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We study possible many body phenomena in the Quantum Anomalous Hall system ofweakly interacting spinor bosons in a square lattice. There are various novelspin-bond correlated superfluids (SF) and quantum or topological phasetransitions among these SF phases. One transition is a first order one drivenby roton droppings ( but with non-zero gaps $ \Delta_R $ ) tuned by the Zeemanfield $ h $. Another is a second order bosonic Lifshitz transition with thedynamic 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 Goldstonemode tuned by the ratio of spin-orbit coupled (SOC) strength over the hoppingstrength. 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 systemhas an anti-unitary $ Z_2 $ Reflection symmetry, there are infinite number ofclassically degenerate family of states on both sides. We perform a systematicorder from quantum disorder analysis to find the quantum ground states, alsocalculate the roton gaps $ \Delta_R $ generated by the order from disordermechanism on both sides of the TT point. The $ N=2 $ and $ N=4 $ SF phases havethe same spin-orbital XY-AFM spin structure, respect the anti-unitary symmetryand break the $ [C_4 \times C_4]_D $ symmetry, so they be distinguished only bythe different topology of the BEC condensation momenta instead of by anydifferences in the symmetry breaking patterns. All these novel quantum ortopological phenomena can be probed in the recent experimentally realizedweakly interacting Quantum Anomalous Hall (QAH) model of $ ^87 Rb $ by Wu, {\slet.al}, Science 354, 83-88 (2016).

Symmetry protected bosonic topological phase transitions: Quantum Anomalous Hall system of weakly interacting spinor bosons in a square lattice