Correlated Dirac particles and superconductivity on the honeycomb lattice

We investigate the properties of the nearest-neighbor singlet pairing and the emergence of $d$-wave superconductivity in the doped honeycomb lattice considering the limit of large interactions and the $t-J_1-J_2$ model. First, by applying a renormalized mean-field procedure as well as slave-boson theories which account for the proximity to the Mott-insulating state, we confirm the emergence of $d$-wave superconductivity, in agreement with earlier works. We show that a small but finite $J_2$ spin coupling between next-nearest neighbors stabilizes $d$-wave symmetry compared to the extended$s$-wave scenario. At small hole doping, to minimize the energy and to gap the whole Fermi surface or all the Dirac points, the superconducting ground state is characterized by a $d+id$ singlet pairing assigned to one valley and a $d-id$ singlet pairing to the other, which then preserves time-reversal symmetry. The slightly doped situation is distinct from the heavily doped case (around 3/8 and 5/8 filling) supporting a pure chiral $d+id$ symmetry and breaking time-reversal symmetry. Then, we apply the functional renormalization group and study in more detail the competition between antiferromagnetism and superconductivity in the vicinity of half filling. We discuss possible applications to strongly correlated compounds with copper hexagonal planes such as In${}_3$Cu${}_2$VO${}_9$. Our findings are also relevant to the understanding of exotic superfluidity with cold atoms.

Figure 1

Form factors of the ES and $d+id$ pairing solutions. Due to the overlap between the Fermi surface and the nodes of the ES solution, the ground state should favor the $d+id$ pairing as discussed in Sec. 3. On the other hand, to gap all the Fermi surface close to half filling, the solution that minimizes the whole energy will be to take the $d+id$ pairing solution in one valley and the $d-id$ in the other (Ref. 130).

Figure 2

Spin gap ${\Delta }_{d+id}$ and ${\Delta }_s$ at $T=0$ [defined in Eqs. (8) and (11)] and superconducting transition temperature $(T_c\sim g_t\Delta )$ when $J_2=0$ for the $d\pm id$ and ES scenarios as a function of the hole doping parameter $\delta =1-n$; the half-filled case here corresponds to one electron per site or $\delta =0$. The order parameters are taken in units of $3g_{s}J/4$. Inset: one can observe that the particle-hole order parameters behave identically for the $d\pm id$ and ES situations.

Figure 3

Spin gaps (at $T=0$) versus doping, for $J_2\ne 0$, from the RMFT. Notations are similar to those in Fig. 2.

Figure 4

Left panel: Interaction vertex labeled with the spin convention (upper diagram). Below, the loop contributions to the flow of the interaction vertex including the particle-particle diagram (a), the crossed particle-hole diagram (b), and the direct particle-hole diagram (c). Right panel: “$N=24$” patching scheme of the Brillouin zone with constant wave-vector dependence within one patch and the representative wave vector chosen on the Fermi line shown here for three different choices of the doping $\delta$. The red dashed line corresponds to Van Hove filling.

Figure 5

FRG critical scale ${\Lambda }_c$ in units of the hopping $t$ vs doping $\delta$ at $T=0$ for various choices of parameters $J_1,J_2$ and on-site interaction $U$. The point $\delta =0.25$ corresponds to Van Hove doping. $J_2>0$ suppresses the AF SDW ordering tendencies, but has only smaller quantitative effects on the $d$-wave pairing at larger couplings. Inclusion of $U$ does not change the qualitative findings.

Figure 6

Typical effective interaction vertex near the critical scale in the regime of the $d$-wave instability in units of $t$. Left panel: Orbital combinations with $o_1=o_2=o_3=o_4$ where $o_i\in \{a,b\}$. The numbers on the axis specify the number of the patch as shown in Fig. 4. On the horizontal axis the wave vector $k_1$ can be read off and on the vertical axis we enumerate $k_2$. $k_3$ is fixed on the first patch, and $k_4$ then follows from momentum conservation. Middle panel: Effective vertex function for the orbital combination where $o_1=o_3,o_2=o_4\ne o_1$. Here, we can clearly identify sharp diagonal structures ($k_1=-k_2$) with a $d$-wave modulation of the amplitude along the diagonal; see Fig. 7. Right panel: Effective vertex function for the orbital combination, where $o_1=o_4,o_2=o_3\ne o_1$. For this orbital combination also a sharp diagonal structure emerges.

Figure 7

Pair scattering $V_{\Lambda }(\vec{k},-\vec{k}\to \vec{p},-\vec{p})$ near the instability for doping $\delta =0.15$, with $\vec{k}$ fixed on one discretization point near the zone boundary, and $\vec{p}$ moving through the other points around the hexagon. The circles are the FRG data, the dashed line is the nearest-neighbor $d$-wave form factor proportional to $-[d_{x^2-y^2}^{*}\left(\vec{k}\right)d_{x^2-y^2}\left(\vec{p}\right)+d_{xy}^{*}\left(\vec{k}\right)d_{xy}\left(\vec{p}\right)]$, and the dotted line is the extended $s$-wave form factor. The left plot is for $J_1=1.5t$ and $J_2=0$, the right plot for $J_1=1.5t$ and $J_2=0.01t$.

Figure 8

Phase diagram obtained with the $\text{U}\left(1\right)$ slave-boson approach for the case $J_2=0$ $(\text{here},J/t=1)$. Details of the theory are presented in the Appendix as well as the definitions of the temperature scales $T_{\text{BE}}$ and $T_{\text{RVB}}$ which correspond respectively to the temperatures associated with the boson (chargon) condensation and spin-gap formation, respectively. Within this approach, an upper bound on the superconducting transition temperature is given by Min$(T_{\text{BE}},T_{\text{RVB}})$.