  Broken symmetries and electronic topology are nicely manifested together in
the second order nonlinear optical responses from topologically nontrivial
materials. While second order nonlinear optical effects from the electric
dipole (ED) contribution have been extensively explored in polar Weyl
semimetals (WSMs) with broken spatial inversion (SI) symmetry, they are rarely
studied in centrosymmetric magnetic WSMs with broken time reversal (TR)
symmetry due to complete suppression of the ED contribution. Here, we report
experimental demonstration of optical second harmonic generation (SHG) in a
magnetic WSM Co$_{3}$Sn$_{2}$S$_{2}$ from the electric quadrupole (EQ)
contribution. By tracking the temperature dependence of the rotation anisotropy
(RA) of SHG, we capture two magnetic phase transitions, with both the SHG
intensity increasing and its RA pattern rotating at $T_{C,1}$=175K and
$T_{C,2}$=120K subsequently. The fitted critical exponents for the SHG
intensity and RA orientation near $T_{C,1}$ and $T_{C,2}$ suggest that the
magnetic phase at $T_{C,1}$ is a 3D Ising-type out-of-plane ferromagnetism
while the other at $T_{C,2}$ is a 3D XY-type all-in-all-out in-plane
antiferromagnetism. Our results show the success of detection and exploration
of EQ SHG in a centrosymmetric magnetic WSM, and hence open the pathway towards
the future investigation of its tie to the band topology.
