4D Quantum hall fluid of Yang's SU(2) monopole, noncommutative geometry and iniegrable models

摘要

In 1983, Laughlin [1] proposed the incompressible quantum Hall fluid (QHF) formulation for the fractional quantization of the Hall effect (FWHE). Soon after, Haldane [2] considered translational (acturally rotational) invariant version of the intergeral quantum Hall fluid (IQHF), a two-dimensional electon gas of N particles moving on a spherical surface S~2 = (S~3～SU(2))/(S~1～U(1)), the first Hopf firbration bundle, of radius R in a radial (Dirac monopole) magnetic field B = (hS)/(eR~2) and the corresponding Hamiltonian of single particle with charge e is H = |Λ-vector|~2/(2MR~2) = (ω_C)|Λ-vector|~2)/(2hS) (1) here M is the effective mass, ω_C = (eB)/M is the cyclotron frequency and 2S is an integer as required by the Dirac's quantization condition.