  We study possible many body phenomena in the Quantum Anomalous Hall system of
weakly interacting spinor bosons in a square lattice. There are various novel
spin-bond correlated superfluids (SF) and quantum or topological phase
transitions among these SF phases. One transition is a first order one driven
by roton droppings ( but with non-zero gaps $ \Delta_R $ ) tuned by the Zeeman
field $ h $. Another is a second order bosonic Lifshitz transition with the
dynamic exponents $ z_x=z_y=2 $ and an accompanying $ [C_4 \times C_4]_D $
symmetry breaking. It is driven by the softening of the superfluid Goldstone
mode tuned by the ratio of spin-orbit coupled (SOC) strength over the hopping
strength. The two phase boundaries meet at a topological tri-critical (TT)
point which separates the $ h=0 $ line into two SF phases with $ N=2 $ and $
N=4 $ condensation momenta respectively. At the $ h=0 $ line where the system
has an anti-unitary $ Z_2 $ Reflection symmetry, there are infinite number of
classically degenerate family of states on both sides. We perform a systematic
order from quantum disorder analysis to find the quantum ground states, also
calculate the roton gaps $ \Delta_R $ generated by the order from disorder
mechanism on both sides of the TT point. The $ N=2 $ and $ N=4 $ SF phases have
the same spin-orbital XY-AFM spin structure, respect the anti-unitary symmetry
and break the $ [C_4 \times C_4]_D $ symmetry, so they be distinguished only by
the different topology of the BEC condensation momenta instead of by any
differences in the symmetry breaking patterns. All these novel quantum or
topological phenomena can be probed in the recent experimentally realized
weakly interacting Quantum Anomalous Hall (QAH) model of $ ^87 Rb $ by Wu, {\sl
et.al}, Science 354, 83-88 (2016).
