First order phase transition in a two-dimensional 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 $q\ensuremath{\approx}{q}_{c}$. In three dimensions, the result is well known: the pairing susceptibility diverges at $q={q}_{c}+0$, and the gap amplitude $\mathrm{\ensuremath{\Delta}}(q)$ gradually increases as $q$ decreases below ${q}_{c}$ and reaches its largest value ${\mathrm{\ensuremath{\Delta}}}_{0}$ at $q=0$. In two dimensions (2D), we find different behavior. Namely, for a parabolic dispersion, the pairing susceptibility also diverges at $q={q}_{c}+0$, but at $q={q}_{c}\ensuremath{-}0$, the gap amplitude jumps to the maximal ${\mathrm{\ensuremath{\Delta}}}_{0}$ and remains equal to it for all $q<{q}_{c}$. For a nonparabolic dispersion ${\ensuremath{\varepsilon}}_{k}=c{k}^{2\ensuremath{\alpha}}$, we find that for $\ensuremath{\alpha}>1$ the transition becomes second order, but the gap evolution is rather sharp, whereas for $\ensuremath{\alpha}<1$ it becomes first order, but $\mathrm{\ensuremath{\Delta}}(q)$ is nonmonotonic. This is similar, but not identical, to the behavior of magnetization near a Stoner transition in 2D.