  We study how the intrinsic anomalous Hall conductivity is modified in
two-dimensional crystals with broken time-reversal symmetry due to weak
inhomogeneity of the applied electric field. Focusing on a clean noninteracting
two-band system without band crossings, we derive the general expression for
the Hall conductivity at small finite wave vector $q$ to order $q^2$, which
governs the Hall response to the second gradient of the electric field. Using
the Kubo formula, we show that the answer can be expressed through the Berry
curvature, Fubini-Study quantum metric, and the rank-3 symmetric tensor which
is related to the quantum geometric connection and physically corresponds to
the gauge-invariant part of the third cumulant of the position operator. We
further compare our results with the predictions made within the semiclassical
approach. By deriving the semiclassical equations of motion, we reproduce the
result obtained from the Kubo formula in some limits. We also find, however,
that the conventional semiclassical description in terms of the definite
position and momentum of the electron is not fully consistent because of
singular terms originating from the Heisenberg uncertainty principle. We thus
present a clear example of a case when the semiclassical approach inherently
suffers from the uncertainty principle, implying that it should be applied to
systems in nonuniform fields with extra care.
