An unusual first order phase transition in a 2D superconductor

A superconductor in equilibrium has Cooper pairs of fermions with zero total momentum. A superconducting state with a phase varying as e^iq·r has a smaller condensation energy E_c(q), same as a superconductor with a fixed phase, but with a finite total momentum q of a Cooper pair. At a critical q_c, the condensation energy changes sign, and a normal state becomes energetically favorable. We analyze how superconductivity emerges if we impose a finite q and gradually decrease it below q_c. In 3D, we find that the evolution is gradual: the gap amplitude Δ increases as q decreases and reaches the equilibrium value Δ₀ at q=0. In 2D, we find that for parabolic dispersion the transition at q_c is first order: the gap amplitude immediately jumps to the equilibrium Δ₀ and remains equal to Δ₀ for all q c. This behavior holds for both BCS and BEC regimes. We find this by solving the set of coupled equations on Δ(q) and the chemical potential and then computing the condensation energy E_c(q) and superfluid stiffness. We analyze the consequences of this behavior. For a non-parabolic dispersion, we find that the transition is again continuous, but the gap evolution is rather sharp.