  We report the first observation of the spin Hall conductivity spectrum in
GaAs at room temperature. Our terahertz polarimetry with a precision of several
$\mu$rads resolves the Faraday rotation of terahertz pulses arising from the
inverse spin Hall effect of optically injected spin-polarized electrons. The
obtained spin Hall conductivity spectrum exhibits an excellent quantitative
agreement with theory, demonstrating a crossover in the dominant origin from
impurity scattering in the DC regime to the intrinsic Berry-curvature mechanism
in the terahertz regime. Our spectroscopic technique opens a new pathway to
analyze anomalous transports related to spin, valley, or orbital degrees of
freedom.
