Paramagnetic auxiliary-field curvature (${\Omega }^{\prime \prime }(0)/N=U^{-1}-\int \nu (ϵ)[-n_F^{\prime }(ϵ)]dϵ$)

For a uniform exchange field with cost $N{M}^2/(2U)$ and spin-band energies ${ϵ}_k-\sigma M$, the grand potential is even in M. Its curvature at zero field is $N[U^{-1}-{\chi }_0]$, with ${\chi }_0=\int \nu (-n_F^{\prime })dϵ$ and spin-summed density of states per site. Negative curvature proves local paramagnetic instability. It does not by itself exclude a first-order transition before that instability, nor determine the stable finite-field amplitude.