Superconductivity near an Ising nematic quantum critical point in two dimensions

Near a two-dimensional Ising-type nematic quantum critical point, the quantum fluctuations of the nematic order parameter are coupled to the electrons, leading to non-Fermi liquid behavior and unconventional superconductivity. The interplay between these two effects has been extensively studied through the Eliashberg equations for the superconducting gap. However, previous studies often rely on various approximations that may introduce uncertainties in the results. Here, we re-visit the issue of how the superconducting transition temperature $T_{c}$ is affected by removing certain common approximations. We numerically solve the self-consistent Dyson-Schwinger equations of the electron propagator $G(p)$, the nematic propagator $D(q)$, and the vertex function $\Gamma_{\mathrm{v}}^{\mathrm{1L}}(p+q,p)$ expanded up to the triangle order, without introducing further approximations. Our calculations reveal that the extended $s$-wave superconducting gap is the only convergent solution to the nonlinear gap equations. We investigate the evolution of $T_{c}$ as the system approaches the nematic quantum critical point from the disordered (tetragonal) phase. Under the bare vertex approximation, $T_{c}$ is monotonically enhanced. However, when vertex corrections are incorporated, $T_{c}$ initially increases but then decreases, with the maximum value of $T_{c}$ occurring at a point away from the quantum critical point. The obtained gap symmetry and the non-monotonic behavior of $T_{c}$ are compared with recent experiments on doped FeSe materials.