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

为了考察有限系统尺寸的影响，我们用计算机模拟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。本研究取得的主要成果如下：1.研究结果。在体扩展态和边缘态之间存在着连续的跃迁。当属于不同Landau子带的体态和边态能量简并时，它们的杂化发生，其程度取决于杂质势的关联长度。离域态具有分维性，且分维值小于2。当能量从边缘态区向体区减小时，分维从各向异性到各向同性变化。数值验证了与霍尔电导量子化密切相关的本征态质心求和规则。霍尔电流的补偿不仅在全球范围内实现，而且在当地也实现了。因此，在霍尔电导率的平台区，即使在无序体系中，霍尔电流的分布也变得均匀。在霍尔电导率随填充因子增加的区域，表现出较大的起伏，并且我们发现霍尔电流分布存在较大的空间起伏。结果表明，由栅势垒控制的电流不仅在实验上而且在理论上都是研究系统内部电流分布的一种非常有用的技术。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，我们还不能明确霍尔电导量子化精度的理论极限。

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.