Unconventional superconductivity on the triangular lattice Hubbard model

Using large-scale dynamical cluster quantum Monte Carlo simulations, we explore the unconventional superconductivity in the hole-doped Hubbard model on the triangular lattice. Due to the interplay of electronic correlations, geometric frustration, and Fermi surface topology, we find a doubly degenerate singlet pairing state at an interaction strength close to the bare bandwidth. Such an unconventional superconducting state is mediated by antiferromagnetic spin fluctuations along the $\Gamma$-$K$ direction, where the Fermi surface is nested. An exact decomposition of the irreducible particle-particle vertex further confirms the dominant component of the effective pairing interaction comes from the spin channel. Our findings suggest the existence of chiral $d+id$ superconductivity in a hole-doped Hubbard triangular lattice in a strongly correlated regime, and provide insight into the superconducting phases of the water-intercalated sodium cobaltates Na${}_x$CoO${}_2·y$H${}_2$O, as well as the organic compounds $\kappa$-(ET)${}_2$X and Pd(dmit)${}_2$.

Figure 1

Cluster spin susceptibility for $N_c=12$, interaction strength $U=8.5t$, temperature $T=0.1t$, and different fillings, $n=0.667$, $0.8$, and 1.

Figure 2

(a) First Brillouin zone, the symmetric path $\Gamma -M-K-\Gamma$, and the noninteracting Fermi surface at different band fillings. (b), (c), (d) Spectral function $A(\mathbf{k},\omega =0)$ on the Fermi surface for $N_c=12$ DCA-CTQMC simulations with $U=8.5t$, $T=0.1t$, and $n=0.667$ in (b), $n=0.8$ in (c), and $n=1$ in (d). The red arrow is the AF ordered wave vector (${\mathbf{Q}}_{\mathbf{AF}}$) and the pink arrow is after shifting its center to $\Gamma$ and rotating it by 60${}^{\circ }$.

Figure 3

Inverse pairing susceptibility $1/{\chi }_{\text{pairing}}$ for $N_c=6$, $U=8.5t$, and $n=0.9$. The singlet $s$ wave and triplet $f$ wave do not diverge, whereas the singlet pairing channels with $d_{x^2-y^2}$- and $d_{xy}$-wave symmetry show a divergency at the same $T_c$. Note that we have multiplied by a factor of 7 the $s$-wave pairing susceptibility in order to use the same vertical scale. Inset: The superconducting transition temperature $T_c$ as a function of doping. FL and SC label the Fermi liquid and superconducting regions, respectively. The nature of the system is still unclear at half filling, hence a question mark in the inset.

Figure 4

Left: $d_{xy}$ projected contributions to the pairing vertex $V_{d_{xy}}$, from the fully irreducible vertex $V_{d_{xy}}^{\Lambda }$, charge $V_{d_{xy}}^C$, and spin $V_{d_{xy}}^S$ cross channels vs $T$ at $n=0.9$, $U=8.5t$. Right: The $d_{x^2-y^2}$-wave projection of the same quantities. In both cases, the contribution to the pairing interaction from spin channel is clearly dominant.