Following the paper exploring the Anderson localization of monochromatically perturbed kicked quantum maps [Phys. Rev. E 97, 012210 (2018)], 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 $t^{\alpha }$ ($t$: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, 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.

中文翻译：

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量子图中动态离域的关键现象：标准图和安德森图

接下来的论文探讨了单色扰动的踢量子图的安德森局部化[物理学报。启ë 97，012210（2018）]，在polychromatically扰动量子地图离域定位过渡现象（QM）进行了研究尤其集中在临界现象的数量的依赖性$\mathrm{中号}$ 的谐波扰动，其中 $\mathrm{中号}+\mathrm{1个}=d$对应于普通无序晶格的空间尺寸。标准图和安德森图被处理和比较。作为分析的基础，我们对系统应用本地化的自洽理论（SCT），对均值-自由路径参数进行了合理的假设，该参数在单色扰动QM的分析中取得了成功。我们通过大幅增加，将数值结果与SCT的预测进行了详细比较$\mathrm{中号}$。数值计算的临界扩散系数${Ť}^{\alpha }$ （$Ť$：time）与一参数缩放理论的预测非常吻合 $\alpha =2/（\mathrm{中号}+\mathrm{1个}）$，但是从数值上获得的局部化长度的临界指数显着偏离了SCT预测。对于STV预测的临界扰动强度，与SCT预测的偏差很大：$\mathrm{中号}$ 是固定的，则SCT会针对临界值的参数依赖性提出合理的预测，但其值为 $\mathrm{1个}/（\mathrm{中号}-\mathrm{1个}）$ 比SCT预测值小1倍，这意味着谐波扰动与主模之间存在很强的协作性。