  We propose to realize the quantum nonlinear Hall effect and the inverse
Faraday effect through the acoustic wave in a time-reversal invariant but
inversion broken Dirac insulator. We focus on the acoustic frequency much lower
than the Dirac gap such that the interband transition is suppressed and these
effects arise solely from the intrinsic valley-contrasting band topology. The
corresponding acoustoelectric conductivity and magnetoacoustic susceptibility
are both proportional to the quantized valley Chern number and independent of
the quasiparticle lifetime. The linear and nonlinear components of the
longitudinal and transverse topological currents can be tuned by adjusting the
polarization and propagation directions of the surface acoustic wave. The
static magnetization generated by a circularly polarized acoustic wave scales
linearly with the acoustic frequency as well as the strain-induced charge
density. Our results unveil a quantized nonlinear topological acoustoelectric
response of gapped Dirac materials, like hBN and transition-metal
dichalcogenide, paving the way toward room-temperature acoustoelectric devices
due to their large band gaps.
