Neutrino as topological Majorana zero modes: the origin of three generations of neutrinos and their mass mixing

Recently, Majorana's spirit returns to modern condensed matter physics -- in the context of topological Majorana zero mode that represents a local half degree of freedom carrying non-Abelian statistics.In this paper, we investigate the topological nature of a Majorana fermion by assuming that it is made up of four Majorana zero modes at cutoff energy scale. First, we show that a pair of Majorana zero modes can realize a $T^4=-1$ time reversal symmetry, a $P^4=-1$ parity symmetry and even a nontrivial $\overline C^4=-1$ charge conjugation symmetry. Next, we propose a $\overline CPT$ super algebra for the Majorana fermion made up of four Majorana zero modes. Furthermore, the origin of three generations of neutrinos(assuming they are Majorana fermions) can be naturally explained as three distinguishable ways to form a pair of (local) complex fermions out of four Majorana zero modes. Finally, we compute the neutrino mass mixing matrix and mass ratios of the three mass eigenstates from a first principle at leading order(in the absence of $CP$ violation and charged lepton corrections). We obtain $\theta_{12} =31.7^\circ$, $\theta_{23}=45^\circ$, $\theta_{13}=0^\circ$ and $m_1/m_3=m_2/m_3=3/\sqrt{5}$. We predict the effective mass in neutrinoless double beta decay to be $m_{\beta\beta}=m_1/\sqrt{5}$.