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

近年来，人们对二维电子在随机磁场中的运动越来越感兴趣。与问题的重要性相反，这一领域仍然存在许多未解决的问题。基于临界波函数的多重分形性，我们研究了二维电子在随机磁场中的局域化性质。首先，我们确定了在平均为零的随机磁场作用下系统中金属-绝缘体相变的存在性及其普适类。结果表明，所有的电子态在这样的系统中是局域化的。我们还研究了在随机磁场中的有限平均系统。当平均场比场涨落强得多时，谱分离成朗道子带，只有子带中心的态被扩展，正如半经典理论所预言的那样。我们证实了该跃迁的普适性类与整数量子Hall系统的普适性类相同。此外，当光场涨落的幅度变大时，扩展态的能量从子带中心向上移动。这些结果对于理解分数量子霍尔系统在v=1/2附近的电子输运是十分重要的。

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.