整数和分数量子霍尔效应的共同物理机制
Common physical mechanism for integer and fractional quantum Hall effects
摘要：
用不同的物理模型研究了量子霍尔效应和分数量子霍尔效应，并从不同的物理机制解释了它们的物理机制。研究了整数和分数量子霍尔效应的共同物理机制，提出了整数和分数量子霍尔效应的统一表述。首先，我们引入了一个具有对称规范的强磁场下的二维理想电子气模型，并引入横向电场$\varepsilon_2$来平衡洛伦兹力。其次，求解了泡利方程，明确给出了波函数和能级。第三，通过对二维理想电子气系统简并密度的计算，得到了系统的霍尔电阻，其中引入了量子霍尔数$ν$。发现新定义的$ν$（文献中称为填充因子）与径向量子数n和角量子数$|m|$有关，不同的$n$和$|m|$对应不同的$ν$ .这为整数和分数量子霍尔效应提供了统一的解释。预言了在没有分数电荷概念的情况下，分数量子霍尔效应还会出现更多的新情况。
英文摘要：
Integer and fractional quantum Hall effects were studied with different physics models and explained by different physical mechanisms. In this paper, the common physical mechanism for integer and fractional quantum Hall effects is studied, where a new unified formulation of integer and fractional quantum Hall effect is presented. Firstly, we introduce a 2-dimensional ideal electron gas model in the presence of strong magnetic field with symmetry gauge, and the transverse electric filed $\varepsilon_2$ is also introduced to balance Lorentz force. Secondly, the Pauli equation is solved where the wave function and energy levels is given explicitly. Thirdly, after the calculation of the degeneracy density for 2-dimensional ideal electron gas system, the Hall resistance of the system is obtained, where the quantum Hall number $ν$ is introduced. It is found that the new defined $ν$, called filling factor in the literature, is related to radial quantum number n and angular quantum number $|m|$, the different $n$ and $|m|$ correspond to different $ν$. This provides unification explaination for integer and fractional quantum Hall effects. It is predicated that more new cases exist of fractional quantum Hall effects without the concept of fractional charge.
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