  We study the influence of the phase relaxation process on Hall resistance and
spin Hall current of a mesoscopic two-dimensional (2D) four-terminal Hall
cross-bar with or without Rashba spin-orbit interaction (SOI) in a
perpendicular uniform magnetic field. We find that the plateaus of the Hall
resistance with even number of edge states can survive for very strong phase
relaxation when the system size becomes much longer than the phase coherence
length. On the other hand, the odd integer Hall resistance plateaus arising
from the SOI are easily destroyed by the weak phase relaxation during the
competition between the magnetic field and the SOI which delocalize the edge
states. In addition, we have also studied the transverse spin Hall current and
found that it exhibits resonant behavior whenever the Fermi level crosses the
Landau band of the system. The phase relaxation process weakens the resonant
spin Hall current and enhances the non-resonant spin Hall current.
