation, which is generated by the modified Bernoulli map, is numerically studied. The map can generate a stationary $(Bl2)$ and nonstationary $(Bg~2)$ sequence by changing the parameter B. We investigate the effect of correlation on localization in the electronic states by the Lyapunov exponent $\ensuremath{\gamma}$ (the inverse localization length) and the localization length $\ensuremath{\xi}$ in the nonstationary regime. At the band center the potential strength, W, dependence of the Lyapunov exponent shows $\ensuremath{\gamma}\ensuremath{\sim}{W}^{2}$ independent of the correlation strength B, while at the band edge the scaling form changes from $\ensuremath{\gamma}\ensuremath{\sim}{W}^{2/3}$ to $\ensuremath{\gamma}\ensuremath{\sim}{W}^{1/2}$ as the correlation increases. It is also numerically shown that the B dependence of the Lyapunov exponent obeys $\ensuremath{\gamma}\ensuremath{\propto}\ensuremath{-}B$ for $Bl~2$ and exponentially decreases for $Bg2.$ Furthermore we investigate the system size, N, dependence of the localization length. It is confirmed that in the strongly correlated cases $(Bl~3.0)$ the exponential localization remains, i.e., $\ensuremath{\gamma}\ensuremath{\sim}{N}^{\ensuremath{\nu}}(\ensuremath{\nu}\ensuremath{\sim}0),$ even for the weak potential strength $(W=0.01)$ in the thermodynamic limit.