  We propose a two-dimensional electron gas (2DEG) system in which an external
magnetic (B) field with a small chirality is applied to provide a topological
U(1) U(1) gauge field that separates conduction electrons of opposite spins in
the transverse direction. Additionally, the vertical electric (E) field in the
2DEG, together with spin-orbit coupling, produces a SU(2) gauge field which
reinforces / opposes the effect of the topological gauge. The system thus
provides a tunable spin Hall effect, where an applied gate voltage on the 2DEG
can be used to modulate the transverse spin current. As this method leads to
the enhancement or cancellation of intrinsic spin Hall, it naturally
distinguishes the extrinsic from the intrinsic effect.
