A Study of Quantum Transport Phenomena by use of Multipole Expansions

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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.