Entanglement dynamics in short and long-range harmonic oscillators

We study the time evolution of the entanglement entropy in the short and long-range coupled harmonic oscillators that have well-defined continuum limit field theories. We first introduce a method to calculate the entanglement evolution in generic coupled harmonic oscillators after quantum quench. Then we study the entanglement evolution after quantum quench in harmonic systems that the couplings decay effectively as $1/r^{d+\alpha}$ with the distance $r$. After quenching the mass from non-zero value to zero we calculate numerically the time evolution of von Neumann and R\'enyi entropies. We show that for $1<\alpha<2$ we have a linear growth of entanglement and then saturation independent of the initial state. For $0<\alpha<1$ depending on the initial state we can have logarithmic growth or just fluctuation of entanglement. We also calculate the mutual information dynamics of two separated individual harmonic oscillators. Our findings suggest that in our system there is no particular connection between having a linear growth of entanglement after quantum quench and having a maximum group velocity or generalized Lieb-Robinson bound.