Critical Phenomena of Dynamical Delocalization in Quantum Maps:standard map and Anderson map

Abstract

Following the paper exploring the Anderson localization of monochromatically perturbed kicked quantum maps [Phys.Rev. E{\bf 97},012210], the delocalization-localization transition phenomena in polychromatically perturbed quantum maps (QM) is investigated focusing particularly on the dependency of critical phenomena upon the number $M$ of the harmonic perturbations, where $M+1=d$ corresponds to the spatial dimension of the ordinary disordered lattice. The standard map and the Anderson map are treated and compared. As the basis of analysis, we apply the self-consistent theory (SCT) of the localization, taking a plausible hypothesis on the mean-free-path parameter which worked successfully in the analyses of the monochromatically perturbed QMs. We compare in detail the numerical results with the predictions of the SCT, by largely increasing $M$. The numerically obtained index of critical subdiffusion $t^{\alpha }$~($t$:time) agrees well with the prediction of one-parameter scaling theory $\alpha =2/(M+1)$, but the numerically obtained critical exponent of localization length significantly deviates from the SCT prediction. Deviation from the SCT prediction is drastic for the critical perturbation strength of the transition: if $M$ is fixed the SCT presents plausible prediction for the parameter dependence of the critical value, but its value is $1/(M-1)$times smaller than the SCT prediction, which implies existence of a strong cooperativity of the harmonic perturbations with the main mode.

15 pages, 14 figures

References in corpus (4)

• Critical properties of the Anderson localization transition and the high dimensional limit
• Dimensional dependence of the metal-insulator transition
• A semiclassical theory of the Anderson transition
• Scaling Properties of Dynamical Localization in Monochromatically Perturbed Quantum Maps: standard map and Anderson map

Cited by in corpus (4)

• Localization and delocalization properties in quasi-periodically driven one-dimensional disordered system
• Presence and Absence of Delocalization-localization Transition in Coherently Perturbed Disordered Lattices
• Localized-Diffusive and Ballistic-Diffusive Transitions in Kicked Incommensurate lattice
• Localization and delocalization properties in quasi-periodically perturbed Kicked Harper and Harper models