三维Anderson模型的谱性质
Spectral properties of three-dimensional Anderson model
摘要：
三维安德森模型是理解安德森局域化转变的范例模型。在这项工作中，我们首先回顾了在过去的50年中获得的一些关键结果，然后从现代数值方法的角度研究它的性质。我们的主要重点是水平敏感性统计量和水平统计量之间的定量比较。前者研究哈密顿本征能级对引入磁通量的敏感性，后者研究未扰动本征能级的性质。我们定义了两个版本的无量纲电导，第一个对应于相对于平均水平间距的水平曲率分布的宽度，和第二个对应于从光谱形状因子获得的海森堡和无量纲时间的比率。我们发现，这两个电导看起来非常相似的本地化过渡，特别是，他们预测一个几乎相同的临界点与其他措施的过渡。然后，我们研究了这些量的一些进一步的性质：对于水平曲率，我们讨论了爱德华兹和里瑟莱斯在他们的开创性工作中研究的水平曲率分布的宽度和特征能量之间的特殊相似性和差异。5，807（1972）]。在背景下的频谱形状因子，我们表明，在临界点，它进入一个广泛的时间无关的制度，在该制度中，它的值是一致的水平压缩性从水平方差。最后，我们测试的平均水平间距比在交叉制度的成本函数最小化方法介绍了[物理。102，064207（2020）]。我们发现，提取的转捩点和标度系数与文献中的数值精度很高。
英文摘要：
The three-dimensional Anderson model represents a paradigmatic model to understand the Anderson localization transition. In this work we first review some key results obtained for this model in the past 50 years, and then study its properties from the perspective of modern numerical approaches. Our main focus is on the quantitative comparison between the level sensitivity statistics and the level statistics. While the former studies the sensitivity of Hamiltonian eigenlevels upon inserting a magnetic flux, the latter studies the properties of unperturbed eigenlevels. We define two versions of dimensionless conductance, the first corresponding to the width of the level curvature distribution relative to the mean level spacing, and the second corresponding to the ratio of the Heisenberg and the Thouless time obtained from the spectral form factor. We show that both conductances look remarkably similar around the localization transition, in particular, they predict a nearly identical critical point consistent with other measures of the transition. We then study some further properties of those quantities: for level curvatures, we discuss particular similarities and differences between the width of the level curvature distribution and the characteristic energy studied by Edwards and Thouless in their pioneering work [J. Phys. C. 5, 807 (1972)]. In the context of the spectral form factor, we show that at the critical point it enters a broad time-independent regime, in which its value is consistent with the level compressibility obtained from the level variance. Finally, we test the scaling solution of the average level spacing ratio in the crossover regime using the cost function minimization approach introduced in [Phys. Rev. B. 102, 064207 (2020)]. We find that the extracted transition point and the scaling coefficient agree with those from the literature to high numerical accuracy.
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