  Although Weyl fermions have proven elusive in high-energy physics, their
existence as emergent quasiparticles has been predicted in certain crystalline
solids in which either inversion or time-reversal symmetry is
broken\cite{WanPRB2011,BurkovPRL2011, WengPRX2015,HuangNatComm2015}. Recently
they have been observed in transition metal monopnictides (TMMPs) such as TaAs,
a class of noncentrosymmetric materials that heretofore received only limited
attention \cite{XuScience2015, LvPRX2015, YangNatPhys2015}. The question that
arises now is whether these materials will exhibit novel, enhanced, or
technologically applicable electronic properties. The TMMPs are polar metals, a
rare subset of inversion-breaking crystals that would allow spontaneous
polarization, were it not screened by conduction electrons
\cite{anderson1965symmetry,shi2013ferroelectric,kim2016polar}. Despite the
absence of spontaneous polarization, polar metals can exhibit other signatures
of inversion-symmetry breaking, most notably second-order nonlinear optical
polarizability, $\chi^{(2)}$, leading to phenomena such as optical
rectification and second-harmonic generation (SHG). Here we report measurements
of SHG that reveal a giant, anisotropic $\chi^{(2)}$ in the TMMPs TaAs, TaP,
and NbAs. With the fundamental and second harmonic fields oriented parallel to
the polar axis, the value of $\chi^{(2)}$ is larger by almost one order of
magnitude than its value in the archetypal electro-optic materials GaAs
\cite{bergfeld2003second} and ZnTe \cite{wagner1998dispersion}, and in fact
larger than reported in any crystal to date.
