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 97, 012210 (2018)2470-004510.1103/PhysRevE.97.012210], the delocalization-localization transition phenomena in polychromatically perturbed quantum maps (QM) is investigated focusing particularly on the dependency of critical phenomena on 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 for our systems, 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^{α} (t:time) agrees well with the prediction of one-parameter scaling theory α=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, then 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.