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THEORETICAL INVESTIGAYTION OF QUANTUM HALL EFFECT

量子霍尔效应的理论研究

基本信息

批准号：
07640522
负责人：
ISHIKAWA Kenzo
金额：
$ 1.41万
依托单位：
HOKKAIDO UNIVERSITY
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
1995
资助国家：
日本
起止时间：
1995 至 1996
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-07640522/
关键词：
Quantum Hall effect fractional Hall effect von Neumann lattice flux phase Hofstadter butterfly duality topological field theory quantum field theory von Neumann格子表現有限サイズ補正抵抗標準

项目摘要

Various problems of Quantum Hall effect are studied based on field theory that is formulated using von Neumann lattice representation and the following new results have been obtained.1. Integer Quantum Hall effectInteger quantum Hall effect is used for standard of resistance and for determining the fine strusture constant. Concerning finite size effect and finite current effect, it was shown in this project that under sufficently strong magnetic field, corrections vanish and the Hall conductance is quantized exactly in realistic two-dimensional systems. The quantum Hall effect disappears, however, if thhe current exceeds a critical value. The critical Hall field is proportional to two halvth of the magnetic field.2. Fractional Hall effectA new mean field theory of the fractional Hall effect based on flux condenced state on von Neumann lattice is proposed. In this theory, one particle spectrum has a fractal structure owing to two scales of the system, lattice constant and flux per plaquette. The latter is connected with the filling factor. It is shown, for the first time, that the fractional Hall effect is understood from Hofstadter butterfly.3. Periodic potentials in the strong magnetic field and dualityOne particle spectra of the systems with periodic short range potentials are obtained by using von Neumann lattice representation. A kind of duality relation is shown to be hold.4. A symmetry breaking of topological field theory by Gribov copies is analyzed.

基于von Neumann格子表示的场论，对量子霍尔效应的各种问题进行了研究，得到了以下新的结果。整数量子霍尔效应整数量子霍尔效应被用来作为电阻的标准和确定精细结构常数。考虑到有限尺寸效应和有限电流效应，在实际的二维系统中，在足够强的磁场下，修正消失，霍尔电导得到精确的量子化。然而，如果电流超过一个临界值，量子霍尔效应就消失了。临界霍尔场与磁场的二分之一成正比。分数霍尔效应提出了一种新的基于von Neumann晶格磁通凝聚态的分数霍尔效应平均场理论。在这个理论中，由于系统的两个尺度，晶格常数和每个斑块的通量，一个粒子谱具有分形结构。后者与填充系数有关。首次证明了分数霍尔效应可以从霍夫施塔特蝴蝶效应中得到解释。用von Neumann格子表示法得到了具有周期短程势系统的单粒子谱。证明了一种对偶关系是成立的。分析了Gribov Copies对拓扑场论的对称性破缺。