Pairing correlation functions in the triangular $t\text{−}J$ model: $d$-wave and $f$-wave superconductivity

Equal-time pairing correlation functions of the two-dimensional $t\text{−}J$ model on the triangular lattice are studied using high-temperature expansions method. We calculate the pairing correlation lengths and the effective pairing interactions down to $T\sim 0.2t$, and find the growth of pairing correlations. When $t>0$ with hole doping, a rapid growth of effective $d$-wave pairing interaction is found that indicates the resonating-valence-bond superconductivity. In contrast, when $t<0$, where the ferromagnetic correlation and antiferromagnetic correlation compete, correlation lengths of triplet pairings as well as $d$-wave pairing grow at low temperatures, although they do not diverge for $T\to 0$ in the competing region, i.e., near half-filling. On the other hand, around $n=0.4$, $f$-wave pairing correlation tends to diverge, although its effective interaction is small. Relation of the present study to the recently discovered superconductor, ${\mathrm{Na}}_x\mathrm{Co}{\mathrm{O}}_2\bullet y{\mathrm{H}}_2\mathrm{O}$, is discussed.

Figure 1

Possible form factors of the pairing correlation functions in the triangular lattice.

Figure 2

(a) Pairing correlation length ${\xi }_{\alpha }$ in the triangular lattice for $t>0$ at $J∕t=0.4$ and $n=0.65$ with $\alpha =s$, $p$, $d$, and $f$. (b) ${\xi }_d$ and the single-particle contribution $\overline{{\xi }_d}$ (see the text), compared with those in the case of square lattice: ${\xi }_d^{\mathit{sq}}$ and $\overline{{\xi }_d^{\mathit{sq}}}$.

Figure 3

Pairing correlation length ${\xi }_d$ and $\overline{{\xi }_d}$ in the triangular lattice for $t>0$ as a function of $n$ at $J∕t=0.4$ and $T∕t=0.4,0.8,1.2,1.6$ (from top to bottom). ${\xi }_d$ is always larger than $\overline{{\xi }_d}$.

Figure 4

Pairing correlation length ${\xi }_{\alpha }$ in the triangular lattice $(t<0)$ for $\alpha =s,p,d,f$ and $J∕\mid t\mid =0.3$ at (a) $n=0.65$ and (b) $n=0.4$.

Figure 5

Pairing correlation lengths, ${\xi }_f$ and $\overline{{\xi }_f}$ in the triangular lattice for $t<0$ as a function of $n$ at $J∕\mid t\mid =0.3$ and $T∕\mid t\mid =0.4,0.8,1.2,1.6$ (from top to bottom) for $f$-wave. ${\xi }_f$ is slightly larger than $\overline{{\xi }_f}$ around $n=0.4$. Note that the error bars are shown for all the data points.