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

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 ξ −1 . We determine the exact value of ξ −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.

Localization under the effect of randomly distributed decoherence

“…It can be related by (11) to a critical phasecoherence length and is a function of the disorder strength σ, see (10). In a completely different way, we obtained (17) already in [8]. Here, its derivation by the new analytical formula (8) and (12) allows to understand the statistical origin of the decoherence-induced insulator-metal transition: Localization is found to survive decoherence, when the exponentially increasing resistance of the long coherent subsystems (8) exceeds their exponentially decreasing frequency of occurrence (12).…”

“…In a completely different way, we obtained (17) already in [8]. Here, its derivation by the new analytical formula (8) and (12) allows to understand the statistical origin of the decoherence-induced insulator-metal transition: Localization is found to survive decoherence, when the exponentially increasing resistance of the long coherent subsystems (8) exceeds their exponentially decreasing frequency of occurrence (12). Any decoherence distribution, for which the number u j of coherent subsystems decreases with their length j faster than exponentially will show only Ohmic behavior.…”

“…However, as shown in figure 3, the decoherence induced transition appears for attached reservoirs (squares) as well as for completely phase randomizing reservoirs (circles). Contributions from 2 Note that our parameter ξ −1 is not the inverse localization length, which is generally defined as limN→∞ 1 2N log TN , but the second-order generalized Lyapunov exponent, defined as limN→∞ 1 N log TN , see [8]. Anyway, both quantities are a measure for the localization in the system.…”

“…In this paper, we address this question using a statistical model for the effects of decoherence [7][8][9]11]. We use virtual decoherence reservoirs [22,23], where the electrons are absorbed and reinjected after randomization of phase and momentum.…”

“…This raises then the fundamental, but up to now only partially answered question, whether decoherence enhances or reduces transport. In this respect tight-binding chains [5][6][7][8][9][10], molecular wires [11][12][13] and aggregates [14][15][16][17] were studied. Surprisingly it was found that the transport can be enhanced by decoherence, a phenomenon referred to as decoherence-assisted transport.…”