A Study of Quantum Transport Phenomena by use of Multipole Expansions

多极展开的核心思想在于对长程和短程动力学进行系统的分类，在物理学的许多分支中都很有用，特别是在描述量子输运现象方面。在过去三年的助学金下，我们利用多极展开研究了与量子霍尔效应基础有关的一些问题。霍尔样品中的电流分布是一个令人感兴趣的主题。我们早些时候指出，很大一部分霍尔电流将沿着样品边缘流动，这是从局域化主导的样品内部驱逐出来的结果。这一理论考虑的关键是对扩展态和局域态之间的霍尔电流进行精确补偿。一篇文章证明了电流补偿定理实际上是建立在规范不变原理上的。对无序有限宽霍尔棒中的电流分布进行了数值实验，结果支持了上述理论考虑。证实了边缘态不是霍尔电流的主要载流子。目前正在准备一篇论文。从上述数值实验中提出的量子霍尔效应的击穿机制目前正在研究中。并对纯净的霍尔棒进行了数值分析。观察到边缘电流随着边缘态的出现而方向交替，这与de Haasvan Alphen振荡密切相关。文中指出，霍尔-电子系统的宏观电磁性质可以用W*规范理论的形式巧妙地概括。

The key idea of multipole expansions lies in a systematic classification of long-and short-distance dynamics and is useful in many branches of physics, especially in the description of quantum transport phenomena. Under a Grant-Aid over the past three years, we have studied some problems pertaining to the foundation of the quantum Hall effect by use of multipole expansions.1. The current distribution in a Hall sample is a subject of interest. We pointed out earlier that a considerable portion of the Hall current would flow along the sample edges as a consequence of expulsion out of the localization-dominated sample interior. Crucial to this theoretical consideration is exact compensation of the Hall current among extended and localized states. A paper has been presented to show that the current-compensation theorem actually rests on the principle of gauge invariance.2. A numerical experiment has been made of the current distributions in finite-width Hall bars with disorder, with the result in support of the theoretical consideration mentioned above. It is confirmed that the edge states are not the principal carriers of the Hall current. A paper is now in preparation.3. A mechanism for the breakdown of the quantum Hall effect, suggested from the above numerical experiment, is now under investigation.4. A numerical analysis has also been made for a clean Hall bar. The edge current is observed to alternate in direction with the emergence of the edge states, in close connection to the de Haasvan Alphen oscillation.5. A paper has been presented to show that macroscopic electromagnetic properties of Hall-electron systems are neatly summarized in the form of a W_* gauge theory.