  Spin Hall effect (SHE), a fundamental transport phenomenon with non-zero spin
current but vanishing charge current, has important applications in spintronics
for the electrical control of spins. Owing to the half-spin nature of
electrons, the rank of spin current (determined by the rank of spin tensors)
has been restricted to 0 and 1 for charge and spin Hall effects. Motivated by
recent studies of pseudospin-1 fermions in solid state and cold atomic systems,
here we introduce and characterize higher-rank ($\geq 2$) SHEs in large spin
($\geq 1$) systems. We find a universal rank-2 spin Hall conductivity $e/{8}\pi
$, with zero rank-0 and 1 conductivities, for a spin-1 model with intrinsic
spin-orbit coupling. Similar rank-2 SHEs can also be found in a spin-3/2
system. An experimental scheme is proposed to realize and measure rank-2 SHEs
with pseudospin-1 ultracold fermionic atoms. Our results reveal novel spin
transport phenomena in large spin systems and may find important applications
in designing innovative spintronic devices.
