Quench dynamics of stripes and phase separation in the two-dimensional $t$-$J$ model

We investigate the fundamental dynamical process of an initial quench of the chemical potential of the two-dimensional $t$-$J$ model. Depending on the ground state phase, sharply different dynamical behavior of the charge distribution and entanglement properties are observed. In the stripe phase, the intertwining of the spin and charge density waves remains stable under time evolution. A ballistic spreading of the charge density is observed with a propagation speed that is only weakly dependent on the coupling ratio, $J\mathrm{/}t$. Moreover, in the phase-separated regime for large $J\mathrm{/}t$, we report a complete dynamical freezing of charge degrees of freedom within, where even under long time evolution the entanglement entropy remains bounded. Our results are obtained by combining large-scale exact diagonalizations and matrix product state techniques for time evolution.