Gap anisotropy and universal pairing scale in a spin-fluctuation model of cuprate superconductors

We consider the evolution of $d_{x^2-y^2}$ pairing, mediated by nearly critical spin fluctuations, with the coupling strength. We show that the onset temperature for pairing, $T^{\ast }$, smoothly evolves between weak and strong couplings passing through a broad maximum at intermediate coupling. At strong coupling, $T^{\ast }$ is of order of the magnetic exchange energy $J$. We argue that for all couplings, pairing is confined to the vicinity of the Fermi surface. We also find that thermal spin fluctuations only modestly reduce $T^{\ast }$, even at criticality, but they substantially smooth the gap anisotropy. The latter evolves with coupling being the largest at weak coupling.

Figure 1

(Color online) The onset temperature $T^{\ast }$ for $d_{x^2-y^2}$ pairing vs the dimensionless coupling $u$ for an Ornstein-Zernike form for the static spin propagator near $Q$ and various values of the magnetic correlation length $\xi$ with $v_F/a=1\text{eV}$. The shading in the insets represents those parts of the Brillouin zone where pairing is confined to. For both small and large $u$, pairing is confined to the vicinity of the Fermi surface.

Figure 2

(Color online) The mass renormalization $Z_k\left(0\right)$ and the pairing gap ${\Delta }_k\left(0\right)$ for small $u=0.2$. Panels (a) and (b) show the results for two different $\xi$; panel (c) shows the angular dependence of the gap compared to the $\text{cos}\left(k_{x}a\right)-\text{cos}\left(k_{y}a\right)$ form; panel (d) shows ${\Delta }_k\left(0\right)$ for $\xi =4.8a$ with and without the pairbreaking contribution from the static spin fluctuations [the term with $n=m$ in Eq. (2)]. Without the static contribution, $Z_k\left(0\right)\equiv 1$.

Figure 3

(Color online) Same as Fig. 2 but for large $u=2$.

Figure 4

(Color online) The onset temperature $T^{\ast }\left(u\right)$ for different forms of the static susceptibility $D_q\left(0\right)$ (see text). The magnitude of $T^{\ast }\left(u\right)$ increases as $D_q\left(0\right)$ drops faster with deviation from $Q$.