Generating CP violation from a modified Fridberg-Lee model

The overall characteristics of the solar and atmospheric neutrino oscillations are approximately consistent with a tribimaximal form of the mixing matrix $U$ of the lepton sector. Exact tribimaximal mixing leads to $\theta_{13}=0$. However, the results from the Daya Bay and RENO experiments have established, such that in comparison to the other neutrino mixing angles, $\theta_{13}$ is small. Moreover, the atmospheric and solar mass splitting differ by two orders of magnitude. These significant differences constitutes the great enthusiasm and main motivation for our research herein reported. Keeping the leading behavior of U as tribimaximal. We would make a response to the following questions: at some level, whether or not the small parameters such as the solar neutrino mass splitting and $U_{e3}$, which vanish in a new framework, can be interpreted as a modified FL neutrino mass model? Subsequently, a minimal single perturbation leads to nonzero values for both of them? Our minimal perturbation matrix is constructed solely from computing the third mass eigenstate, using the rules of perturbation theory. Let us point out that in contrast with~\cite{my}, this matrix is not ad hoc assumed, but is instead built following a series of steps we will outline. Also in compared to the original FL neutrino mass model which generalize it by inserting phase factors, our work is more accurate. Subsequently, we produce the following results that add new contributions to the literature: a) we obtain a realistic neutrino mixing matrix with $\delta\neq0$ and $\theta_{23}=45^\circ$; b) the solar mass splitting term is dominated by an imaginary term, which could induce the existence of Majorana neutrinos, along with explaining a large CP violation in nature.