An unusual first order phase transition in a 2D superconductor

We consider a superconductor under external perturbation, which forces Cooper pairs to develop with a finite total momentum $q$. The condensation energy of such a state decreases with $q$ and vanishes at a critical $q_c$. We analyze how superconducting order evolves at $q \approx q_c$. In 3D, the result is well-known: the pairing susceptibility diverges at $q = q_c +0$, and the gap amplitude $\Delta (q)$ gradually increases as $q$ decreases below $q_c$ and reaches its largest value $\Delta_0$ at $q=0$. In 2D, we find different behavior. Namely, for a parabolic dispersion, the pairing susceptibility also diverges at $q = q_c +0$, but at $q = q_c -0$, the gap amplitude jumps to the maximal $\Delta_0$ and remains equal to it for all $q <q_c$. For a non-parabolic dispersion $\varepsilon_{k}= c k^{2\alpha}$, we find that for $\alpha>1$ the transition becomes second-order, but the gap evolution is rather sharp, whereas for $\alpha<1$ it becomes first-order, but $\Delta (q)$ is non-monotonic. This is similar, but not identical, to the behavior of magnetization near a Stoner transition in 2D.