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The existence of three generations of neutrinos(charged leptons/quarks) andtheir mass mixing are deep mysteries of our universe. Recently, Majorana'sspirit returns in modern condensed matter physics -- in the context oftopological Majorana zero modes in certain classes of topologicalsuperconductors(TSCs). In this paper, we attempt to investigate the topologicalnature of the neutrino by assuming that a relativistic Majorana fermion can bedivided into four topological Majorana zero modes at cutoff energy scale, e.g.planck scale. We begin with an exactly solvable $1$D lattice model whichrealizes a $T^2=-1$ time reversal symmetry protected TSC, and show that a pairof topological Majorana zero modes can realize a $T^4=-1$ time reversalsymmetry.Moreover, we find that a pair of topological Majorana zero modes canalso realize a $P^4=-1$ parity symmetry and even a nontrivial $\overlineC^4=-1$ charge conjugation symmetry. Next, we argue that the origin of threegenerations of neutrinos(charged leptons and quarks) can be naturally explainedas three distinguishable ways of forming a pair of complex fermions(withopposite spin polarizations) out of four topological Majorana zero modes,characterized by the $T^4=-1$, $(TP)^4=-1$ and $(T \overline C)^4=-1$fractionalized symmetries that each complex fermion carries at cutoff energyscale. Finally, we use a semiclassical approach to compute the neutrino massmixing matrix at leading order(LO), e.g., in the absence of $CP$ violationcorrection. We obtain $\theta_{12}=31.7^\circ, \theta_{23}=45^\circ$ and$\theta_{13}=0^\circ$(the golden ratio pattern), which is consistent with an$A_5$ flavor symmetry pattern. We further predict an exact mass ratio of thethree mass eigenstates of neutrinos with $m_1/m_3=m_2/m_3=3/\sqrt{5}$ and aneffective mass of neutrinoless double beta decay$m_{0\nu\beta\beta}=m_1/\sqrt{5}$.

physical review research

Fractionalized time reversal, parity and charge conjugation symmetry in  topological superconductor: a possible origin of three generations of  neutrinos and mass mixing