  Second harmonic generation (SHG) is a fundamental nonlinear optical
phenomenon widely used both for experimental probes of materials and for
application to optical devices. Even-order nonlinear optical responses
including SHG generally require breaking of inversion symmetry, and thus have
been utilized to study noncentrosymmetric materials. Here, we study
theoretically the SHG in inversion-symmetric Dirac and Weyl semimetals under a
DC current which breaks the inversion symmetry by creating a nonequilibrium
steady state. Based on analytic and numerical calculations, we find that Dirac
and Weyl semimetals exhibit strong SHG upon application of finite current. Our
experimental estimation for a Dirac semimetal Cd$_3$As$_2$ and a magnetic Weyl
semimetal Co$_3$Sn$_2$S$_2$ suggests that the induced susceptibility
$\chi^{(2)}$ for practical applied current densities can reach
$10^5~\mathrm{pm}\cdot\mathrm{V}^{-1}$ with mid-IR or far-IR light. This value
is 10$^2$-10$^4$ times larger than those of typical nonlinear optical
materials. We also discuss experimental approaches to observe the
current-induced SHG and comment on current-induced SHG in other topological
semimetals in connection with recent experiments.
