Abstract
Dynamics of two dimensional electrons under the strong perpendicular magnetic field is shown to be described by a multi-component fermion theory. The electric conductance has a remarkable property known as the quantum Hall effect. The Hall conductance is quantized in units ofe 2/h in the gap region and in the localized state region. The proof of exactness is presented in general cases using quantum field theory.
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A more complete form of the Lagrangian obtained without any approximation than that discussed here can be treated by the same way and the following discussions include these cases
A simple example is presented here. Smoother functions can be used
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Ishikawa, K., Matsuyama, T. Magnetic field induced multi-component QED3 and quantum Hall effect. Z. Phys. C - Particles and Fields 33, 41–45 (1986). https://doi.org/10.1007/BF01410451
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DOI: https://doi.org/10.1007/BF01410451