We investigate the dynamics of the Hubbard–Holstein model after a sudden switch-on of the electron–phonon interaction within nonequilibrium dynamical mean-field theory (DMFT). Using the self-consistent Migdal approximation for the electron–phonon coupling together with self-consistent second-order perturbation theory for the electron–electron interaction, we show that the relaxation dynamics exhibits a crossover between electron-dominated and phonon-dominated regimes, extending to finite Hubbard interaction the scenario previously identified in the Holstein model. To investigate the emergence of a self-consistent thermalization of the electron–phonon system, we analyze the DMFT dynamics by enforcing the self-consistency after each global time-evolution, rather than at every time step. This allows us to follow the build-up of a self-consistent thermal bath. In the plane of real time and DMFT iteration number, thermalization is marked by a sharp propagating front. This front appears in electronic observables already for weak quenches within the simulated time window, whereas the phononic sector exhibits a visible front only at sufficiently strong coupling. Thus, at weak coupling the local dispersionless phonons show a delayed onset of front formation, while near and beyond the crossover the front develops on comparable timescales in both the electronic and phononic sectors. Whenever both fronts are resolved, they propagate with the same velocity, showing that thermalization spreads coherently through the coupled electron–phonon system.

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• Electron-phonon coupling
• Lattice dynamics
• Nonequilibrium lattice models
• Strongly correlated systems
• Dynamical mean field theory
• Holstein model
• Hubbard model