Insensitivity of Quantized Hall Conductance to Disorder and Interactions
Abstract
A two-dimensional quantum Hall system is studied for a wide class of potentials including single-body random potentials and repulsive electron-electron interactions. We assume that there exists a non-zero excitation gap above the ground state(s), and then the conductance is derived from the linear perturbation theory with a sufficiently weak electric field. Under these two assumptions, we proved that the Hall conductance
σ
xy
and the diagonal conductance
σ
yy
satisfy
|
σ
xy
+
e
2
ν/h|≤const.
L
−1/12
and
|
σ
yy
|≤const.
L
−1/12
. Here
e
2
/h
is the universal conductance with the charge
−e
of electron and the Planck constant
h
;
ν
is the filling factor of the Landau level, and
L
is the linear dimension of the system. In the thermodymanic limit, our results show
σ
xy
=−
e
2
ν/h
and
σ
yy
=0
. The former implies that integral and fractional filling factors
ν
with a gap lead to, respectively, integral and fractional quantizations of the Hall conductance.