AI 中文总结
研究量子粒子在无限凯莱树上跳跃，利用种群动力学求解腔方程分析局域态密度分布，发现其有半分形性，对称性质随幂律跳跃分布指数连续变化，系统会从半分形过渡到局域，转变处波函数呈半局域化。
AI 中文摘要
我们研究了一个量子粒子在无限凯莱树上跳跃的情况，其最近邻跳跃幅度在弱连接附近服从$|t|^{-a}$形式的奇异分布且无在位无序。由于该图是二分图，模型具有手性对称性，这强烈影响谱中心本征态的统计。利用种群动力学求解传播子的腔方程，我们分析了局域态密度的分布，发现其呈现出宽幂律尾。这些尾意味着一种特殊的波函数统计形式，即半分形性：本征态占据系统的很大一部分，但它们的高阶矩表现为多重分形态。我们发现局域态密度分布的对称性质不仅由对称类决定，还随控制幂律跳跃分布的指数连续变化。随着该指数变化，系统从半分形区域过渡到局域区域。在转变处，波函数呈现出一种极端的中间形式，即半局域化，在其支撑上同时扩展但根据高阶矩局域化。
英文摘要
We study a quantum particle hopping on an infinite Cayley tree with nearest-neighbor hopping amplitudes drawn from a distribution singular as $|t|^{-a}$ near weak links and no on-site disorder. Because the graph is bipartite, the model has chiral symmetry, which strongly affects the statistics of eigenstates at the center of the spectrum. Using population dynamics to solve the cavity equations for the propagator, we analyze the distribution of the local density of states and show that it develops broad power-law tails. These tails imply an unusual form of wave-function statistics, which we call semi-fractality: the eigenstates occupy an extensive fraction of the system, but their higher moments behave as in a multifractal state. We find that the symmetry properties of the local-density-of-states distribution are not fixed only by the symmetry class, but vary continuously with the exponent controlling the power-law hopping distribution. As this exponent is changed, the system crosses from a semi-fractal regime to a localized one. At the transition, the wave functions realize an extreme intermediate form that we call semi-localized, simultaneously extended in their support but localized according to higher moments.
Comments13 pages, 14 figures. Comments are welcome!
