Competing many-body instabilities and unconventional superconductivity in graphene

The band structure of graphene exhibits van Hove singularities (VHSs) at dopings $x=\pm 1/8$ away from the Dirac point. Near the VHS, interactions effects, enhanced due to the large density of states, can give rise to various many-body phases. We study the competition between many-body instabilities in graphene using the functional renormalization group. We predict a rich phase diagram, which, depending on band structure as well as the range and scale of Coulomb interactions, contains a $d+id$-wave superconducting (SC) phase, or a spin-density-wave phase at the VHS. The $d+id$ state is expected to exhibit quantized charge and spin Hall response, as well as Majorana modes bound to vortices. Nearby the VHS, we find singlet $d+id$-wave and triplet $f$-wave SC phases.

Figure 1

Phase diagram for graphene around van Hove filling as obtained by FRG. ${\Lambda }_c$ serves as an upper bound for $T_c$ as a function of doping. Left inset: Dominant $d+id$ instability at van Hove filling for $U_0=10$ eV and the band structure in Ref. 5. Away from van Hove filling [dark shaded (blue) area], ${\Lambda }_c$ drops. Whether $d+id$- or $f$-wave SC instability is preferred depends on the long rangedness of the interaction (right inset: $U_1/U_0=0.45$ and $U_2/U_0=0.15$).

Figure 2

(a) Band structure of graphene for $t_1=2.8$ eV (red) and for $t_1=2.8$, $t_2=0.7$, $t_3=0.02$ eV (black). (b) Fermi surface near the van Hove point [dashed blue level in (a)], 96 patches ($\mathbf{k}$ points) used in the FRG and the nesting vector, and the partial nesting vectors. (c) DOS for both band structures in (a). Inset: Position shift of Fermi surface nesting (dashed vertical lines) vs the VHS peak.

Figure 3

$d_{x^2-y^2}$- and $d_{xy}$-wave solutions for (a) $U_0=10$ eV only and (b) $U_1/U_0=0.45$, $U_2/U_0=0.15$. (a1),(a2) and (b1),(b2) show the form factors of $d_{x^2-y^2}$ and $d_{xy}$ plotted along the Fermi surface according to patch indices defined in Fig. 2b, as well as the real space pair amplitude patterns. The solutions change from (a) to (b). The analytic form factors given in the text (red) fit the numerical data (black). (a3) and (b3) show the gap profile of $d+id$ along the Fermi surface (the actual connection to experimental energy scale can still vary by a global factor). The gap anisotropy increases from (a) to (b).

Figure 4

Pairing amplitudes, form factors, and gap profiles for the $f$-wave phase as defined in Fig. 3, representative for larger fillings than the VHS for (a) $U_1/U_0=0.6$ to (b) $U_1/U_0=0.6$, $U_2/U_0=0.2$. The gap profile is nodal, and the nodal points shift from (a) to (b).