Fluctuating Pair Density Wave in Finite-temperature Phase Diagram of the $t$-$t^\prime$ Hubbard Model

Abstract

The Hubbard model and its extensions are canonical theoretical frameworks for understanding correlated electronic states, including those in high-$T_c$ cuprates. Here, we use state-of-the-art thermal tensor network method to map out the temperature-doping phase diagram of the $t$-$t^\prime$ Hubbard model. On the electron-doped side, we find a $d$-wave superconducting (dSC) regime, supporting the scenario of high-$T_c$ superconductivity. In contrast, on the hole-doped side, no robust dSC phase is detected. Instead, a finite-temperature regime dominated by strong pair-density-wave (PDW) fluctuations emerges, which may eventually give way to charge density wave order upon further cooling. The PDW state exhibits inter-arc pairing with net momentum near $(0, \pi)$, distinct from the zero-momentum pairing in conventional dSC. Furthermore, these fluctuating PDW states occupy the lower portion of the pseudogap regime on the hole-doped side. We provide a comprehensive finite-temperature perspective consistent with previous ground-state studies, shedding new light on pairing instabilities and exotic electronic states in high-$T_c$ superconductors.