Charge density wave state in extremely overdoped cuprates driven by phonons

Recent resonant x-ray scattering (RXS) experiments revealed a novel charge order in extremely overdoped ${\mathrm{La}}_{2\ensuremath{-}x}{\mathrm{Sr}}_{x}{\mathrm{CuO}}_{4}$ (LSCO) [Li et al., Phys. Rev. Lett. 131, 116002 (2023)]. The observed charge order appears around the $(\ensuremath{\pi}/3,0)$ wave vector, distinct from the well-known stripe fluctuations near 1/8 doping, and persists from cryogenic temperatures to room temperature. To investigate the origin of this charge order in the overdoped regime, we use determinant quantum Monte Carlo (DQMC) simulations to examine correlated models with various interactions. We demonstrate that this distinctive CDW originates from remnant correlations in extremely overdoped cuprates, with its specific pattern shaped by interactions beyond the Hubbard model, particularly electron-phonon couplings. The persistence of the $(\ensuremath{\pi}/3,0)$ wave vector across different doping levels indicates the presence of nonlocal couplings. Our study reveals the significant role of phonons in cuprates, which assist correlated electrons in the formation of unconventional phases.