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Recently, a robust $d$-wave superconductivity has been unveiled in the groundstate of the 2D $t$-$t'$-$J$ model -- with both nearest-neighbor ($t$) andnext-nearest-neighbor ($t'$) hoppings -- through the density matrixrenormalization group calculations in the ground state. In this study, weexploit the state-of-the-art thermal tensor network approach to accuratelysimulate the finite-temperature electron states of the $t$-$t'$-$J$ model oncylinders with widths up to $W=6$. Our analysis suggests that in the dome-likesuperconducting phase, the $d$-wave pairing susceptibility exhibits a divergentbehavior with $\chi_\textrm{SC} \propto 1/T^\alpha$ below the onset temperature$T_c^*$. Near the optimal doping, $T_c^*$ reaches its highest value of about$0.05 t$ ($\equiv 0.15 J$). Above $T_c^*$ yet below a higher crossovertemperature $T^*$, the magnetic susceptibility is suppressed, and the Fermisurface also exhibits node-antinode structure, resembling the pseudogapbehaviors observed in cuprates. Our unbiased and accurate thermal tensornetwork calculations obtain the phase diagram of the $t$-$t'$-$J$ model with$t'/t>0$, shedding light on the $d$-wave superconducting and pseudogap phasesin the enigmatic cuprate phase diagram.

$d$-wave Superconductivity, Pseudogap, and the Phase Diagram of $t$-$t'$-$J$ Model at Finite Temperature