Weak-coupling instabilities of SU(N) fermions on the Bernal-stacked honeycomb bilayer in presence of on-site Hubbard Interactions

Bernal-stacked bilayer graphene hosts an interesting 'non-relativistic' semi-metallic dispersion different from monolayer graphene. At this quadratic band touching, short-range interactions are marginal and hence cause instabilities to a variety of ground states. In this work we consider the instabilities of even $N$ species of fermions on the Bernal bilayer with an $SU(N)$-symmetric contact interaction. For $SU(2)$ fermions with an on-site Hubbard interaction the ground state has been found to be to a magnetic Néel state for all strengths of the interaction. In contrast, the leading weak coupling instability for $N>2$ is a non-magnetic ground state, which is gapped and odd under time reversal. On the other hand, at strong coupling we expect Néel or VBS ground states of the effective self-conjugate $SU(N)$ spin models. Motivated by this observation, we investigate the phase diagram for even $N>2$ using determinantal quantum Monte Carlo computations.