分数量子霍尔效应的有效哈密顿量只包含电子相互作用项，为研究强关联电子体系提供了一个重要的平台。另一方面，基于分数量子霍尔效应的拓扑量子计算具有高度的容错性，引起了实验和理论物理学家的极大关注。申请人将使用算符代数方法，在二次量子化框架下研究填充数为1/2的p波配对Pfaffian分数量子霍尔态及其赝势哈密顿量。申请人将通过构造Pfaffian态的二次量子化迭代公式和研究此赝势哈密顿量的所有零能量基态，来揭示配对在分数量子霍尔效应中起到的作用。算符代数方法避免了数值方法中的有限粒子数效应，可以提供全新的思路，并且和坐标表象中研究一次量子化波函数的方法互补。申请人还将把Pfaffian态的赝势哈密顿量投影到特定的朗道能级来构造新的赝势哈密顿量，在理论上寻找新的拓扑量子霍尔态。预期此研究除了揭示配对在分数量子霍尔效应中起到的作用外，还会对研究更一般的配对态乃至发现新的拓扑量子态起到启发作用。

The effective Hamiltonian for the fractional quantum Hall effect only involves the interaction term between electrons, thus providing an important platform for studying strongly correlated electron systems. On the other hand, topological quantum computation based on fractional quantum Hall effect has greatly intrigued both experimentalists and theorists, for it possesses high fault tolerance. In this project, the applicant will study ν=1/2 p-wave paired Pfaffian fractional quantum Hall state and its pseudopotential Hamiltonian with the method of operator algebra in the framework of second quantization. The applicant seeks to reveal the role that pairing plays in the fractional quantum Hall effect by constructing second-quantized recursive formula for the Pfaffian state and studying all the zero energy ground states of this pseudopotential Hamiltonian. The method of operator algebra is not susceptible to finite particle number effect of the numerical method, can provide new ideas to the field and complement the method of studying first-quantized wave function and pseudopotential in the coordinate representation. The applicant will also project the pseudopotential Hamiltonian for the Pfaffian state to certain Landau levels to construct new pseudopotential Hamiltonians and will look for new topological quantum Hall states from a theoretical perspective. It is expected that this research will shed new light on studying general paired states and discovering new topological quantum states besides revealing the important role that pairing plays in the fractional quantum Hall effect.