A Study of Quantum Transport Phenomena by use of Multipole Expansions

多极展开的关键思想在于对长距离和短距离动力学的系统分类，并且在物理学的许多分支中都很有用，特别是在量子输运现象的描述中。在过去三年的资助下，我们研究了与利用多极展开建立量子霍尔效应有关的一些问题。霍尔样品中的电流分布是感兴趣的主题。我们在前面指出，霍尔电流的相当一部分将沿着样品边缘流动，这是从局域化主导的样品内部排出的结果。这一理论考虑的关键是扩展和本地化状态之间的霍尔电流的精确补偿。本文证明了电流补偿定理实际上是建立在规范不变性原理的基础上的。本文对有限宽无序霍尔条中的电流分布进行了数值实验，实验结果支持了上述理论分析。证实了边缘态不是霍尔电流的主要载流子。一份文件现在正在审查中。从上述数值实验中提出的量子霍尔效应的击穿机制，现在正在研究中。还对一个干净的霍尔杆进行了数值分析。观察到边缘电流随着边缘态的出现而改变方向，与de Haasvan Alphen振荡密切相关。一篇论文表明霍尔电子系统的宏观电磁性质可以用W * 规范理论的形式简洁地概括。

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.