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

近年来，人们对随机磁场中的二维电子越来越感兴趣。这一领域还存在着许多尚未解决的问题，而不是问题的重要性。基于临界波函数的多重分形，研究了二维电子在随机磁场中的局域性。首先，我们确定了在随机磁场作用下金属-绝缘子跃迁的存在性及其普适性。结果表明，在这样的系统中，所有的电子态都是局域化的。我们还研究了具有有限均值的随机磁场中的系统。当平均场的强度远远大于场波动的强度时，光谱分成朗道子带，并且只有子带中心的态按照半经典理论的预测得到扩展。我们证实了该跃迁的普适性类与整数量子霍尔系统的普适性类相同。当场涨落强度增大时，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具有重要意义。

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.