Unconventional discontinuous transitions in a 2D system with spin and valley degrees of freedom

We analyze the transition into the most favorable ordered state for a system of 2D fermions with spin and valley degrees of freedom. We show that for a range of rotationally invariant dispersions, the ordering transition is highly unconventional: the associated susceptibility diverges (or almost diverges) at the transition, yet immediately below it the system jumps discontinuously into a fully polarized state. We analyze the dispersion of the longitudinal and transverse collective modes in different parameter regimes above and below the transition. Additionally, we consider ordering in a system with full $\mathrm{SU(4)}$ symmetry and show that there is a cascade of discontinuous transitions into a set of states, which includes a quarter-metal, a half-metal and a three-quarter metal. We compare our results with the data for biased bilayer and tri-layer graphene.