Superconducting correlations and thermodynamic properties in 2D square and triangular $t$-$J$ model

Equal-time superconducting correlation functions of the two-dimensional $t$-$J$ model on the square lattice are studied using high-temperature expansion method.[1] The sum of the pairing correlation, its spatial dependence and correlation length are obtained down to $T \simeq 0.2t$. By comparison of single-particle contributions in the correlation functions, we find effective attractive interactions between quasi-particles in $d_{x^2-y^2}$-wave channel. It is shown that $d$-wave correlation grows rapidly at low temperatures for the doping $0.1 < \delta < 0.5$. The temperature for this growth is roughly scaled by $J/2$. This is in sharp contrast to the Hubbard model in a weak or intermediate coupling region, where there are few numerical evidences of superconductivity. We also study the possible $d$- and $f$-wave pairing in the triangular $t$-$J$ model.[2] When $t>0$ with hole doping, a rapid growth of effective $d$-wave paring interaction is found that indicates the resonating-valence-bond superconductivity. In contrast, when $t<0$, where the ferromagnetic- and antiferromagnetic correlation compete, correlation lengths of the $f$-wave triplet paring tends to diverge around $\delta=0.6$, although its effective interaction is small. This result is compared and discussed with the recently discovered superconductor, Na$_x$CoO$_2$$\cdot y$H$_2$O, where Co atoms form a triangular lattice. Specific heat in low temperatures are also obtained in the high-temperature expansion method. We will discuss that the doping dependence of the specific heat coefficient, $\gamma$, agrees with experimental data. \par\noindent [1] T. Koretsune and M. Ogata, J.\ Phys.\ Soc.\ Japan {\bf 74}, 1390 (2005). [2] T. Koretsune and M. Ogata, Phys.\ Rev.\ Lett.\ {\bf 89}, 116401 (2002), and Phys.\ Rev.\ B{\bf 72}, 134513 (2005).