Superconducting Gap in the Hubbard Model and the Two-Gap Energy Scales of High-$T_c$ Cuprate Superconductors

Recent experiments (angle-resolved photoemission spectroscopy and Raman) suggest the presence of two distinct energy gaps in high-temperature superconductors (HTSC), exhibiting different doping dependences. The results of a variational cluster approach to the superconducting state of the two-dimensional Hubbard model are presented which show that this model qualitatively describes this gap dichotomy. The antinodal gap increases with less doping, a behavior long considered as reflecting the general gap behavior of the HTSC. On the other hand, the near-nodal gap does even slightly decrease with underdoping. An explanation of this unexpected behavior is given which emphasizes the crucial role of spin fluctuations in the pairing mechanism.

Figure 1

(a) Spectral function of the Hubbard model for $U=8t$, $t^{\prime }=-0.3t$, and $x=0.07$ hole doping, calculated with the VCA on a $3\times 3$ cluster in the SC symmetry-broken phase. The arrow denotes the antinodal (AN) region. (b) The corresponding Fermi surface. The antinodal region is marked by AN (blue), the nodal region by N (red). Solid lines indicate the momentum scans, along which the gap is determined. Numbers refer to Fig. 2.

Figure 2

(a) “Normalized” SC gap ${\Delta }_0\equiv 2{\Delta }_{\mathbf{k}}/(\cos k_x-\cos k_y)$ for $\mathbf{k}$ near the antinodal (AN) and near the nodal FS point (N) as a function of doping for the Hubbard model (parameters are given in the text). (b) SC gap as a function of the $d$-wave factor ($\cos k_x-\cos k_y$) on the Fermi arc for $x=0.07$ hole doping. The numbers correspond to the momentum scans marked in Fig. 1. Inset: $A(\mathbf{k},\omega )$ in a small window around the FS for points 0 to 6.

Figure 3

Spectral function from $(\pi ,0)$ to $(\pi ,\pi )$ at $h=0.07$ doping. Left: $A(\mathit{k},\omega )$ in the SC broken phase. Middle: SC phase, but ${\Sigma }^{\mathrm{sc}}=0$ (see the text). Right: Paramagnetic solution.

Figure 4

The doping dependence of $v(q)$ at $q=(\pi ,\pi )$, $(\frac{\pi }{2},\frac{\pi }{2})$, and $(\pi ,0)$ for $U=8t$, calculated with the QMC method on an $8\times 8$ cluster for inverse temperature $\beta =2.5t^{-1}$. Notice that the interaction hardly depends on the fermionic momentum $\mathbf{k}$, so only the dependence on $\mathbf{q}$ is shown for $\mathbf{k}=(-\pi ,0)$.

Figure 5

Contributions to the gap $\Delta ({\mathbf{k}}_F)$ in Eq. (3) as a function of ${\mathbf{k}}^{\prime }$, divided by the $d$-wave factor $\cos (k_{F,x})-\cos (k_{F,y})$ for normalization reasons. Typical parameters have been used, $\mu =-1.0t$, $t^{\prime }=-0.3t$. Left: Fermi momentum ${\mathbf{k}}_{F,N}$ near the nodal direction. Arrows indicate transfer vectors with large weights. Right: Fermi momentum ${\mathbf{k}}_{F,\mathrm{AN}}$ near the antinodal direction. The arrow is the AF momentum ${\mathbf{Q}}_{\mathrm{AF}}$. The color scale for both plots is given in the right plot.