  We study non-perturbatively the anomalous Hall current and its high harmonics
generated in Weyl and Dirac semimetals by strong elliptically polarized laser
fields, in the context of kinetic theory. We find a novel crossover between
perturbative and non-perturbative regimes characterized by the electric field
strength $\mathcal{E}^{*}= \frac{\mu \omega}{ 2 e v_F}$ ($\omega$: laser
frequency, $\mu$: Fermi energy, $v_F$: Fermi velocity). In the perturbative
regime, the anomalous Hall current quadratically depends on the field strength
($\mathcal{E}$), whereas the higher order corrections, as well as high
harmonics, vanish at zero temperature. In the non-perturbative regime, the
anomalous Hall current saturates and decays as
$(\log{\mathcal{E}})/\mathcal{E}$, while even-order high harmonics are
generated when inplane rotational symmetry is broken. Based on the analytical
solution of the Boltzmann equation, we reveal the topological origin of the
sharp crossover: the Weyl monopole stays inside or moves outside of the Fermi
sphere, respectively, during its fictitious motion in the pertubative or
non-pertubative regimes. Our findings establish a new non-linear response
intrinsically connected to topology, characteristic to Weyl and Dirac
semimetals.
