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

Abstract: Recently, robust $d$-wave superconductive (SC) order has been unveiled in the ground state of the 2D $t$-$t&#39;$-$J$ model -- with both nearest-neighbor ($t$) and next-nearest-neighbor ($t&#39;$) hoppings -- by density matrix renormalization group studies. However, there is currently a debate on whether the $d$-wave SC holds up strong on both $t&#39;/t&gt;0$ and $t&#39;/t&lt;0$ cases for the $t$-$t&#39;$-$J$ model, which correspond to the electron- and hole-doped sides of the cuprate phase diagram, respectively. Here we exploit state-of-the-art thermal tensor network approach to accurately obtain the phase diagram of the $t$-$t&#39;$-$J$ model on cylinders with widths up to $W=6$ and down to low temperature as $T/J \simeq 0.06$, pushing the boundaries of contemporary finite-$T$ calculations. For $t&#39;/t&gt;0$, we find a dome-like SC regime with a diverging $d$-wave pairing susceptibility, $\chi_\textrm{SC} \propto 1/T<sup>\alpha$ below a characteristic temperature $T_c<sup>*$. Near optimal doping, $T_c<sup>*$ reaches its highest value of about $0.15 J$. Above $T_c<sup>*$ yet below a higher crossover temperature $T<sup>*$, the magnetic susceptibility becomes suppressed, which can be related to the onset of pseudogap (PG) behaviors. On the other hand, for $t&#39;/t&lt;0$ we find the pairing correlations are much weaker, although there exhibits a node-antinode structure in the PG regime as observed in the hole-doped cuprates. The thermal tensor network calculations of the $t$-$t&#39;$-$J$ model underscore both the similarities and differences in the finite-temperature phase diagram between the fundamental model and cuprates, yielding unique insights into their intricate behaviors.