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

Electrons on honeycomb or pi-flux lattices obey 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 can be organized into 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 $U(1)\times SU(4)$ 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 argue it turns superconducting at the critical value of the chemical potential through a ``flop" between the tensor components. The theory is exactly solvable when the $SU(4)$ is generalized to $SU(N)$ and $N$ taken large. A lattice Hamiltonian that may exhibit this transition, parallels with the Gross-Neveu model, and applicability to related electronic systems are briefly discussed.