  An intriguing magneto-transport property is demonstrated by tight-binding
lattice electrons with Rashba spin-orbit coupling (SOC) in a magnetic field.
With the flux strength $\phi={2\pi/N}$ ($N$ is an integer) and the Zeeman
splitting fixed, when increasing the Rashba SOC $\lambda$, the spin-Hall and
charge-Hall conductances (SHC and CHC) undergo four-step evolutions: the SHC
shows size-dependent resonances and jumps at three critical $\lambda_{c}$'s,
and changes its sign at $\lambda_{c1}$ and $\lambda_{c3}$; while the CHC
exhibits three quantum jumps by $-Ne^2/h$, $+2Ne^2/h$ and $-Ne^2/h$. Such
four-step evolutions are also reflected in topological characters and spin
polarizations of edge states of a cylindrical system, and are robust against
weak disorder.
