We present an analytical method, based on a real space decimation scheme, to extract the exact eigenvalues of a macroscopically large set of pinned localized excitations in a Cayley tree fractal network. Within a tight binding scheme we exploit the above method to scrutinize the effect of a deterministic deformation of the network, first through a hierarchical distribution in the values of the nearest neighbor hopping integrals, and then through a radial Aubry–André-Harper quasiperiodic modulation. With increasing generation index, the inflating loop-less tree structure hosts pinned eigenstates on the peripheral sites that spread from the outermost rings into the bulk of the sample, resembling the spread of a forest fire, ‘lighting up’ a predictable set of sites and leaving the rest ‘un-ignited’. The penetration depth of the envelope of amplitudes can be precisely engineered. The quasiperiodic modulation yields hitherto unreported quantum butterflies, which have further been investigated by calculating the inverse participation ratio for the eigenstates, and a multifractal analysis. The applicability of the scheme to photonic fractal waveguide networks is discussed at the end.

我们提出了一种基于实空间抽取方案的分析方法，以提取 Cayley 树分形网络中宏观大的固定局部激发集的精确特征值。在紧绑定方案中，我们利用上述方法来检查网络确定性变形的影响，首先通过最近邻跳跃积分值的分层分布，然后通过径向 Aubry-André-Harper 准周期调制。随着生成指数的增加，膨胀的无环树结构在外围站点上承载固定本征态，从最外环扩散到样本的大部分，类似于森林火灾的蔓延，“点亮”一组可预测的站点和让其余的“未点燃”。可以精确设计振幅包络的穿透深度。准周期调制产生迄今为止未报道的量子蝴蝶，通过计算本征态的反参与比和多重分形分析进一步研究了量子蝴蝶。该方案对光子分形的适用性最后讨论波导网络。