Theoretical Study of Two-Dimensional Electronic States in a Random Magnetic Field

近年来，人们对随机磁场中二维电子的兴趣与日俱增。在这一领域，不仅没有问题的重要性，还存在着许多悬而未决的问题。利用一种新的基于临界波函数多重分形性的有限尺寸标度，研究了二维电子在随机磁场中的局域化性质。首先，我们确定了随机磁场作用下的金属-绝缘体相变的存在性及其普适性类。结果表明，所有电子态都定域在这样的系统中。我们还研究了具有有限平均值的随机磁场中的系统。在平均场比场涨落幅度大得多的情况下，谱被分成朗道子带，按照半经典理论的预测，只有子带中心的态被扩展。我们证实了这种跃迁的普适类与整数量子霍尔系统的普适类相同。此外，当场涨落幅度变大时，扩展态的能量从子带中心向上移动。这些结果对于理解v=1/2附近分数量子霍尔系统中的电子输运是非常重要的。

In recent years there has been a growing interest in two-dimensional electrons in a random magnetic field. Instead of importance of the problem, there still exist many unsolved questions in this field. We have studies the localization property of two-dimensional electrons in random magnetic fields in terms of a novel finite-size scaling based on the multifractality of the critical wave function. At first, we have determined the existence of the metal-insulator transition in a system subject to a random magnetic field with vanishing mean and its universality class. The result suggests that all electronic states are localized in such systems. We also investigated systems in a random magnetic field with finite mean. In the case that the average field is much stronger than the magnitude of field fluctuations, the spectrum separates into Landau subbands and only the states at subband centers are extended as predicted by a semiclassical theory. We confirmed that the universality class of this transition is the same with that of the integer quantum Hall system. Furthermore, the energy of the extended state shifts upward from the subband center when the magnitude of field fluctuation becomes stronger. These results are quite important for understanding electron transport in a fractional quantum Hall system near v=1/2.