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

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.