论文标题
基态波在球体和atiyah-singer索引定理上的量子大厅效应的功能
Ground state wave functions for the quantum Hall effect on a sphere and the Atiyah-Singer index theorem
论文作者
论文摘要
使用Dirac操作员在背景磁场中使用Dirac Operator在球形几何形状中研究了量子大厅的效应，该效果是由球形中心的Wu-Yang磁性单极管提供的。波函数是非平凡$ u（1）$束的横截面，零点能量随后消失，没有扰动可以降低能量。 Atiyah-Singer索引定理限制了基态的退化。 在复合费米模型中还研究了分数量子厅效应。统计量规场的涡流由与单极场相关的狄拉克字符串提供。只有在涡流具有均匀数量的通量单元并采取抵消背景场的情况下，才能达到独特的基态，从而减少了复合费米子看到的有效场。有一个独特的间隙基态，对于大粒子，$ν= \ frac {1} {2 k+1} $被恢复。
The quantum Hall effect is studied in a spherical geometry using the Dirac operator for non-interacting fermions in a background magnetic field, which is supplied by a Wu-Yang magnetic monopole at the centre of the sphere. Wave functions are cross-section of a non-trivial $U(1)$ bundle, the zero point energy then vanishes and no perturbations can lower the energy. The Atiyah-Singer index theorem constrains the degeneracy of the ground state. The fractional quantum Hall effect is also studied in the composite Fermion model. Vortices of the statistical gauge field are supplied by Dirac strings associated with the monopole field. A unique ground state is attained only if the vortices have an even number of flux units and act to counteract the background field, reducing the effective field seen by the composite fermions. There is a unique gapped ground state and, for large particle numbers, fractions $ν=\frac{1}{2 k+1}$ are recovered.
