Quantum Thermalization via Travelling Waves

Isolated quantum many-body systems which thermalize under their own dynamics are expected to act as their own thermal baths, thereby losing memory of initial conditions and bringing their local subsystems to thermal equilibrium. Here we show that the infinite-dimensional limit of a quantum lattice model, as described by dynamical mean-field theory (DMFT), provides a natural framework to understand this self-consistent thermalization process. Using the Fermi-Hubbard model as a working example, we demonstrate that the emergence of a self-consistent bath occurs via a sharp thermalization front, moving ballistically and separating the initial condition from the long time thermal fixed point. We characterize the full DMFT dynamics through an effective temperature for which we derive a traveling wave equation of the Fisher-Kolmogorov-Petrovsky-Piskunov type. This equation allows for predicting the asymptotic shape of the front and its velocity, which match perfectly the full DMFT numerics. Our results provide a new angle to understand the onset of quantum thermalization in closed isolated systems.