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

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