Theoretical Study of the Quantum Hall Effect in Systems with Finite Width.

为了研究有限系统尺寸的影响，我们用计算机模拟的方法研究了二维无序电子系统在垂直强磁场作用下的电子态和电流分布。本研究的主要结果如下：1.在体扩展态和边缘态之间存在连续跃迁。当属于不同朗道子带的体态和边态能量简并时，它们会发生杂化，其程度取决于杂质势的关联长度.离域态具有分形特征，并以小于2的分形维数扩展。随着能量从边态区到体区的降低，分形维数从各向异性到各向同性变化.数值计算证实了与霍尔电导量子化密切相关的本征态质心求和规则.霍尔电流的补偿已被发现不仅在全局上而且在局部上实现。因此，在霍尔电导率的平台区，霍尔电流分布变得均匀，即使在无序系统.在霍尔电导率作为填充因子的函数上升的区域中，它显示出大的波动，并且我们发现霍尔电流分布中存在大的空间波动。研究表明，利用栅势垒控制电流是研究系统内部电流分布的一种非常有用的方法，库仑相互作用的引入在一定程度上减小了霍尔电导的表观尺寸效应。然而，我们不能成功地澄清霍尔电导率量子化的准确性的理论极限。

In order to investigate the effect of the finite system size we studied the electronic states and the current distribution in two dimensional disordered electron systems subject to perpendicular strong magnetic fields, by using computer simulations. The main results obtained in this study are as follows.1. There exist continuous transition between the bulk extended states and the edge states. When the bulk and edge states belonging to different Landau subbands are energetically degenerate, their hybridization occurs and its extent depends on the correlation length of impurity potentials.2. The delocalized states have a fractal character and extend with fractal dimensions smaller than two. As the energy decreases from the edge state region to the bulk region, the fractal dimensions vary from the anisotropic to the isotropic one.3. The sum rule about the center of mass of eigenstates, which is intimately related to the quantization of the Hall conductivity, has been confirmed numerically.4. The compensation of the Hall current has been found to be realized not only globally but also locally. Therefore, in the plateau region of the Hall conductivity, the Hall current distribution becomes uniform even in the disordered systems.5. In the region where the Hall conductivity as a function of the filling factor rises up, it shows large fluctuations and we have found large spatial fluctuations in the Hall current distribution.6. It has been found that the current control by the gate barrier is a quite useful technique in studying the current distribution inside the system not only experimentally but also theoretically.The inclusion of the Coulomb interaction has been found to reduce the apparent size effect on the Hall conductivity to some extent. However we could not succeed to clarity the theoretical limit of the accuracy of the Hall conductivity quantization.