We study theoretically the zero temperature intrinsic anomalous Hall effect
in an experimentally realized 2D spin-orbit coupled Bose gas. For anisotropic
atomic interactions and as the spin-orbit coupling strength increases, the
system undergoes a ground state phase transition from states exhibiting a total
in-plane magnetization to those with a perpendicular magnetization along the
z direction. We show that finite frequency, or ac, Hall responses exist in
both phases in the absence of an artificial magnetic field, as a result of
finite inter-band transitions. However, the characteristics of the anomalous
Hall responses are drastically different in these two phases because of the
different symmetries preserved by the corresponding ground states. In
particular, we find a finite dc Hall conductivity in one phase but not the
other. The underlying physical reasons for this are analyzed further by
exploring relations of the dc Hall conductivity to the system's chirality and
Berry curvatures of the Bloch bands. Finally, we discuss an experimental method
of probing the anomalous Hall effect in trapped systems.Comment: 13 pages, 11 figure