Strong $CP$ problem and spontaneous generation of the $CP$ violating phase in the CKM matrix

We show that in a complementary two-Higgs doublet model (C2HDM) the $CP$ violating phase in the Cabibbo-Kobayashi-Maskawa matrix can be generated spontaneously, dangerous flavor changing neutral currents (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 ${10}^{-12}\sim {10}^{-10}$ which can be tested in future experiments.