  A model with both Rashba and Dresselhaus spin orbit (SO) couplings and
Hubbard electron-electron interaction is studied on planar strips at quarter
filling at zero temperature in the clean limit. In the absence of Hubbard
repulsion and at equilibrium, within linear response theory, a nonmonotonic
behavior of the spin Hall conductivity as a function of the ratio of the Rashba
(alpha) and Dresselhaus (beta) strengths was found for large enough SO
strengths. This behavior is signalled by a peak or a cusp, depending on the
strip width, at intermediate values of beta/alpha in the interval [0,1]. This
behavior of the spin Hall conductivity was correlated with the one for the
longitudinal spin conductivity. This study was then extended to the
out-of-equilibrium regime that arises by imposing a finite voltage bias between
the two ends of an open strip. This system, in the presence of a Hubbard term
with coupling U, was treated with the density matrix renormalization group
technique and with the Landauer-Buttiker formalism. It was found that relevant
properties to the spin Hall effect, such as the transversal spin current and
the spin accumulation, present a similar nonmonotonic behavior as the one found
for the spin conductivities. More importantly, it was also found that these
properties are enhanced by the repulsive Hubbard interaction up to a moderate
value of U.
