Competing pairing channels in the doped honeycomb lattice Hubbard model

Proposals for superconductivity emerging from correlated electrons in the doped Hubbard model on the honeycomb lattice range from chiral $d+id$ singlet to $p+ip$ triplet pairing, depending on the considered range of doping and interaction strength, as well as the approach used to analyze the pairing instabilities. Here, we consider these scenarios using large-scale dynamic cluster approximation (DCA) calculations to examine the evolution in the leading pairing symmetry from weak to intermediate coupling strength. These calculations focus on doping levels around the van Hove singularity (VHS) and are performed using DCA simulations with an interaction-expansion continuous-time quantum Monte Carlo cluster solver. We calculated explicitly the temperature dependence of different uniform superconducting pairing susceptibilities and found a consistent picture emerging upon gradually increasing the cluster size: while at weak coupling the $d+id$ singlet pairing dominates close to the VHS filling, an enhanced tendency towards $p$-wave triplet pairing upon further increasing the interaction strength is observed. The relevance of these systematic results for existing proposals and ongoing pursuits of odd-parity topological superconductivity are also discussed.

Figure 1

Real-space clusters with (a) $N_c=3$, (b) 4, and (c) 12 unit cells along with their corresponding momentum patches within the Brillouin zone. The clusters $N_c=3$ and 12 already retain the lattice $D_{6h}$ symmetry, while for the $N_c=4$ cluster we enforce this symmetry, as shown in (b), where the four outer hollow sites and the two outer black sites are equivalent due to this symmetry. (d) The phase factors for difference nearest-neighbor pairing channels, where $s,d_{xy},d_{x^2-y^2}$, and $d\pm id$ correspond to singlet pairing states, while $p_x,p_y,p\pm ip$ and $f$ are triplet states. $w=exp(\pm i2\pi /3)$ here.

Figure 2

Noninteracting $\left(U=0\right)$ pairing susceptibilities for various channels at the VHS filling. All channels will diverge at low temperature due to the logarithmic divergence of the DOS at the VHS. The $p$-wave susceptibilities exhibit the strongest increases, as indicated by the fit line. The inset shows the peak in the DOS at the VHS for the density $n=3/4$ (gray).

Figure 3

(a) Effective pairing susceptibilities for $N_c=3,U=2t$ at the VHS density $n=3/4$. (b) Effective pairing susceptibilities for $N_c=3,U=6t$ at the VHS density. (c) Density dependence of the effective pairing susceptibilities at $\beta t=20 \left(T/t=0.05\right)$. All $d$-wave channels are degenerate, as are the $p$ waves. The $s$ and $f$ waves are not divergent and hence are not shown here.

Figure 4

(a) Effective pairing susceptibilities for $N_c=4,U=2t$ at the VHS density $n=3/4$. (b) Effective pairing susceptibilities for $N_c=4,U=4t$ at the VHS density. Note that the data in (a) and (b) share labels. (c) Density dependence of the effective pairing susceptibility at $\beta t=20 \left(T/t=0.05\right)$.

Figure 5

(a) Effective pairing susceptibility for $N_c=12,U=t$ at the VHS density $n=3/4$. (b) Effective pairing susceptibility for $N_c=12,U=2t$ at the VHS density. Note that the data in (a) and (b) share labels. (c) Density dependence of the effective pairing susceptibility at $\beta t=40 \left(T/t=0.025\right)$.