Kekulé superconductivity and antiferromagnetism on the graphene lattice

We investigate superconducting order in the extended Hubbard model on the two-dimensional graphene lattice using the variational cluster approximation with an exact diagonalization solver at zero temperature. Building on the results of Faye et al. [Phys. Rev. B 92, 085121 (2015)], which identified triplet $p$- and $\left(p+ip\right)$-wave superconductivity as the most favored pairing symmetries in that model, we place uniform superconducting solutions in competition with a nonuniform Kekulé $(p+ip$-K) superconducting pattern, similar to those proposed by Roy and Herbut [Phys. Rev. B 82, 035429 (2010)]. We find that the $p+ip$-K solution is in fact the most favored pairing in most of the phase diagrams. In addition, we show that antiferromagnetism can coexist with the $p+ip$-K state and that both orders are enhanced by their coexistence.

Figure 1

Tiling of the honeycomb lattice by six-site clusters (gray shading) used in VCA. The three colors (red, green, and blue) of the links correspond to the three different phases of the bond triplet pairing in the Kekulé pattern defined as $p+ip$-K in the text. The $A$ and $B$ sublattices are indicated, as well as the three elementary vectors ${\mathbf{e}}_{1,2,3}$.

Figure 2

Spectral function of (left) BCS-like $\left(U=0\right)$ solutions and (right) interacting, VCA solutions for (top) $p+ip$ and (bottom) $p+ip$-K pairing. Doping is set at 5%. The BCS solutions were giving pairing amplitudes $\mathrm{\Delta }=0.1$, but this exaggerates the order parameter compared to the VCA solutions. Notice that the $p+ip$-K state is gapless (hidden) at $U=0$ but has a gap in the presence of interactions, whereas the opposite is true of the $p+ip$ state. Also, the asymmetry between the two distinct Dirac points $\mathbf{K}$ and ${\mathbf{K}}^{\prime }$ is very strong in the noninteracting case, but much weaker in the presence of interactions, to the extent that it is barely visible here. A fixed Lorenzian broadening $\eta =0.05$ has been used at $U=0$, whereas $\eta$ increases linearly from $0.03$ to $0.2$ as a function of $\left|\omega \right|$ in the interacting case.

Figure 3

Condensation energy as a function of doping $\delta =1-n$ at various values of $U$ and $V$, for $p,p+ip$, and $p+ip$-K superconductivity.

Figure 4

$p+ip$-K order parameter as a function of doping at (top) $U=3t$ and (bottom) $U=6t$, and several values of $V$.

Figure 5

Superconducting $p+ip$-K and AF order parameters as a function of the on-site repulsion $U$ at half filling. The dashed curves are the pure solutions and the solid curves the coexistence solutions.

Figure 6

Superconducting order parameters $p+ip$-K and antiferromagnetic (AF) versus doping at $U=4t$ and $U=6t$ and $V=0$ (top and bottom panels, respectively). At $U=4t$ the antiferromagnetism disappears around $3\%$ of doping and $10\%$ at $U=6t$. The dotted blue curve is the pure superconducting order parameter, i.e., when the Weiss field for antiferromagnetism $h_{\mathrm{AF}}=0$ and the red curve corresponds to the coexistence solution with AF. Notice the increase of the amplitude of the superconducting order parameter in the presence of AF.

Figure 7

Phase diagram on the $U\text{−}\delta$ plane at $V=0$. The red curve corresponds to the critical line where the antiferromagnetic (AF) solution that coexists with the chiral $p+ip$-K solution disappears. In the region below the blue curve, the uniform $p+ip$ solution is preferred over the $p+ip$-K solution when antiferromagnetism is suppressed, but the $\mathrm{AF}\oplus p+ip$-K solution has lower energy and therefore dominates in that region.