  Recently, a robust $d$-wave superconductivity has been unveiled in the ground
state of the 2D $t$-$t'$-$J$ model -- with both nearest-neighbor ($t$) and
next-nearest-neighbor ($t'$) hoppings -- through the density matrix
renormalization group calculations in the ground state. In this study, we
exploit the state-of-the-art thermal tensor network approach to accurately
simulate the finite-temperature electron states of the $t$-$t'$-$J$ model on
cylinders with widths up to $W=6$. Our analysis suggests that in the dome-like
superconducting phase, the $d$-wave pairing susceptibility exhibits a divergent
behavior 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 crossover
temperature $T^*$, the magnetic susceptibility is suppressed, and the Fermi
surface also exhibits node-antinode structure, resembling the pseudogap
behaviors observed in cuprates. Our unbiased and accurate thermal tensor
network calculations obtain the phase diagram of the $t$-$t'$-$J$ model with
$t'/t>0$, shedding light on the $d$-wave superconducting and pseudogap phases
in the enigmatic cuprate phase diagram.
