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 time 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 well-established 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 Edwards and Thouless (1972), in which the hopping at one lattice edge is changed from periodic to antiperiodic boundary conditions. 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 recently in Šuntajs et al. (2020). The latter approach seeks for the optimal scaling solution in the vicinity of the crossing point, while at the same time it allows for the crossing point to drift due to finite-size corrections. We find that the extracted transition point and the scaling coefficient agree with those from the literature to high numerical accuracy.

三维安德森模型代表理解安德森本地化过渡的范式模型。在这项工作中，我们首先回顾一下该模型在过去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其性质。我们的主要重点是水平灵敏度统计数据与水平统计数据之间的定量比较。前者研究插入磁通量时哈密顿本征水平的敏感性，而后者研究未受干扰本征水平的性质。我们定义了无因次电导的两个版本，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，第二个对应于从频谱形状因子中获得的海森堡时间与无用时间之比。我们显示，两种电导在本地化过渡过程中看起来都非常相似，特别是，它们预测的临界点与过渡过程中其他公认的措施几乎一致。然后，我们研究这些量的其他一些属性：对于水准曲率，我们讨论了水准曲率分布的宽度与Edwards和Thouless在其开创性工作Edwards和Thouless（1972）中研究的特征能量之间的特殊相似之处和不同之处。一个晶格边缘的跳变从周期性边界条件变为反周期性边界条件。在频谱形状因数的背景下，我们表明，在临界点它进入了一个与时间无关的宽泛范围，其值与从水平方差获得的水平可压缩性一致。最后，我们使用Šuntajs等人最近引入的成本函数最小化方法，测试了交叉状态下平均水平间距比的缩放解。（2020）。后一种方法在交叉点附近寻求最佳缩放比例解决方案，同时又允许交叉点由于有限大小的校正而发生漂移。我们发现，提取的过渡点和比例系数与文献中的数值精度高相吻合。我们使用最近在Šuntajs等人中引入的成本函数最小化方法，测试了交叉状态下平均水平间距比的缩放解。（2020）。后一种方法在交叉点附近寻求最佳缩放比例解决方案，同时又允许交叉点由于有限大小的校正而发生漂移。我们发现，提取的过渡点和比例系数与文献中的数值精度高相吻合。我们使用最近在Šuntajs等人中引入的成本函数最小化方法，测试了交叉状态下平均水平间距比的缩放解。（2020）。后一种方法在交叉点附近寻求最佳缩放比例解决方案，同时又允许交叉点由于有限大小的校正而发生漂移。我们发现，提取的过渡点和比例系数与文献中的数值精度高相吻合。