Dirac CP phase in the neutrino mixing matrix and the Froggatt-Nielsen mechanism with $\mathbf{d}\mathbf{e}\mathbf{t}[{\mathbf{M}}_{\mathbf{\nu }}]=\mathbf{0}$

We discuss the Dirac CP violating phase ${\delta }_{CP}$ in the Froggatt-Nielsen model for a neutrino mass matrix $M_{\nu }$ imposing a condition ${\rm det} [M_\nu]=0$\mathrm{.}ThisadditionalconditionrestrictstheCPviolatingphase$\delta_{CP}$drastically\mathrm{.}Wefindthatthephase$\delta_{CP}$ is predicted in the region of $\pm (0.4-2.9)$ radian, which is consistent with the recent T2K and NO$\nu$A data. There is a remarkable correlation between ${\delta }_{CP}$ and ${\sin }^2{\theta }_{23}$. The phase ${\delta }_{CP}$ converges on $\sim \pm \pi \mathrm{/}2$ if ${\sin }^2{\theta }_{23}$ is larger than $0.5$. Thus, accurate measurements of ${\sin }^2{\theta }_{23}$ are important for a test of our model. The effective mass $m_{ee}$ for the neutrinoless double beta decay is predicted in the rage of $3.3-4.0$ meV.