An unusual first order phase transition in a 2D superconductor

Abstract

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 $\hat{I}”(q)$ gradually increases as $q$ decreases below $q_c$ and reaches its largest value $\hat{I}{”}_0$ 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 $\hat{I}{”}_0$ and remains equal to it for all $q<q_c$. For a non-parabolic dispersion ${\epsilon }_k=c{k}^{2\hat{I}\pm }$, we find that for $\hat{I}\pm >1$ the transition becomes second-order, but the gap evolution is rather sharp, whereas for $\hat{I}\pm <1$ it becomes first-order, but $\hat{I}”(q)$ is non-monotonic. This is similar, but not identical, to the behavior of magnetization near a Stoner transition in 2D.

26 pages + 9 figures