Fractional quantization of Hall resistance as a consequence of mesoscopic feedback

Abstract

A nonlinear single-particle model is introduced, which captures the characteristic of systems in the quantum Hall regime. The model entails the magnetic Shr\"odinger equation with spatially variable magnetic flux density. The distribution of flux is prescribed via the postulates of the mesoscopic mechanics (MeM) introduced in my previous articles [cf. J. Phys. Chem. Solids, 65 (2004), 1507-1515; J. Geom. Phys., Vol. 55/1 (2005), 1-18]. The model is found to imply exact integer and fractional quantization of the Hall conductance. In fact, Hall resistance is found to be $R_H = \frac{h}{2e^2}\frac{M}{N}$ at the filling factor value $N/M$. The assumed geometry of the Hall plate is rectangular. Special properties of the magnetic Shr\"odinger equation with the mesoscopic feedback loop allow us to demonstrate quantization of Hall resistance as a direct consequence of charge and flux quantization. I believe results presented here shed light at the overall status of the MeM in quantum physics, confirming its validity.