Unconventional discontinuous transitions in isospin systems

We show that two-dimensional fermions with dispersion $k^2$ or $k^4$ undergo a first-order Stoner transition to a fully spin-polarized state despite that the spin susceptibility diverges at the critical point. We extend our analysis to systems with dispersion $k^{2\alpha }$ and spin and valley isospin and show that there is a cascade of instabilities into fractional-metal states with some electron bands fully depleted; narrow intermediate ranges of partially-depleted bands exist for $\alpha <1$ or $\alpha >2$. The susceptibility becomes large near each transition. We discuss applications to biased bilayer and tri-layer graphene and moiré systems.