准周期链中的局部化：基于局部传播子收敛性的理论
Localisation in quasiperiodic chains: a theory based on convergence of local propagators
摘要：
准周期系统作为肥沃的土壤研究本地化，由于他们的倾向已经在一个维度上表现出丰富的流动性边缘的相图。然而，准周期势的确定性和强相关性的性质提供了不同于无序系统的挑战。出于这一动机，我们提出了一个理论的本地化在准周期链与最近的邻居跳跃，基于本地传播的收敛性;利用的事实，即相关的自能的虚部作为一个概率序参量本地化过渡，重要的是，承认一个连续的分数表示。通过分析这些连续分数的收敛性，可以确定局部化或其不存在，从而产生临界点和流动性边缘。有趣的是，我们发现在临界点的序参量与系统大小的异常标度，符合临界本征态的分形特征。自洽理论在高阶也被认为是概念上连接到理论的基础上继续分数，并发现在实践中收敛到相同的结果。结果举例说明，通过分析三个准周期模型的理论，涵盖了一系列的行为。
英文摘要：
Quasiperiodic systems serve as fertile ground for studying localisation, due to their propensity already in one dimension to exhibit rich phase diagrams with mobility edges. The deterministic and strongly-correlated nature of the quasiperiodic potential nevertheless offers challenges distinct from disordered systems. Motivated by this, we present a theory of localisation in quasiperiodic chains with nearest-neighbour hoppings, based on the convergence of local propagators; exploiting the fact that the imaginary part of the associated self-energy acts as a probabilistic order parameter for localisation transitions and, importantly, admits a continued-fraction representation. Analysing the convergence of these continued fractions, localisation or its absence can be determined, yielding in turn the critical points and mobility edges. Interestingly, we find anomalous scalings of the order parameter with system size at the critical points, consistent with the fractal character of critical eigenstates. Self-consistent theories at high orders are also considered, shown to be conceptually connected to the theory based on continued fractions, and found in practice to converge to the same result. Results are exemplified by analysing the theory for three quasiperiodic models covering a range of behaviour.
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