  Quantum dots placed along a vibrating nanotube provide a quantum simulation
platform that can directly address the electron-phonon interaction. This offers
promising prospects for the search of new quantum materials and the study of
strong correlation effects. As this platform is naturally operated by coupling
the dots to an electronic reservoir, state preparation is straightforwardly
achieved by driving into the steady state. Here we show that for intermediate
electron-phonon coupling strength, the system with spin-polarized quantum dots
undergoes a Peierls transition into an insulating regime which exhibits
charge-density wave order in the steady state as a consequence of the
competition between electronic Coulomb repulsive interactions and
phonon-induced attractive interactions. The transport phenomena can be directly
observed as fingerprints of electronic correlations. We also present powerful
methods to numerically capture the physics of such an open electron-phonon
system at large numbers of phonons. Our work paves the way to study and detect
correlated electron-phonon physics in the nanotube quantum simulator with
current experimentally accessible techniques.
