Dynamical evolution of entanglement in disordered oscillator systems

We study the non-equilibrium dynamics of a disordered quantum system consisting of harmonic oscillators in a d-dimensional lattice. If the system is sufficiently localized, we show that, starting from a broad class of initial product states that are associated with a tiling (decomposition) of the d-dimensional lattice, the dynamical evolution of entanglement follows an area law in all times. Moreover, the entanglement bound reveals a dependency on how the subsystems are located within the lattice in dimensions d >= 2. In particular, the entanglement grows with the maximum degree of the dual graph associated with the lattice tiling.