  The spin Hall effect (SHE) is a generation of spin polarization for moving
spin carriers in materials under an external electric field and has been
observed in semiconductors, metals, and insulators at or below room
temperature. Recent theoretical analyses show that spin Hall current can be
induced by the baryon chemical potential gradient which plays the role of the
analogous electric field and which becomes sizable in the fireballs created in
heavy-ion collisions at beam energy of ${\cal O}(10)$~GeV. In this letter, we
study this important mechanism for spin polarization generation that has not
been systematically explored before and predict the signature of the SHE in
those collisions using a (3+1)~D viscous hydrodynamic model MUSIC with AMPT
initial condition. We propose to use the second Fourier coefficients of the net
spin polarization of Lambda hyperon as sensitive probes to search for the SHE.
Those SHE observables show a qualitative difference in both the sign and beam
energy dependence for the situations with and without the SHE. Future
experimental observation of these distinct qualitative features would provide
strong evidence for the existence of the SHE in the hot and dense QCD matter at
trillions of degrees.
