Superconductivity near an Ising nematic quantum critical point in two dimensions

TL;DR

This work investigates superconductivity near a two-dimensional Ising nematic quantum critical point by solving self-consistent Dyson–Schwinger equations for the electron propagator, the nematic propagator, and a triangle-order vertex without common Eliashberg approximations. By retaining full momentum integration, Landau damping feedback, and vertex corrections, the authors find that an extended $s$-wave gap is the only stable pairing state, while $T_c$ behavior changes qualitatively when vertex corrections are included: $T_c$ is enhanced toward the QCP under bare vertex but becomes non-monotonic with a maximum away from the QCP when vertex corrections are accounted for. The non-monotonic $T_c(r)$ is attributed to the intricate balance between NFL decoherence and Cooper pairing, offering a potential explanation for anomalous Tc trends observed in FeSe$_{1-x}$Te$_x$ materials. The results emphasize the importance of vertex corrections and point to future directions including higher-order vertex effects and multi-orbital extensions to better capture real materials.

Abstract

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 $Γ_{\mathrm{v}}^{\mathrm{1L}}(p+q,p)$ 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.

Figures (5)

• Figure 1: Diagrams for the DS equations of $G(p)$, $D(q)$, and $\Gamma_{\mathrm{v}}^{\mathrm{1L}}(p+q,p)$. Thin and thick solid (dashed) lines represent free and renormalized electron (boson) propagators, respectively.
• Figure 2: Angular dependence of $A_1(\theta)$, $\Phi(\theta)$, and $\Delta(\theta)$ at the frequency $\epsilon_{n}=\pi T$ and the Fermi momentum $\mathbf{p}_{\mathrm{F}}$. Other model parameters are $g=0.45$ and $T=0.01$. Blue and orange-red curves correspond to $r = 10^{-1}$ and $r = 10^{-2}$, respectively.
• Figure 3: Superconducting transition temperature $T_{c}$ as a function of the parameter $r$ for $g=0.45$. The blue, orange-red, and yellow curves correspond to the MEFS, ME, and VERTEX approximations, respectively. $T_{c}=0$ if $r$ becomes sufficiently large.
• Figure 4: Curves of $T_{c}(r)$ with error bars. Here, $g=0.45$.
• Figure 5: The frequency dependence of $A_1(\epsilon_{n})$, $\Phi(\epsilon_{n})$, and $\Delta(\epsilon_{n}) = \Phi(\epsilon_{n})/A_1(\epsilon_{n})$ in the $\theta=0$ direction for electrons restricted on the Fermi surface. The results are obtained for $g=0.45$ and $T=0.02$.