SO(8) unification and the large- N theory of superconductor-insulator transition of two-dimensional Dirac fermions

Electrons on honeycomb or pi-flux lattices obey the effective massless Dirac equation at low energies and at the neutrality point, and should suffer quantum phase transitions into various Mott insulators and superconductors at strong two-body interactions. We show that 35 out of 36 such order parameters that provide Lorentz-invariant mass gaps to Dirac fermions belong to a single irreducible tensor representation of the $SO(8)$ symmetry of the two-dimensional Dirac Hamiltonian for the spin-1/2 lattice fermions. The minimal interacting Lagrangian away from the neutrality point has the $SO(8)$ symmetry reduced to $U(1)\ifmmode\times\else\texttimes\fi{}SU(4)$ by finite chemical potential, and it allows only two independent interaction terms. When the Lagrangian is nearly $SO(8)$ symmetric and the ground state insulating at the neutrality point, we show it turns superconducting at the critical value of the chemical potential through a ``flop'' between the tensor components, by exactly solving the $SU(4)\ensuremath{\rightarrow}SU(N)$ generalization in the large-$N$ limit. A lattice Hamiltonian that may exhibit this transition, parallels with and the consequences for the Gross-Neveu model, and applicability to related electronic systems are briefly discussed.