二维安德森局部化：局部化景观理论，精确对角化和时间依赖模拟的见解
Anderson localisation in two dimensions: insights from Localisation Landscape Theory, exact diagonalisation, and time-dependent simulations
摘要：
受冷原子系统实验进展的启发，我们使用并改进了局域化景观理论（LLT），来研究具有点状随机散射体的二维系统。我们首先表明，准确的本征态不能有效地用于提取本地化长度。然后，我们提供了一个全面的审查已知的LLT，并确认与LLT的有效势的哈密顿量具有非常相似的低能本征态的物理势。接下来，我们使用LLT来计算非常低能量的最大局部化本征态的局部化长度，并测试我们的方法对精确对角化的影响。此外，我们提出了一个传输实验，最佳检测安德森本地化，并演示如何可以提取的长度尺度与（一般小于）的本地化长度。此外，我们还研究了从一维到二维的维度交叉，为已有的趋势提供了新的解释。来自LLT的流动性边缘的预测进行了测试，直接薛定谔时间演化，被发现是非物理的。此外，我们调查扩大波包，发现这些可以是有用的检测和量化安德森本地化的传输实验中，唯一的缺点是无法解决的本地化长度的能量依赖。然后，我们利用LLT揭示了离散无序晶格和连续二维无序系统的安德森模型之间的联系，这提供了强大的新见解。从这里，我们表明，本地化可以区分从其他效果的比较，在一个有序的潜在的所有其他属性不变的动态。最后，我们彻底调查的影响，加速和排斥粒子间的相互作用，为当前的实验相关。
英文摘要：
Motivated by experimental progress in cold atomic systems, we use and advance Localisation Landscape Theory (LLT), to examine two-dimensional systems with point-like random scatterers. We begin by showing that exact eigenstates cannot be efficiently used to extract the localisation length. We then provide a comprehensive review of known LLT, and confirm that the Hamiltonian with the effective potential of LLT has very similar low energy eigenstates to that with the physical potential. Next, we use LLT to compute the localisation length for very low-energy, maximally localised eigenstates and test our method against exact diagonalisation. Furthermore, we propose a transmission experiment that optimally detects Anderson localisation, and demonstrate how one may extract a length scale which is correlated with (and in general smaller than) the localisation length. In addition, we study the dimensional crossover from one to two dimensions, providing a new explanation to the established trends. The prediction of a mobility edge coming from LLT is tested by direct Schrödinger time evolution and is found to be unphysical. Moreover, we investigate expanding wavepackets, to find that these can be useful in detecting and quantifying Anderson localisation in a transmission experiment, with the only disadvantage being the inability to resolve the energy dependence of the localisation length. Then, we utilise LLT to uncover a connection between the Anderson model for discrete disordered lattices and continuous two-dimensional disordered systems, which provides powerful new insights. From here, we demonstrate that localisation can be distinguished from other effects by a comparison to dynamics in an ordered potential with all other properties unchanged. Finally, we thoroughly investigate the effect of acceleration and repulsive interparticle interactions, as relevant for current experiments.
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