Applications of gauge theories to low-dimensional electron systems

规范场在凝聚态物理和粒子物理中都起着重要的作用，这一点现在已被广泛接受。特别是，从对量子霍尔系统中Skyrmion激发的观察来看，低维电子系统为检验和应用规范理论的各种思想提供了一个很好的实用场所。在过去四年的资助下，我们利用规范理论研究了一些与量子霍尔效应基础有关的问题。霍尔电流分布：霍尔样品中的电流分布是一个非常重要的课题。我们之前已经指出，相当一部分霍尔电流将沿着样品边缘流动，这是由于从定位主导的样品内部排出的结果。我们进行了数值实验来验证这幅霍尔电流的图像。量子霍尔效应的击穿：从上述数值实验中提出的量子霍尔效应击穿的机制，通过进一步的数值分析进行了检验。指出所观察到的击穿现象可以用朗道内过程、无序场和霍尔场之间的竞争来解释。分数量子霍尔效应（FQHE）：一种研究分数量子霍尔效应的有效理论的新方法，它利用了朗道能级的投影和玻色子化，依赖于量子霍尔态的不可压缩特性，而不涉及标准的复合玻色子或复合费米子图像。特别是，对于双层系统，我们成功地构建了一个有效的规范理论，该理论适当地结合了集体激发谱，这是从一般磁转子理论中知道的，但传统的chen - simons理论无法获得。

It is now widely accepted that gauge fields play an important role in condensed-matter physics as well as in particle physics. In particular, as seen from the observation of Skyrmion excitations in quantum Hall systems, low-dimensional electron systems offer a good and practical place to test and apply various ideas of gauge theories. Under a Grant-in-Aid over the past four years, we have thus studied some problems pertaining to the foundation of the quantum Hall effect by use of gauge theories.1. Hall-current distribution : The current distribution in a Hall sample is a subject of basic importance. We had earlier pointed out 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. We carried out a numerical experiment to verify this picture of the bulk-edge Hall current.2. Breakdown of the quantum Hall effect : A mechanism for the breakdown of the quantum Hall effect, suggested from the above numerical experiment, was examined by a further numerical analysis. It was pointed out that the observed breakdown phenomena may be explained by an intra-Landau-level process, competition between disorder and the Hall field.3. Fractional quantum Hall effect (FQHE) : A new approach to the effective theory of the FQHE was developed, that makes use of projection to Landau levels and bosonization and that relies on the incompressible nature of the quantum Hall states without referring to the standard composite-boson or composite- fermion picture. In particular, for double-layer systems we succeeded in constructing an effective gauge theory that properly incorporates the collective-excitation spectrum, known from a general magneto-roton theory but inaccessible by the conventional Chern-Simons theories.