Pattern formation and auto-parametric instability in charge density wave states subject to a quantum

The t-V model on a two-dimensional square lattice exhibits a $(\pi,\pi)$ charge density wave (CDW) at half filling. Following a quantum quench of the repulsive interaction, the CDW order parameter manifests three distinct dynamical regimes: phase-locking, Landau-damping, and overdamped behavior. We show that even in the phase-locking regime, the oscillatory dynamics of the CDW order parameter becomes unstable due to a parametric resonance mechanism. This instability enables individual electrons to simultaneously develop multiple modes contributing to both $\mathbf{q}$ and $-\mathbf{q}$ orders, ultimately driving the emergence of spatial inhomogeneities and complex pattern formation.