we choose the frustrated triangular lattice to avoid the unnecessary complexity due to spontaneous symmetry-breaking or structural transition. As discussed in Sec. \[sec:intro\], the lattice degrees of freedom here serve to introduce the stochastic dynamics for the order parameter of the Mott transition.

The experimental time scale of temperature-driven Mott transition, which is also the time scale $\tau_\Phi$ for the order-parameter dynamics, is usually much longer than the characteristic time of lattice fluctuations. On the other hand, the relaxation time of electrons $\tau_e$, which ranges from a few tens to hundred femtoseconds, is short compared with the lattice time scale $\tau_L$ of the picosecond order. This separation of time scales allows us to apply the Born-Oppenheimer (BO) approximation in our MD simulations. Indeed, BO approximation is often employed in [*ab inito*]{} or quantum MD method [@marx09; @plasienka17]. The state-of-the-art DFT-based quantum MD, however, cannot describe the Mott transition. In order to capture the electron correlation effects, here we employ a new quantum MD scheme in which the atomic forces are computed on-the-fly from the Gutzwiller solution of Hubbard-type models [@chern17]. Applying the Gutzwiller MD to a Hubbard liquid model, we have recently demonstrated for the first time a Mott MIT 