Strong CP Problem and spontaneous generation of CP violating phase in CKM matrix

We show that in a Complementary two-Higgs doublet model(C2HDM) the CP violating phase in the CKM matrix can be generated spontaneously, dangerous FCNC can be naturally suppressed and the strong CP problem can also be avoided. The two Higgs doublets in the model are complementary in the sense that none of them is enough to describe masses of a given type of quarks. We find that the strength of FCNC is suppressed by the strength of Yukawa couplings of the first generation quark and the tree-level FCNC is sufficiently small. Using an explicit example, we show that radiative correction to the assumed Yukawa couplings can modify the discussion about the strong $\theta$. The correction to the strong $\theta$ is estimated to be less than around $10^{-12}\sim10^{-10}$ which can be tested in future experiment.