  In quantum anomalous Hall (QAH) systems, the Hall conductance is quantized
and the corresponding effective topological theory of the system is the
Chern-Simons theory. The conductance quantum is given by the universal constant
$e^2/h$ -- the inverse von Klitzing constant -- that is independent of the bulk
gap, as well as the size of the system. This picture relies on the assumption
that the edge modes are sharply localized at the edge, i.e. they have zero
width. We show that considering the physical case where the edge modes have
finite localization length $b$, the effective action would not be topological
in bulk direction anymore. Due to non-zero $b$ the conductance quantum will be
corrected as $(1-\varepsilon)e^2/h$ where $\varepsilon$ encompasses the
non-universal (i.e. material/sample dependent) part that is determined by the
dimensionless ratios $\frac{gb}{\hbar v_F}$ and $\frac{b}{L}$ where $g,v_F,L$
are the bulk gap, Fermi velocity and sample length. To compute the
non-universal correction $\varepsilon$ we use anomaly inflow framework
according to which the bulk action produces the correct amount of anomaly
inflow that would cancel the anomaly of the chiral edge modes. These
corrections place limits on the precision of measurable quantization in units
of the inverse von Klitzing constant for QAH systems with smaller sizes and/or
smaller bulk gaps. Our result suggests that the failure of precision
measurements to reproduce the exact conductance quantum $e^2/h$ is not an
annoying sample quality issue, but it contains the quantitative physics of
anomaly inflow that can be inferred by the systematic study of such
corrections.
