Chiral $d$-wave superconductivity on the honeycomb lattice close to the Mott state

We study superconductivity on the honeycomb lattice close to the Mott state at half filling. Due to the sixfold lattice symmetry and disjoint Fermi surfaces at opposite momenta, we show that several different fully gapped superconducting states naturally exist on the honeycomb lattice, of which the chiral $d+i{d}^{\prime }$-wave state has previously been shown to appear when superconductivity appears close to the Mott state. Using renormalized mean-field theory to study the $t-J$ model and quantum Monte Carlo calculations of the Hubbard-$U$ model we show that the $d+i{d}^{\prime }$-wave state is the favored superconducting state for a wide range of on-site repulsion $U$, from the intermediate to the strong coupling regime. We also investigate the possibility of a mixed chirality $d$-wave state, where the overall chirality cancels. We find that a state with $d+i{d}^{\prime }$-wave symmetry in one valley but $d-i{d}^{\prime }$-wave symmetry in the other valley is not possible in the $t-J$ model without reducing the translational symmetry, due to the zero-momentum and spin-singlet nature of the superconducting order parameter. Moreover, any extended unit cells result either in disjoint Dirac points, which cannot harbor this mixed chirality state, or the two valleys are degenerate at the zone center, where valley hybridization prevents different superconducting condensates. We also investigate extended unit cells where the overall chirality cancels in real space. For supercells containing up to eight sites, including the Kekulé distortion, we find no energetically favorable $d$-wave solution with an overall zero chirality within the restriction of the $t-J$ model.

Figure 1

The superconducting order parameter ${\Delta }_0$ in units of $\frac{3}{4}{g}_{J}J$ for $d+i{d}^{\prime }$-wave and extended $s$-wave (ES) symmetries as a function of the doping level $\delta =1-n$ for $J/t=0.8$. Within RMFT $T_c$ can be approximated by the quantity $g_t{\Delta }_0$ which is also plotted.

Figure 2

Lowest positive quasiparticle energies for the illustrative cases of $\tilde{t}=1$, ${\Delta }_0=0.4\tilde{t}$, and $\tilde{\mu }=0$ (a) and $\tilde{\mu }=0.5\tilde{t}$ (b) for the $d\pm i{d}^{\prime }$-wave superconducting states. The color scale is chosen such that only $E_{QP}=0$ is black, which is only present in (a). In (b) there is always a finite energy gap.

Figure 3

Pairing susceptibility $\chi$ (solid lines) and its bubble contribution ${\chi }_0$ (dashed lines) as a function of electron filling $n$ for four different values of the on-site Hubbard $U/t=1,2,4,6$ and calculated for the temperature $T/t=0.2$. The connected pairing susceptibility $\chi -{\chi }_0$ reveals the impact of the many-body effects from $U$, whether to enhance ($\chi -{\chi }_0>0$) or suppress ($\chi -{\chi }_0<0$) pair formation. Error bars are smaller than the symbols.

Figure 4

Different sized unit cells on the honeycomb lattice indicated by the blue unit vectors and with the complex phase of the chiral $d$-wave bond order parameter displayed on each bond. Red (half) circular plaquette-centered arrows mark the winding (increasing angles) of the order parameter phase around each plaquette. Green circular site-centered arrows mark the winding of the order parameter phase around each site. Original two-site unit cell (a), four-site unit cell (b), and eight-site unit cell (c) with the phase of the order parameter on the horizontal bonds fixed to zero. (d) Six-site unit cell with a Kekulé distortion.