Pair-density-wave fluctuations in the lightly doped $t-J$ model: density matrix renormalization group and finite-size project-entangled pair-state studies

Cuprate superconductors host intertwined electronic orders—$d$-wave superconductivity, pair-density-wave (PDW), charge-density-wave (CDW), and spin order. The $t$-$J$ model is a candidate minimal description, yet obtaining controlled results at low doping on two-dimensional clusters remains challenging. Here we combine state-of-the-art density matrix renormalization group (DMRG) with finite-size projected entangled pair states (PEPS) equipped with Monte Carlo sampling to study the lightly doped ($\delta\lesssim 0.10$) $t$-$J$ model on open-boundary stripe geometries and $C_4$-symmetric clusters. We scan the next-nearest-neighbor hopping $t_2$ across electron-like ($t_2>0$), hole-like ($t_2<0$), and $t_2=0$ regimes. On small clusters ($\le 8\times 8$), DMRG and PEPS agree quantitatively on both energies and local observables; on larger systems our PEPS calculations reach up to $16\times 16$ sites. In this doping range we find signatures of a $4a_0\times 4a_0$ CDW coexisting with antiferromagnetic backgrounds. Superconductivity is predominantly $d$-wave and is enhanced for $t_2>0$ but suppressed for $t_2<0$ relative to $t_2=0$, which is consistent with many recent relevant numerical studies. Strikingly, on top of the uniform $d$-wave component we resolve a plaquette PDW with period two lattice spacings; this PDW is markedly strengthened for $t_2<0$ and strongly suppressed for $t_2>0$, in contrast to the trend of the uniform $d$-wave component. Taken together, our results highlight a plaquette PDW with period two as a feature of the lightly doped $t$-$J$ model—strengthened for $t_2<0$ and suppressed for $t_2>0$—and provide a concrete benchmark for future theory and experiment.