Coexistence of strong nematic and superconducting correlations in a two-dimensional Hubbard model

Using a dynamic cluster quantum Monte Carlo approximation, we study a two-dimensional Hubbard model with a small orthorhombic distortion in the nearest-neighbor hopping integrals. We find a large nematic response in the low-frequency single-particle scattering rate which develops with decreasing temperature and doping as the pseudogap region is entered. At the same time, the $d$-wave superconducting gap function develops an $s$-wave component and its amplitude becomes anisotropic. The strength of the pairing correlations, however, is found to be unaffected by the strong anisotropy, indicating that $d$-wave superconductivity can coexist with strong nematicity in the system.

Figure 1

Plots of the gradient of the $k$-space occupation, $|\nabla n_k|$ in the first quadrant of the Brillouin zone, calculated for the system with small directional anisotropy. On the left-hand side, results are shown for the $4\times 4$ cluster with $U=6t$ and $t^{\prime }=0$ in (a)–(c) for a fixed temperature $T=0.074t$ but varying doping as indicated, and in (d) for a lower temperature $T=0.05t$ and a doping $\delta =0.05$. The right-hand side illustrates the effects of cluster size and varying parameters for a doping $\delta =0.05$. (e) and (f) show results for a $2\times 2$ cluster for different values of $U$ and $t^{\prime }$ as indicated. (g) and (f) show results for the $4\times 4$ cluster for $t^{\prime }=0$ for different values of $U$.

Figure 2

Real and imaginary parts of the self-energy $\Sigma (\mathbf{K},{\omega }_n)$ vs frequency ${\omega }_n$ for $\mathbf{K}=(\pi ,0)$ and $(0,\pi )$ for a doping $\delta =0.05$ and temperature $T=0.125t$. Results for both the anisotropic and isotropic cases are shown.

Figure 3

Color map of the anisotropic ratio of the quasiparticle weight ${\sigma }_Z$ over the temperature-doping plane, for $U=6t$. The solid blue curve indicates the pseudogap temperature $T^{*}\left(\delta \right)$ which is obtained as the temperature at which the uniform magnetic susceptibility ${\chi }_m[q=(0,0),T]$ has a maximum.

Figure 4

(a) Leading $d$-wave eigenvalues ${\lambda }_d$ of the Bethe-Salpeter Eq.  (4) in the particle-particle channel vs temperature. The eigenvalues for both the isotropic and anisotropic models are plotted. (b) The momentum dependence of the leading eigenvector ${\phi }_d(K,\pi T)$ at the lowest Matsubara frequency along the path in Brillouin zone shown in the inset. (c) The relative $s$-wave contribution to the $d$-wave contribution, $|[{\Phi }_d(\pi ,0)+{\Phi }_d(0,\pi )]/[{\Phi }_d(\pi ,0)-{\Phi }_d(0,\pi )]|$, as a function of temperature for different dopings.