Density Waves and Cooper Pairing on the Honeycomb Lattice

Motivated by the surge in research activities on graphene, we investigate instabilities of electrons on the honeycomb lattice, interacting by onsite and nearest-neighbor terms, using a renormalization group scheme. Near half band filling, critical minimal interaction strengths are required for instabilities toward antiferromagnetic or charge-density-wave order. Away from half-filling, $f$-wave triplet-pairing and $d+id$ singlet-pairing instabilities are found to emerge out of density-wave regimes.

Figure 1

Left: Brillouin zone and $18\times 3$ points used for discretizing the wave-vector dependent interaction. The solid lines are at constant band energy. The lattice constant (minimal distance between 2 A-sublattice sites) is set to 1. Right: Critical temperature $T_c$ for the flow to strong coupling vs interaction parameters $U$, $V$ at half-filling $\mu =0$, $J=0$. In the region with $T_c=0$, the semimetal is stable. For small $U$ and $V>1.2t$, the flow is toward a CDW instability, for small $V$ and $U>3.8t$ toward a SDW instability.

Figure 2

Upper plots: Flow of susceptibilities, solid line CDW, dotted line SDW, for (a) $U=4t$, $V=0$, (b) $U=0$, $V=1.5t$. Lower plots: Interactions $V_T(k_1,k_2,k_3)$ very close to the SDW instability at $T_c\approx 0.05t$ for $U=4t$, $V=0$. The colorbar indicates the values of the couplings. The incoming wave vectors $k_1$ and $k_2$ are on the inner rings near the Dirac points, for $k_{1/2}=1$ to 12 on sublattice A, and $k_{1/2}=13$ to 24 for SL B. The 1st outgoing particle $k_3$ is at point 1 and SL A. In (c), the 2nd outgoing is on SL A, in (d) on SL B.

Figure 3

(a) Flow of pairing susceptibilities in $f$-wave (solid lines) and CDW channel (dashed lines) for chemical potentials $\mu =0.1t$ (thick lines) to $\mu =0.75t$ (thin lines). (b) $T_c\mathrm{s}$ for the flow to strong coupling vs $\mu$. Crosses: $U=1.2t$, $V=2.4t$ with a CDW instability for $\mu <0.7t$ and a triplet Cooper instability for $\mu >0.7t$. Circles: $U=1.2t$, $J=2.4t$ with a SDW instability for $\mu <0.65t$ and a singlet $d$-wave Cooper instability for $\mu \ge 0.65t$. (c) Interaction at low scales for $U=1.2t$, $V=2.4t$, $\mu =0.75t$, and outgoing wave vectors fixed on SL $A$, $k_3=2$ for a $18\times 3$ discretization. Points $k_1$ and $k_2$ are on the middle of the three rings nearest to the FS and on SL $A$ for index 1 to 18 and on $B$ for 19 to 36. The color bar indicates the values of the couplings. (d) Crosses: Pair scattering $V_T(k_1,{\overline{k}}_1,k_2)$ for $U=1.2t$, $V=2.4t$, $\mu =0.75t$ with incoming and outgoing particles on SL $A$ and ${\overrightarrow{k}}_1+\overrightarrow{k}({\overline{k}}_1)=0$ vs $k_2$ around the Brillouin zone hexagon. The dotted line is the ansatz $-V_f{d}_f^{*}({\overrightarrow{k}}_1)d_f({\overrightarrow{k}}_2)$. Circles: Same data for the $d$-wave instability at $U=1.2t$, $J=2.4t$, $\mu =0.75t$. Here the dashed line is $-V_d[d_{x^2-y^2}^{*}({\overrightarrow{k}}_1)d_{x^2-y^2}({\overrightarrow{k}}_2)+d_{xy}^{*}({\overrightarrow{k}}_1)d_{xy}({\overrightarrow{k}}_2)]$.