Revisiting the Charge Transport in Quantum Hall Systems

We reexamine the charge transport induced by a weak electric field in two-dimensional quantum Hall systems in a finite, periodic box at very low temperatures. The resulting linear response coefficients consist of the time-independent term $\sigma_{xy}$ corresponding to the Hall conductance and the linearly time-dependent term $\gamma_{sy}\cdot t$ in the transverse and longitudinal directions $s=x,y$ in a slow switching limit for adiabatically applying the initial electric field. The latter terms $\gamma_{sy}\cdot t$ are due to the acceleration of the electrons by the uniform electric field in the finite and isolated system, and so the time-independent term $\sigma_{yy}$ corresponding to the diagonal conductance always vanishes. The well known topological argument yields the integral and fractional quantization of the averaged Hall conductance $\bar{\sigma_{xy}}$ over gauge parameters under the assumption on the existence of a spectral gap above the ground state. In addition to this fact, we show that the averaged acceleration coefficients $\bar{\gamma_{sy}}$ are vanishing under the same assumption. In the non-interacting case, the spectral gap between the neighbouring Landau levels persists if the vector and the electrostatic potentials together satisfy a certain condition, and then the Hall conductance $\sigma_{xy}$ without averaging exhibits the exact integral quantization in the infinite volume limit with the vanishing acceleration coefficients. We also estimate their finite size corrections. In the interacting case, the averaged Hall conductance $\bar{\sigma_{xy}}$ for a non-integer filling of the electrons is quantized to a fraction not equal to an integer under the assumption that the potentials satisfy certain conditions in addition to the gap assumption.