  The nonlinear Hall effect is an unconventional response, in which a voltage
can be driven by two perpendicular currents in the Hall-bar measurement.
Unprecedented in the family of the Hall effects, it can survive time-reversal
symmetry but is sensitive to the breaking of discrete and crystal symmetries.
It is a quantum transport phenomenon that has deep connection with the Berry
curvature. However, a full quantum description is still absent. Here we
construct a quantum theory of the nonlinear Hall effect by using the
diagrammatic technique. Quite different from nonlinear optics, nearly all the
diagrams account for the disorder effects, which play decisive role in the
electronic transport. After including the disorder contributions in terms of
the Feynman diagrams, the total nonlinear Hall conductivity is enhanced but its
sign remains unchanged for the 2D tilted Dirac model, compared to the one with
only the Berry curvature contribution. We discuss the symmetry of the nonlinear
conductivity tensor and predict a pure disorder-induced nonlinear Hall effect
for point groups $C_{3}$, $C_{3h}$, $C_{3v}$, $D_{3h}$, $D_{3}$ in 2D, and $T$,
$T_{d}$, $C_{3h}$, $D_{3h}$ in 3D. This work will be helpful for explorations
of the topological physics beyond the linear regime.
