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Ground state wave functions for the quantum Hall effect on a sphere and the Atiyah–Singer index theorem
Journal of Physics A: Mathematical and Theoretical ( IF 2.1 ) Pub Date : 2020-05-11 , DOI: 10.1088/1751-8121/ab85e1
Brian P Dolan 1, 2 , Aonghus Hunter-McCabe 1
Affiliation  

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 center 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 ##IMG## [http://ej.iop.org/images/...] {$\nu =\frac{1}{2k+1}$}

中文翻译:

基态波函数对球体上的量子霍尔效应和Atiyah-Singer指数定理

使用Dirac算子对球形磁场中的量子霍尔效应进行了研究,用于背景磁场中的非相互作用费米子,该磁场由位于球体中心的Wu-Yang磁单极子提供。波动函数是非平凡的U(1)束的横截面,然后零点能量消失,并且没有扰动可以降低能量。Atiyah-Singer指数定理限制了基态的退化。在复合费米子模型中还研究了分数量子霍尔效应。统计量规场的涡旋由与单极场相关的狄拉克弦提供。仅当涡流具有偶数个磁通单位并起到抵消本底场的作用,从而减小复合费米子看到的有效场时,才能获得唯一的基态。
更新日期:2020-05-11
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